SDP Product Equations¶
This notebook generates an overview of the formulas used to calculate product properties by the parametric model.
In [1]:
import sys
from IPython.display import display, Math, Latex, HTML, Markdown
from sympy import latex, Lambda, Symbol
sys.path+=['..']
from sdp_par_model import evaluate
from sdp_par_model import reports as iapi
from sdp_par_model.config import PipelineConfig
from sdp_par_model.parameters.definitions import *
from sdp_par_model.parameters.container import BLDep
# Product properties to show
exprs = [('Rflop', 'PFLOP/s', Constants.peta),
('Rout', 'TB/s', Constants.tera)]
Python code to generate formulas:
In [2]:
def show_formulas(sym_defs, group=1):
maths = ""
for i, sym in enumerate(sorted(sym_defs)):
if isinstance(sym_defs[sym], BLDep):
maths += r'%s(%s) &= %s %s' % (
latex(Symbol(sym)),
",".join(sym_defs[sym].pars.keys()),
latex(sym_defs[sym].term),
r"\\" if i % group == 0 else " & ")
else:
maths += r'%s &= %s %s' % (latex(Symbol(sym)), latex(sym_defs[sym]),
r"\\" if i % group == group-1 else " & ")
display(Math(r"\begin{aligned}%s\end{aligned}" % maths))
def formulas_for_configuration(telescope, band, pipeline):
cfg = PipelineConfig(telescope=telescope,band=band,pipeline=pipeline)
if not cfg.is_valid()[0]:
return
# Heading
display(Markdown("## Equations for %s %s:" % (telescope, pipeline)))
# Loop through configuration to show formulas for
free_syms = set()
for expr, unit, mult in exprs:
# Determine and show product formulas
values = cfg.eval_expression_products(expr)
formulas = cfg.eval_products_symbolic(expr)
# Show equations for products
maths = ""
for product in sorted(values.keys()):
maths += (r'%s_{%s} &= %s \\ &= %f\,\text{%s} \\ \\' %
(expr, product,
latex(formulas.get(product, 0)),
values.get(product, 0) / mult, unit))
maths = (r"\begin{aligned}%s %s &= %f\,\text{%s}\end{aligned}" % \
(maths, expr, sum(values.values()) / mult, unit))
display(Math(maths))
# Collect free symbols
free_syms = free_syms.union(evaluate.collect_free_symbols(formulas.values()))
# Look up helper equations
display(Markdown("### Intermediate equations"))
sym_defs = cfg.eval_symbols(free_syms, symbolify='helper')
show_formulas(sym_defs)
free_syms2 = evaluate.collect_free_symbols(sym_defs.values()).union(free_syms).difference(sym_defs)
sym_defs2 = cfg.eval_symbols(free_syms2, symbolify='all')
show_formulas(sym_defs2)
# And finally all remaining constants
display(Markdown("### Constants"))
free_syms3 = evaluate.collect_free_symbols(sym_defs2.values()).\
union(free_syms, free_syms2).difference(sym_defs, sym_defs2)
const_defs = cfg.eval_symbols(free_syms3, optimize_expression='Rflop')
show_formulas(const_defs, 3)
In [3]:
for telescope, band in [(Telescopes.SKA1_Low, Bands.Low),
(Telescopes.SKA1_Mid, Bands.Mid1)]:
for pipeline in [Pipelines.Ingest, Pipelines.ICAL,Pipelines.DPrepA_Image]:
formulas_for_configuration(telescope, band, pipeline)
Equations for SKA1_Low Ingest:¶
$\displaystyle \begin{aligned}Rflop_{Average} &= 8 N_{beam} N_{pp} R_{vis,ingest} \\ &= 0.000306\,\text{PFLOP/s} \\ \\Rflop_{Demix} &= N_{beam} N_{f,min} R_{vis,ingest} \left(154 N_{Ateam} + 84 + \frac{N_{f,min} T_{int,used} \left(N_{Ateam}^{2} + N_{Ateam} \left(24 N_{solve} + 33\right) + 64\right)}{N_{f,max} T_{ion}}\right) \\ &= 0.621274\,\text{PFLOP/s} \\ \\Rflop_{Flag} &= 278 N_{beam} N_{pp} R_{vis,ingest} \\ &= 0.010634\,\text{PFLOP/s} \\ \\Rflop_{Receive} &= N_{beam} N_{f,min} \left(\frac{1000 N_{a}}{T_{int,used}} + \frac{4 N_{pp} R_{vis,ingest}}{N_{f,min}}\right) \\ &= 0.000153\,\text{PFLOP/s} \\ \\ Rflop &= 0.632367\,\text{PFLOP/s}\end{aligned}$
$\displaystyle \begin{aligned}Rout_{Average} &= M_{vis} N_{beam} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.457235\,\text{TB/s} \\ \\Rout_{Demix} &= M_{vis} N_{beam} N_{pp} R_{vis,ingest} \\ &= 0.459025\,\text{TB/s} \\ \\Rout_{Flag} &= M_{vis} N_{beam} N_{pp} R_{vis,ingest} \\ &= 0.459025\,\text{TB/s} \\ \\Rout_{Receive} &= M_{vis} N_{beam} N_{pp} R_{vis,ingest} \\ &= 0.459025\,\text{TB/s} \\ \\ Rout &= 1.834309\,\text{TB/s}\end{aligned}$
Intermediate equations¶
$\displaystyle \begin{aligned}R_{vis}(b,bcount) &= \frac{bcount \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{B_{max} Q_{bw} \Theta_{fov,total}} \right)}}\right)\right)}{T_{int,used}} \\R_{vis,ingest} &= \frac{N_{f,max} \left(N_{a} + N_{bl}\right)}{T_{int,used}} \\\end{aligned}$
$\displaystyle \begin{aligned}Q_{bw} &= \frac{0.600124986981879}{\sqrt{1 - \frac{1}{amp_{f max}}}} \\T_{int,used} &= \max\left(T_{int,min}, \min\left(T_{snap}, \frac{1.0046082220589 D_{s} \sqrt{1 - \frac{1}{amp_{f max}}}}{B_{max} \Omega_{E} Q_{\mathcal{fov,I}} \sqrt{Q_{subband}}}\right)\right) \\\Theta_{fov,total} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda}{D_{s}} \\\end{aligned}$
Constants¶
$\displaystyle \begin{aligned}B_{max} &= 65000 & D_{s} &= 38 & M_{vis} &= 12 \\N_{Ateam} &= 10 & N_{a} &= 512 & N_{beam} &= 1 \\N_{bl} &= 130816.0 & N_{f,max} &= 65536 & N_{f,min} &= 40 \\N_{f,out} &= 65536 & N_{pp} &= 4 & N_{solve} &= 10 \\\Omega_{E} &= 7.292115 \cdot 10^{-5} & Q_{fov} &= 1.0 & Q_{\mathcal{fov,I}} &= 2.7 \\Q_{subband} &= 1.32046924775612 & T_{int,min} &= 0.9 & T_{ion} &= 10.0 \\T_{snap} &= 600 & amp_{f max} &= 1.08 & \lambda &= 3.42619952 \\\lambda_{max} &= 5.99584916 & \lambda_{min} &= 0.85654988 & \end{aligned}$
Equations for SKA1_Low ICAL:¶
$\displaystyle \begin{aligned}Rflop_{Correct} &= 8 N_{Ipatches} N_{beam} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.185943\,\text{PFLOP/s} \\ \\Rflop_{DFT} &= \frac{N_{beam} N_{majortotal} \left(32 N_{a}^{2} N_{source} + 128 N_{a}^{2} + 266 N_{a} N_{source}\right) \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)}}{N_{bl}} \\ &= 3.167950\,\text{PFLOP/s} \\ \\Rflop_{Degrid} &= 8 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,predict}{\left(B_{max}{\left(i \right)} \right)} N_{kernel2,predict}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.174045\,\text{PFLOP/s} \\ \\Rflop_{Degridding Kernel Update} &= \frac{5.0 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,predict}{\left(B_{max}{\left(i \right)} \right)} N_{gcf,used,predict}{\left(B_{max}{\left(i \right)} \right)} N_{kernel,AW,predict}^{2}{\left(B_{max}{\left(i \right)} \right)} \log{\left(N_{kernel,AW,predict}^{2}{\left(B_{max}{\left(i \right)} \right)} \right)}}{T_{kernel,predict}{\left(B_{max}{\left(i \right)} \right)}}}{\log{\left(2 \right)}} \\ &= 0.111305\,\text{PFLOP/s} \\ \\Rflop_{FFT} &= \frac{2.5 N_{beam} N_{f,FFT,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp} \log{\left(N_{pix,linear}^{2} \right)} \max\left(1, \frac{\Delta_{W max}{\left(B_{max} \right)}}{\Delta_{W stack}}\right)}{T_{snap} \log{\left(2 \right)}} \\ &= 0.082433\,\text{PFLOP/s} \\ \\Rflop_{Flag} &= 279 N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.054040\,\text{PFLOP/s} \\ \\Rflop_{Grid} &= 8 N_{beam} N_{facet}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)} N_{kernel2,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.174045\,\text{PFLOP/s} \\ \\Rflop_{Gridding Kernel Update} &= \frac{5.0 N_{beam} N_{facet}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,backward}{\left(B_{max}{\left(i \right)} \right)} N_{gcf,used,backward}{\left(B_{max}{\left(i \right)} \right)} N_{kernel,AW,backward}^{2}{\left(B_{max}{\left(i \right)} \right)} \log{\left(N_{kernel,AW,backward}^{2}{\left(B_{max}{\left(i \right)} \right)} \right)}}{T_{kernel,backward}{\left(B_{max}{\left(i \right)} \right)}}}{\log{\left(2 \right)}} \\ &= 0.111305\,\text{PFLOP/s} \\ \\Rflop_{IFFT} &= \frac{2.5 N_{beam} N_{f,FFT,predict} N_{facet,predict}^{2} N_{majortotal} N_{pix,linear,predict}^{2} N_{pp} \log{\left(N_{pix,linear,predict}^{2} \right)} \max\left(1, \frac{\Delta_{W max}{\left(B_{max} \right)}}{\Delta_{W stack}}\right)}{T_{snap} \log{\left(2 \right)}} \\ &= 0.412166\,\text{PFLOP/s} \\ \\Rflop_{Identify Component} &= \frac{N_{beam} N_{facet}^{2} N_{majortotal} N_{scales} N_{subbands} N_{tt} \left(2 N_{minor} N_{patch}^{2} + 2 N_{pix,linear}^{2}\right)}{T_{obs}} \\ &= 0.000298\,\text{PFLOP/s} \\ \\Rflop_{Phase Rotation} &= \frac{28 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 0.000000\,\text{PFLOP/s} \\ \\Rflop_{Phase Rotation Predict} &= \frac{28 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 0.000000\,\text{PFLOP/s} \\ \\Rflop_{Reprojection} &= \frac{50.0 N_{beam} N_{f,proj,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.000295\,\text{PFLOP/s} \\ \\Rflop_{Reprojection Predict} &= \frac{50.0 N_{beam} N_{f,proj,predict} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.001474\,\text{PFLOP/s} \\ \\Rflop_{Solve} &= N_{beam} N_{subbands} \left(N_{\mathcal{self}} + 1\right) \left(\frac{48 N_{a}^{2} N_{pp} N_{solve} \left(N_{Ipatches} \max\left(1, \frac{T_{solve}}{t_{ICAL,I}}\right) + \max\left(1, \frac{N_{B,parameters}}{N_{subbands}}\right) \max\left(1, \frac{T_{solve}}{t_{ICAL,B}}\right) + \max\left(1, \frac{T_{solve}}{t_{ICAL,G}}\right)\right)}{T_{solve}} + \frac{40 N_{pp} \left(N_{Ipatches} + 2\right) \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)}}{N_{subbands}}\right) \\ &= 0.110247\,\text{PFLOP/s} \\ \\Rflop_{Source Find} &= \frac{600 N_{majortotal} N_{source} N_{source,find,iterations}}{T_{obs}} \\ &= 0.000000\,\text{PFLOP/s} \\ \\Rflop_{Subtract Image Component} &= \frac{2 N_{beam} N_{majortotal} N_{minor} N_{patch}^{2} N_{pp} N_{scales} N_{subbands} N_{tt}}{T_{obs}} \\ &= 0.001175\,\text{PFLOP/s} \\ \\Rflop_{Subtract Visibility} &= 2 N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.000387\,\text{PFLOP/s} \\ \\Rflop_{Visibility Weighting} &= 16 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.000568\,\text{PFLOP/s} \\ \\ Rflop &= 4.587674\,\text{PFLOP/s}\end{aligned}$
$\displaystyle \begin{aligned}Rout_{Correct} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 2.324289\,\text{TB/s} \\ \\Rout_{DFT} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 2.324289\,\text{TB/s} \\ \\Rout_{Degrid} &= M_{vis} N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,predict}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.425793\,\text{TB/s} \\ \\Rout_{Degridding Kernel Update} &= 8 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{pp} Q_{gcf}^{3} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,predict}{\left(B_{max}{\left(i \right)} \right)} N_{gw,predict}^{3}{\left(B_{max}{\left(i \right)} \right)}}{T_{kernel,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 429.540789\,\text{TB/s} \\ \\Rout_{FFT} &= \frac{M_{px} N_{beam} N_{f,FFT,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.047169\,\text{TB/s} \\ \\Rout_{Flag} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 2.324289\,\text{TB/s} \\ \\Rout_{Grid} &= \frac{M_{cpx} N_{beam} N_{facet}^{2} N_{majortotal} N_{pix,linear} N_{pp} \left(\frac{N_{pix,linear}}{2} + 1\right) \sum_{i=1}^{N_{bl}} N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{snap}} \\ &= 84.579581\,\text{TB/s} \\ \\Rout_{Gridding Kernel Update} &= 8 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} Q_{gcf}^{3} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,backward}{\left(B_{max}{\left(i \right)} \right)} N_{gw,backward}^{3}{\left(B_{max}{\left(i \right)} \right)}}{T_{kernel,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 429.540789\,\text{TB/s} \\ \\Rout_{IFFT} &= \frac{M_{cpx} N_{beam} N_{f,FFT,predict} N_{facet,predict}^{2} N_{majortotal} N_{pix,linear,predict} N_{pp} \left(\frac{N_{pix,linear,predict}}{2} + 1\right)}{T_{snap}} \\ &= 0.235856\,\text{TB/s} \\ \\Rout_{Identify Component} &= \frac{M_{cpx} N_{beam} N_{facet}^{2} N_{majortotal} N_{minor} N_{subbands}}{T_{obs}} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Phase Rotation} &= M_{vis} N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} R_{vis,backward}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Phase Rotation Predict} &= \frac{M_{vis} N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Reprojection} &= \frac{M_{px} N_{beam} N_{f,proj,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.047169\,\text{TB/s} \\ \\Rout_{Reprojection Predict} &= \frac{M_{px} N_{beam} N_{f,proj,predict} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.235847\,\text{TB/s} \\ \\Rout_{Solve} &= \frac{M_{jones} N_{a} N_{beam} N_{subbands} \left(N_{\mathcal{self}} + 1\right) \left(N_{Ipatches} \max\left(1, \frac{T_{solve}}{t_{ICAL,I}}\right) + \max\left(1, \frac{N_{B,parameters}}{N_{subbands}}\right) \max\left(1, \frac{T_{solve}}{t_{ICAL,B}}\right) + \max\left(1, \frac{T_{solve}}{t_{ICAL,G}}\right)\right)}{T_{solve}} \\ &= 0.000004\,\text{TB/s} \\ \\Rout_{Source Find} &= 100 M_{cpx} N_{majortotal} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Subtract Image Component} &= \frac{M_{px} N_{beam} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{subbands} N_{tt}}{T_{obs}} \\ &= 0.001720\,\text{TB/s} \\ \\Rout_{Subtract Visibility} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 2.324289\,\text{TB/s} \\ \\Rout_{Visibility Weighting} &= M_{vis} N_{beam} N_{f,min,gran} N_{facet}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{1}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.018647\,\text{TB/s} \\ \\ Rout &= 953.970523\,\text{TB/s}\end{aligned}$
Intermediate equations¶
$\displaystyle \begin{aligned}\Delta_{W max} &= \Delta_{W max} \\N_{f,FFT,backward} &= N_{f,out} \\N_{f,FFT,predict} &= \min\left(N_{f,max}, N_{f,out} N_{tt}\right) \\N_{f,gcf,backward}(b) &= \max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \\N_{f,gcf,predict}(b) &= \max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \\N_{f,proj,backward} &= N_{f,out} \\N_{f,proj,predict} &= \min\left(N_{f,max}, N_{f,out} N_{tt}\right) \\N_{f,vis}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{B_{max} Q_{bw} \Theta_{fov,total}} \right)}}\right)\right) \\N_{f,vis,backward}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right) \\N_{f,vis,predict}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov,predict} b} \right)}}\right)\right) \\N_{gcf,used,backward}(b) &= Q_{gcf}^{2} \cdot \left(1 - \left(1 - \frac{1}{Q_{gcf}^{2}}\right)^{\frac{T_{ion} \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right)}{\max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)}}\right) \\N_{gcf,used,predict}(b) &= Q_{gcf}^{2} \cdot \left(1 - \left(1 - \frac{1}{Q_{gcf}^{2}}\right)^{\frac{T_{ion} \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov,predict} b} \right)}}\right)\right)}{\max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)}}\right) \\N_{gw,backward}(b) &= 1.0 \Theta_{fov} \sqrt{\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}} \\N_{gw,predict}(b) &= 1.0 \Theta_{fov,predict} \sqrt{\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}} \\N_{kernel,AW,backward}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5} \\N_{kernel,AW,predict}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5} \\N_{kernel2,backward}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{1.0} \\N_{kernel2,predict}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{1.0} \\R_{vis}(b,bcount) &= \frac{bcount \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{B_{max} Q_{bw} \Theta_{fov,total}} \right)}}\right)\right)}{T_{int,used}} \\R_{vis,backward}(b,bcount) &= \frac{bcount \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right)}{\min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)} \\T_{coal,backward}(b) &= \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right) \\T_{coal,predict}(b) &= \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right) \\T_{kernel,backward}(b) &= T_{ion} \\T_{kernel,predict}(b) &= T_{ion} \\\end{aligned}$
$\displaystyle \begin{aligned}N_{facet,predict} &= N_{facet} \\N_{pix,linear} &= \frac{3.25100497089045 B_{max} Q_{fov} Q_{pix} Q_{subband} \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\N_{pix,linear,predict} &= \frac{3.25100497089045 B_{max} Q_{fov} Q_{pix} Q_{subband} \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\Q_{bw} &= \frac{0.600124986981879}{\sqrt{1 - \frac{1}{amp_{f max}}}} \\T_{int,used} &= \max\left(T_{int,min}, \min\left(T_{snap}, \frac{1.0046082220589 D_{s} \sqrt{1 - \frac{1}{amp_{f max}}}}{B_{max} \Omega_{E} Q_{\mathcal{fov,I}} \sqrt{Q_{subband}}}\right)\right) \\\Theta_{fov} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\\Theta_{fov,predict} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\\Theta_{fov,total} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda}{D_{s}} \\\epsilon_{f approx} &= 2.44948974278318 \sqrt{1 - \frac{1}{amp_{f max}}} \\\end{aligned}$
Constants¶
$\displaystyle \begin{aligned}B_{max} &= 65000 & D_{s} &= 38 & \Delta_{W stack} &= 2.31428911159104 \\M_{cpx} &= 16.0 & M_{jones} &= 64.0 & M_{px} &= 8.0 \\M_{vis} &= 12 & N_{B,parameters} &= 500 & N_{Ipatches} &= 30 \\N_{a} &= 512 & N_{aa} &= 9 & N_{beam} &= 1 \\N_{bl} &= 130816.0 & N_{f,max} &= 65536 & N_{f,min} &= 40 \\N_{f,min,gran} &= 800 & N_{f,out} &= 40 & N_{facet} &= 1 \\N_{majortotal} &= 9 & N_{minor} &= 10000 & N_{mm} &= 4 \\N_{patch} &= 4097 & N_{pp} &= 4 & N_{scales} &= 10 \\N_{\mathcal{self}} &= 3 & N_{solve} &= 10 & N_{source} &= 1000 \\N_{source,find,iterations} &= 10 & N_{subbands} &= 7 & N_{tt} &= 5 \\\Omega_{E} &= 7.292115 \cdot 10^{-5} & Q_{fov} &= 2.7 & Q_{\mathcal{fov,I}} &= 2.7 \\Q_{gcf} &= 8.0 & Q_{kernel} &= 10.0 & Q_{pix} &= 2.5 \\Q_{subband} &= 1.32046924775612 & T_{int,min} &= 0.9 & T_{ion} &= 10.0 \\T_{obs} &= 3600.0 & T_{snap} &= 600 & T_{solve} &= 600 \\amp_{f max} &= 1.08 & \epsilon_{w} &= 0.01 & \lambda &= 3.42619952 \\\lambda_{max} &= 5.99584916 & \lambda_{min} &= 0.85654988 & r_{facet base} &= 0.2 \\t_{ICAL,B} &= 3600.0 & t_{ICAL,G} &= 1.0 & t_{ICAL,I} &= 10.0 \\\end{aligned}$
Equations for SKA1_Low DPrepA_Image:¶
$\displaystyle \begin{aligned}Rflop_{Correct} &= 8 N_{Ipatches} N_{beam} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.042088\,\text{PFLOP/s} \\ \\Rflop_{DFT} &= \frac{N_{beam} N_{majortotal} \left(32 N_{a}^{2} N_{source} + 128 N_{a}^{2} + 266 N_{a} N_{source}\right) \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)}}{N_{bl}} \\ &= 1.434116\,\text{PFLOP/s} \\ \\Rflop_{Degrid} &= 8 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,predict}{\left(B_{max}{\left(i \right)} \right)} N_{kernel2,predict}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.009110\,\text{PFLOP/s} \\ \\Rflop_{Degridding Kernel Update} &= \frac{5.0 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,predict}{\left(B_{max}{\left(i \right)} \right)} N_{gcf,used,predict}{\left(B_{max}{\left(i \right)} \right)} N_{kernel,AW,predict}^{2}{\left(B_{max}{\left(i \right)} \right)} \log{\left(N_{kernel,AW,predict}^{2}{\left(B_{max}{\left(i \right)} \right)} \right)}}{T_{kernel,predict}{\left(B_{max}{\left(i \right)} \right)}}}{\log{\left(2 \right)}} \\ &= 0.017089\,\text{PFLOP/s} \\ \\Rflop_{FFT} &= \frac{2.5 N_{beam} N_{f,FFT,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp} \log{\left(N_{pix,linear}^{2} \right)} \max\left(1, \frac{\Delta_{W max}{\left(B_{max} \right)}}{\Delta_{W stack}}\right)}{T_{snap} \log{\left(2 \right)}} \\ &= 0.002541\,\text{PFLOP/s} \\ \\Rflop_{Flag} &= 279 N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.012232\,\text{PFLOP/s} \\ \\Rflop_{Grid} &= 8 N_{beam} N_{facet}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)} N_{kernel2,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.009110\,\text{PFLOP/s} \\ \\Rflop_{Gridding Kernel Update} &= \frac{5.0 N_{beam} N_{facet}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,backward}{\left(B_{max}{\left(i \right)} \right)} N_{gcf,used,backward}{\left(B_{max}{\left(i \right)} \right)} N_{kernel,AW,backward}^{2}{\left(B_{max}{\left(i \right)} \right)} \log{\left(N_{kernel,AW,backward}^{2}{\left(B_{max}{\left(i \right)} \right)} \right)}}{T_{kernel,backward}{\left(B_{max}{\left(i \right)} \right)}}}{\log{\left(2 \right)}} \\ &= 0.017089\,\text{PFLOP/s} \\ \\Rflop_{IFFT} &= \frac{2.5 N_{beam} N_{f,FFT,predict} N_{facet,predict}^{2} N_{majortotal} N_{pix,linear,predict}^{2} N_{pp} \log{\left(N_{pix,linear,predict}^{2} \right)} \max\left(1, \frac{\Delta_{W max}{\left(B_{max} \right)}}{\Delta_{W stack}}\right)}{T_{snap} \log{\left(2 \right)}} \\ &= 0.012704\,\text{PFLOP/s} \\ \\Rflop_{Identify Component} &= \frac{N_{beam} N_{facet}^{2} N_{majortotal} N_{scales} N_{subbands} N_{tt} \left(2 N_{minor} N_{patch}^{2} + 2 N_{pix,linear}^{2}\right)}{T_{obs}} \\ &= 0.000037\,\text{PFLOP/s} \\ \\Rflop_{Image Spectral Fitting} &= \frac{N_{beam} N_{majortotal} N_{pix,linear,fov,total}^{2} N_{pp} N_{tt} \left(2.0 N_{f,FFT,backward} + 2.0 N_{f,FFT,predict}\right)}{T_{obs}} \\ &= 0.000005\,\text{PFLOP/s} \\ \\Rflop_{Phase Rotation} &= \frac{28 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 0.000000\,\text{PFLOP/s} \\ \\Rflop_{Phase Rotation Predict} &= \frac{28 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 0.000000\,\text{PFLOP/s} \\ \\Rflop_{Reprojection} &= \frac{50.0 N_{beam} N_{f,proj,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.000003\,\text{PFLOP/s} \\ \\Rflop_{Reprojection Predict} &= \frac{50.0 N_{beam} N_{f,proj,predict} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.000003\,\text{PFLOP/s} \\ \\Rflop_{Subtract Image Component} &= \frac{2 N_{beam} N_{majortotal} N_{minor} N_{patch}^{2} N_{pp} N_{scales} N_{subbands} N_{tt}}{T_{obs}} \\ &= 0.000072\,\text{PFLOP/s} \\ \\Rflop_{Subtract Visibility} &= 2 N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.000088\,\text{PFLOP/s} \\ \\Rflop_{Visibility Weighting} &= 16 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.000048\,\text{PFLOP/s} \\ \\ Rflop &= 1.556332\,\text{PFLOP/s}\end{aligned}$
$\displaystyle \begin{aligned}Rout_{Correct} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.526097\,\text{TB/s} \\ \\Rout_{DFT} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.526097\,\text{TB/s} \\ \\Rout_{Degrid} &= M_{vis} N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,predict}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.035899\,\text{TB/s} \\ \\Rout_{Degridding Kernel Update} &= 8 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{pp} Q_{gcf}^{3} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,predict}{\left(B_{max}{\left(i \right)} \right)} N_{gw,predict}^{3}{\left(B_{max}{\left(i \right)} \right)}}{T_{kernel,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 17.945080\,\text{TB/s} \\ \\Rout_{FFT} &= \frac{M_{px} N_{beam} N_{f,FFT,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.003954\,\text{TB/s} \\ \\Rout_{Flag} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.526097\,\text{TB/s} \\ \\Rout_{Grid} &= \frac{M_{cpx} N_{beam} N_{facet}^{2} N_{majortotal} N_{pix,linear} N_{pp} \left(\frac{N_{pix,linear}}{2} + 1\right) \sum_{i=1}^{N_{bl}} N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{snap}} \\ &= 2.626481\,\text{TB/s} \\ \\Rout_{Gridding Kernel Update} &= 8 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} Q_{gcf}^{3} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,backward}{\left(B_{max}{\left(i \right)} \right)} N_{gw,backward}^{3}{\left(B_{max}{\left(i \right)} \right)}}{T_{kernel,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 17.945080\,\text{TB/s} \\ \\Rout_{IFFT} &= \frac{M_{cpx} N_{beam} N_{f,FFT,predict} N_{facet,predict}^{2} N_{majortotal} N_{pix,linear,predict} N_{pp} \left(\frac{N_{pix,linear,predict}}{2} + 1\right)}{T_{snap}} \\ &= 0.019773\,\text{TB/s} \\ \\Rout_{Identify Component} &= \frac{M_{cpx} N_{beam} N_{facet}^{2} N_{majortotal} N_{minor} N_{subbands}}{T_{obs}} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Image Spectral Fitting} &= \frac{M_{px} N_{beam} N_{majortotal} N_{pix,linear,fov,total}^{2} N_{pp} N_{tt}}{T_{obs}} \\ &= 0.000082\,\text{TB/s} \\ \\Rout_{Phase Rotation} &= M_{vis} N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} R_{vis,backward}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Phase Rotation Predict} &= \frac{M_{vis} N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Reprojection} &= \frac{M_{px} N_{beam} N_{f,proj,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.000494\,\text{TB/s} \\ \\Rout_{Reprojection Predict} &= \frac{M_{px} N_{beam} N_{f,proj,predict} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.000494\,\text{TB/s} \\ \\Rout_{Subtract Image Component} &= \frac{M_{px} N_{beam} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{subbands} N_{tt}}{T_{obs}} \\ &= 0.000288\,\text{TB/s} \\ \\Rout_{Subtract Visibility} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.526097\,\text{TB/s} \\ \\Rout_{Visibility Weighting} &= M_{vis} N_{beam} N_{f,min,gran} N_{facet}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{1}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.004748\,\text{TB/s} \\ \\ Rout &= 40.686763\,\text{TB/s}\end{aligned}$
Intermediate equations¶
$\displaystyle \begin{aligned}\Delta_{W max} &= \Delta_{W max} \\N_{f,FFT,backward} &= N_{f,out} \\N_{f,FFT,predict} &= \min\left(N_{f,max}, N_{f,out} N_{tt}\right) \\N_{f,gcf,backward}(b) &= \max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \\N_{f,gcf,predict}(b) &= \max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \\N_{f,proj,backward} &= N_{tt} \\N_{f,proj,predict} &= N_{tt} \\N_{f,vis}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{B_{max} Q_{bw} \Theta_{fov,total}} \right)}}\right)\right) \\N_{f,vis,backward}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right) \\N_{f,vis,predict}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov,predict} b} \right)}}\right)\right) \\N_{gcf,used,backward}(b) &= Q_{gcf}^{2} \cdot \left(1 - \left(1 - \frac{1}{Q_{gcf}^{2}}\right)^{\frac{T_{ion} \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right)}{\max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)}}\right) \\N_{gcf,used,predict}(b) &= Q_{gcf}^{2} \cdot \left(1 - \left(1 - \frac{1}{Q_{gcf}^{2}}\right)^{\frac{T_{ion} \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov,predict} b} \right)}}\right)\right)}{\max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)}}\right) \\N_{gw,backward}(b) &= 1.0 \Theta_{fov} \sqrt{\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}} \\N_{gw,predict}(b) &= 1.0 \Theta_{fov,predict} \sqrt{\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}} \\N_{kernel,AW,backward}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5} \\N_{kernel,AW,predict}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5} \\N_{kernel2,backward}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{1.0} \\N_{kernel2,predict}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{1.0} \\R_{vis}(b,bcount) &= \frac{bcount \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{B_{max} Q_{bw} \Theta_{fov,total}} \right)}}\right)\right)}{T_{int,used}} \\R_{vis,backward}(b,bcount) &= \frac{bcount \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right)}{\min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)} \\T_{coal,backward}(b) &= \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right) \\T_{coal,predict}(b) &= \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right) \\T_{kernel,backward}(b) &= T_{ion} \\T_{kernel,predict}(b) &= T_{ion} \\\end{aligned}$
$\displaystyle \begin{aligned}N_{facet,predict} &= N_{facet} \\N_{pix,linear} &= \frac{3.25100497089045 B_{max} Q_{fov} Q_{pix} Q_{subband} \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\N_{pix,linear,fov,total} &= \frac{3.25100497089045 B_{max} Q_{fov} Q_{pix} Q_{subband}}{D_{s}} \\N_{pix,linear,predict} &= \frac{3.25100497089045 B_{max} Q_{fov} Q_{pix} Q_{subband} \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\Q_{bw} &= \frac{0.600124986981879}{\sqrt{1 - \frac{1}{amp_{f max}}}} \\T_{int,used} &= \max\left(T_{int,min}, \min\left(T_{snap}, \frac{1.0046082220589 D_{s} \sqrt{1 - \frac{1}{amp_{f max}}}}{B_{max} \Omega_{E} Q_{\mathcal{fov,I}} \sqrt{Q_{subband}}}\right)\right) \\\Theta_{fov} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\\Theta_{fov,predict} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\\Theta_{fov,total} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda}{D_{s}} \\\epsilon_{f approx} &= 2.44948974278318 \sqrt{1 - \frac{1}{amp_{f max}}} \\\end{aligned}$
Constants¶
$\displaystyle \begin{aligned}B_{max} &= 65000 & D_{s} &= 38 & \Delta_{W stack} &= 5.71616422372295 \\M_{cpx} &= 16.0 & M_{px} &= 8.0 & M_{vis} &= 12 \\N_{Ipatches} &= 30 & N_{a} &= 512 & N_{aa} &= 9 \\N_{beam} &= 1 & N_{bl} &= 130816.0 & N_{f,max} &= 65536 \\N_{f,min} &= 40 & N_{f,min,gran} &= 800 & N_{f,out} &= 40 \\N_{facet} &= 1 & N_{majortotal} &= 11 & N_{minor} &= 1000 \\N_{mm} &= 4 & N_{patch} &= 4097 & N_{pp} &= 2 \\N_{scales} &= 10 & N_{source} &= 1000 & N_{subbands} &= 7 \\N_{tt} &= 5 & \Omega_{E} &= 7.292115 \cdot 10^{-5} & Q_{fov} &= 1.0 \\Q_{\mathcal{fov,I}} &= 2.7 & Q_{gcf} &= 8.0 & Q_{kernel} &= 10.0 \\Q_{pix} &= 2.5 & Q_{subband} &= 1.32046924775612 & T_{int,min} &= 0.9 \\T_{ion} &= 10.0 & T_{obs} &= 3600.0 & T_{snap} &= 600 \\amp_{f max} &= 1.08 & \epsilon_{w} &= 0.01 & \lambda &= 3.42619952 \\\lambda_{max} &= 5.99584916 & \lambda_{min} &= 0.85654988 & r_{facet base} &= 0.2 \\\end{aligned}$
Equations for SKA1_Mid Ingest:¶
$\displaystyle \begin{aligned}Rflop_{Average} &= 8 N_{beam} N_{pp} R_{vis,ingest} \\ &= 0.000292\,\text{PFLOP/s} \\ \\Rflop_{Demix} &= N_{beam} N_{f,min} R_{vis,ingest} \left(154 N_{Ateam} + 84 + \frac{N_{f,min} T_{int,used} \left(N_{Ateam}^{2} + N_{Ateam} \left(24 N_{solve} + 33\right) + 64\right)}{N_{f,max} T_{ion}}\right) \\ &= 0.593070\,\text{PFLOP/s} \\ \\Rflop_{Flag} &= 278 N_{beam} N_{pp} R_{vis,ingest} \\ &= 0.010152\,\text{PFLOP/s} \\ \\Rflop_{Receive} &= N_{beam} N_{f,min} \left(\frac{1000 N_{a}}{T_{int,used}} + \frac{4 N_{pp} R_{vis,ingest}}{N_{f,min}}\right) \\ &= 0.000146\,\text{PFLOP/s} \\ \\ Rflop &= 0.603660\,\text{PFLOP/s}\end{aligned}$
$\displaystyle \begin{aligned}Rout_{Average} &= M_{vis} N_{beam} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.433796\,\text{TB/s} \\ \\Rout_{Demix} &= M_{vis} N_{beam} N_{pp} R_{vis,ingest} \\ &= 0.438222\,\text{TB/s} \\ \\Rout_{Flag} &= M_{vis} N_{beam} N_{pp} R_{vis,ingest} \\ &= 0.438222\,\text{TB/s} \\ \\Rout_{Receive} &= M_{vis} N_{beam} N_{pp} R_{vis,ingest} \\ &= 0.438222\,\text{TB/s} \\ \\ Rout &= 1.748463\,\text{TB/s}\end{aligned}$
Intermediate equations¶
$\displaystyle \begin{aligned}R_{vis}(b,bcount) &= \frac{bcount \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{B_{max} Q_{bw} \Theta_{fov,total}} \right)}}\right)\right)}{T_{int,used}} \\R_{vis,ingest} &= \frac{N_{f,max} \left(N_{a} + N_{bl}\right)}{T_{int,used}} \\\end{aligned}$
$\displaystyle \begin{aligned}Q_{bw} &= \frac{0.600124986981879}{\sqrt{1 - \frac{1}{amp_{f max}}}} \\T_{int,used} &= \max\left(T_{int,min}, \min\left(T_{snap}, \frac{1.0046082220589 D_{s} \sqrt{1 - \frac{1}{amp_{f max}}}}{B_{max} \Omega_{E} Q_{\mathcal{fov,I}} \sqrt{Q_{subband}}}\right)\right) \\\Theta_{fov,total} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda}{D_{s}} \\\end{aligned}$
Constants¶
$\displaystyle \begin{aligned}B_{max} &= 150000 & D_{s} &= 13.5 & M_{vis} &= 12 \\N_{Ateam} &= 10 & N_{a} &= 197 & N_{beam} &= 1 \\N_{bl} &= 19306.0 & N_{f,max} &= 65536 & N_{f,min} &= 40 \\N_{f,out} &= 65536 & N_{pp} &= 4 & N_{solve} &= 10 \\\Omega_{E} &= 7.292115 \cdot 10^{-5} & Q_{fov} &= 1.0 & Q_{\mathcal{fov,I}} &= 2.7 \\Q_{subband} &= 1.31607401295249 & T_{int,min} &= 0.14 & T_{ion} &= 10.0 \\T_{snap} &= 600 & amp_{f max} &= 1.034 & \lambda &= 0.571033253333333 \\\lambda_{max} &= 0.85654988 & \lambda_{min} &= 0.285516626666667 & \end{aligned}$
Equations for SKA1_Mid ICAL:¶
$\displaystyle \begin{aligned}Rflop_{Correct} &= 8 N_{Ipatches} N_{beam} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.010411\,\text{PFLOP/s} \\ \\Rflop_{DFT} &= \frac{N_{beam} N_{majortotal} \left(32 N_{a}^{2} N_{source} + 128 N_{a}^{2} + 266 N_{a} N_{source}\right) \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)}}{N_{bl}} \\ &= 5.473809\,\text{PFLOP/s} \\ \\Rflop_{Degrid} &= 8 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,predict}{\left(B_{max}{\left(i \right)} \right)} N_{kernel2,predict}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.590383\,\text{PFLOP/s} \\ \\Rflop_{Degridding Kernel Update} &= \frac{5.0 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,predict}{\left(B_{max}{\left(i \right)} \right)} N_{gcf,used,predict}{\left(B_{max}{\left(i \right)} \right)} N_{kernel,AW,predict}^{2}{\left(B_{max}{\left(i \right)} \right)} \log{\left(N_{kernel,AW,predict}^{2}{\left(B_{max}{\left(i \right)} \right)} \right)}}{T_{kernel,predict}{\left(B_{max}{\left(i \right)} \right)}}}{\log{\left(2 \right)}} \\ &= 0.541087\,\text{PFLOP/s} \\ \\Rflop_{FFT} &= \frac{2.5 N_{beam} N_{f,FFT,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp} \log{\left(N_{pix,linear}^{2} \right)} \max\left(1, \frac{\Delta_{W max}{\left(B_{max} \right)}}{\Delta_{W stack}}\right)}{T_{snap} \log{\left(2 \right)}} \\ &= 0.294690\,\text{PFLOP/s} \\ \\Rflop_{Flag} &= 279 N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.090772\,\text{PFLOP/s} \\ \\Rflop_{Grid} &= 8 N_{beam} N_{facet}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)} N_{kernel2,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.590383\,\text{PFLOP/s} \\ \\Rflop_{Gridding Kernel Update} &= \frac{5.0 N_{beam} N_{facet}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,backward}{\left(B_{max}{\left(i \right)} \right)} N_{gcf,used,backward}{\left(B_{max}{\left(i \right)} \right)} N_{kernel,AW,backward}^{2}{\left(B_{max}{\left(i \right)} \right)} \log{\left(N_{kernel,AW,backward}^{2}{\left(B_{max}{\left(i \right)} \right)} \right)}}{T_{kernel,backward}{\left(B_{max}{\left(i \right)} \right)}}}{\log{\left(2 \right)}} \\ &= 0.541087\,\text{PFLOP/s} \\ \\Rflop_{IFFT} &= \frac{2.5 N_{beam} N_{f,FFT,predict} N_{facet,predict}^{2} N_{majortotal} N_{pix,linear,predict}^{2} N_{pp} \log{\left(N_{pix,linear,predict}^{2} \right)} \max\left(1, \frac{\Delta_{W max}{\left(B_{max} \right)}}{\Delta_{W stack}}\right)}{T_{snap} \log{\left(2 \right)}} \\ &= 1.473449\,\text{PFLOP/s} \\ \\Rflop_{Identify Component} &= \frac{N_{beam} N_{facet}^{2} N_{majortotal} N_{scales} N_{subbands} N_{tt} \left(2 N_{minor} N_{patch}^{2} + 2 N_{pix,linear}^{2}\right)}{T_{obs}} \\ &= 0.013744\,\text{PFLOP/s} \\ \\Rflop_{Phase Rotation} &= \frac{28 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 0.737887\,\text{PFLOP/s} \\ \\Rflop_{Phase Rotation Predict} &= \frac{28 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 0.737887\,\text{PFLOP/s} \\ \\Rflop_{Reprojection} &= \frac{50.0 N_{beam} N_{f,proj,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.017794\,\text{PFLOP/s} \\ \\Rflop_{Reprojection Predict} &= \frac{50.0 N_{beam} N_{f,proj,predict} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.088968\,\text{PFLOP/s} \\ \\Rflop_{Solve} &= N_{beam} N_{subbands} \left(N_{\mathcal{self}} + 1\right) \left(\frac{48 N_{a}^{2} N_{pp} N_{solve} \left(N_{Ipatches} \max\left(1, \frac{T_{solve}}{t_{ICAL,I}}\right) + \max\left(1, \frac{N_{B,parameters}}{N_{subbands}}\right) \max\left(1, \frac{T_{solve}}{t_{ICAL,B}}\right) + \max\left(1, \frac{T_{solve}}{t_{ICAL,G}}\right)\right)}{T_{solve}} + \frac{40 N_{pp} \left(N_{Ipatches} + 2\right) \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)}}{N_{subbands}}\right) \\ &= 0.017353\,\text{PFLOP/s} \\ \\Rflop_{Source Find} &= \frac{600 N_{majortotal} N_{source} N_{source,find,iterations}}{T_{obs}} \\ &= 0.000000\,\text{PFLOP/s} \\ \\Rflop_{Subtract Image Component} &= \frac{2 N_{beam} N_{majortotal} N_{minor} N_{patch}^{2} N_{pp} N_{scales} N_{subbands} N_{tt}}{T_{obs}} \\ &= 0.000671\,\text{PFLOP/s} \\ \\Rflop_{Subtract Visibility} &= 2 N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.000651\,\text{PFLOP/s} \\ \\Rflop_{Visibility Weighting} &= 16 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.002090\,\text{PFLOP/s} \\ \\ Rflop &= 11.223115\,\text{PFLOP/s}\end{aligned}$
$\displaystyle \begin{aligned}Rout_{Correct} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 3.904163\,\text{TB/s} \\ \\Rout_{DFT} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 3.904163\,\text{TB/s} \\ \\Rout_{Degrid} &= M_{vis} N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,predict}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 1.567427\,\text{TB/s} \\ \\Rout_{Degridding Kernel Update} &= 8 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{pp} Q_{gcf}^{3} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,predict}{\left(B_{max}{\left(i \right)} \right)} N_{gw,predict}^{3}{\left(B_{max}{\left(i \right)} \right)}}{T_{kernel,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 1248.289933\,\text{TB/s} \\ \\Rout_{FFT} &= \frac{M_{px} N_{beam} N_{f,FFT,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 2.846965\,\text{TB/s} \\ \\Rout_{Flag} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 3.904163\,\text{TB/s} \\ \\Rout_{Grid} &= \frac{M_{cpx} N_{beam} N_{facet}^{2} N_{majortotal} N_{pix,linear} N_{pp} \left(\frac{N_{pix,linear}}{2} + 1\right) \sum_{i=1}^{N_{bl}} N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{snap}} \\ &= 9105.310769\,\text{TB/s} \\ \\Rout_{Gridding Kernel Update} &= 8 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} Q_{gcf}^{3} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,backward}{\left(B_{max}{\left(i \right)} \right)} N_{gw,backward}^{3}{\left(B_{max}{\left(i \right)} \right)}}{T_{kernel,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 1248.289933\,\text{TB/s} \\ \\Rout_{IFFT} &= \frac{M_{cpx} N_{beam} N_{f,FFT,predict} N_{facet,predict}^{2} N_{majortotal} N_{pix,linear,predict} N_{pp} \left(\frac{N_{pix,linear,predict}}{2} + 1\right)}{T_{snap}} \\ &= 14.235491\,\text{TB/s} \\ \\Rout_{Identify Component} &= \frac{M_{cpx} N_{beam} N_{facet}^{2} N_{majortotal} N_{minor} N_{subbands}}{T_{obs}} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Phase Rotation} &= M_{vis} N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} R_{vis,backward}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 1.567427\,\text{TB/s} \\ \\Rout_{Phase Rotation Predict} &= \frac{M_{vis} N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 316.237205\,\text{TB/s} \\ \\Rout_{Reprojection} &= \frac{M_{px} N_{beam} N_{f,proj,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 2.846965\,\text{TB/s} \\ \\Rout_{Reprojection Predict} &= \frac{M_{px} N_{beam} N_{f,proj,predict} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 14.234825\,\text{TB/s} \\ \\Rout_{Solve} &= \frac{M_{jones} N_{a} N_{beam} N_{subbands} \left(N_{\mathcal{self}} + 1\right) \left(N_{Ipatches} \max\left(1, \frac{T_{solve}}{t_{ICAL,I}}\right) + \max\left(1, \frac{N_{B,parameters}}{N_{subbands}}\right) \max\left(1, \frac{T_{solve}}{t_{ICAL,B}}\right) + \max\left(1, \frac{T_{solve}}{t_{ICAL,G}}\right)\right)}{T_{solve}} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Source Find} &= 100 M_{cpx} N_{majortotal} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Subtract Image Component} &= \frac{M_{px} N_{beam} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{subbands} N_{tt}}{T_{obs}} \\ &= 0.059312\,\text{TB/s} \\ \\Rout_{Subtract Visibility} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 3.904163\,\text{TB/s} \\ \\Rout_{Visibility Weighting} &= M_{vis} N_{beam} N_{f,min,gran} N_{facet}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{1}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.141676\,\text{TB/s} \\ \\ Rout &= 11971.244580\,\text{TB/s}\end{aligned}$
Intermediate equations¶
$\displaystyle \begin{aligned}\Delta_{W max} &= \Delta_{W max} \\N_{f,FFT,backward} &= N_{f,out} \\N_{f,FFT,predict} &= \min\left(N_{f,max}, N_{f,out} N_{tt}\right) \\N_{f,gcf,backward}(b) &= \max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \\N_{f,gcf,predict}(b) &= \max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \\N_{f,proj,backward} &= N_{f,out} \\N_{f,proj,predict} &= \min\left(N_{f,max}, N_{f,out} N_{tt}\right) \\N_{f,vis}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{B_{max} Q_{bw} \Theta_{fov,total}} \right)}}\right)\right) \\N_{f,vis,backward}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right) \\N_{f,vis,predict}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov,predict} b} \right)}}\right)\right) \\N_{gcf,used,backward}(b) &= Q_{gcf}^{2} \cdot \left(1 - \left(1 - \frac{1}{Q_{gcf}^{2}}\right)^{\frac{T_{ion} \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right)}{\max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)}}\right) \\N_{gcf,used,predict}(b) &= Q_{gcf}^{2} \cdot \left(1 - \left(1 - \frac{1}{Q_{gcf}^{2}}\right)^{\frac{T_{ion} \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov,predict} b} \right)}}\right)\right)}{\max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)}}\right) \\N_{gw,backward}(b) &= 1.0 \Theta_{fov} \sqrt{\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}} \\N_{gw,predict}(b) &= 1.0 \Theta_{fov,predict} \sqrt{\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}} \\N_{kernel,AW,backward}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5} \\N_{kernel,AW,predict}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5} \\N_{kernel2,backward}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{1.0} \\N_{kernel2,predict}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{1.0} \\R_{vis}(b,bcount) &= \frac{bcount \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{B_{max} Q_{bw} \Theta_{fov,total}} \right)}}\right)\right)}{T_{int,used}} \\R_{vis,backward}(b,bcount) &= \frac{bcount \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right)}{\min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)} \\T_{coal,backward}(b) &= \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right) \\T_{coal,predict}(b) &= \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right) \\T_{kernel,backward}(b) &= T_{ion} \\T_{kernel,predict}(b) &= T_{ion} \\\end{aligned}$
$\displaystyle \begin{aligned}N_{facet,predict} &= N_{facet} \\N_{pix,linear} &= \frac{3.25100497089045 B_{max} Q_{fov} Q_{pix} Q_{subband} \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\N_{pix,linear,predict} &= \frac{3.25100497089045 B_{max} Q_{fov} Q_{pix} Q_{subband} \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\Q_{bw} &= \frac{0.600124986981879}{\sqrt{1 - \frac{1}{amp_{f max}}}} \\T_{int,used} &= \max\left(T_{int,min}, \min\left(T_{snap}, \frac{1.0046082220589 D_{s} \sqrt{1 - \frac{1}{amp_{f max}}}}{B_{max} \Omega_{E} Q_{\mathcal{fov,I}} \sqrt{Q_{subband}}}\right)\right) \\\Theta_{fov} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\\Theta_{fov,predict} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\\Theta_{fov,total} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda}{D_{s}} \\\epsilon_{f approx} &= 2.44948974278318 \sqrt{1 - \frac{1}{amp_{f max}}} \\\end{aligned}$
Constants¶
$\displaystyle \begin{aligned}B_{max} &= 150000 & D_{s} &= 13.5 & \Delta_{W stack} &= 533.649617084138 \\M_{cpx} &= 16.0 & M_{jones} &= 64.0 & M_{px} &= 8.0 \\M_{vis} &= 12 & N_{B,parameters} &= 500 & N_{Ipatches} &= 1 \\N_{a} &= 197 & N_{aa} &= 9 & N_{beam} &= 1 \\N_{bl} &= 19306.0 & N_{f,max} &= 65536 & N_{f,min} &= 40 \\N_{f,min,gran} &= 800 & N_{f,out} &= 40 & N_{facet} &= 9 \\N_{majortotal} &= 9 & N_{minor} &= 10000 & N_{mm} &= 4 \\N_{patch} &= 4097 & N_{pp} &= 4 & N_{scales} &= 10 \\N_{\mathcal{self}} &= 3 & N_{solve} &= 10 & N_{source} &= 1000 \\N_{source,find,iterations} &= 10 & N_{subbands} &= 4 & N_{tt} &= 5 \\\Omega_{E} &= 7.292115 \cdot 10^{-5} & Q_{fov} &= 2.7 & Q_{\mathcal{fov,I}} &= 2.7 \\Q_{gcf} &= 8.0 & Q_{kernel} &= 10.0 & Q_{pix} &= 2.5 \\Q_{subband} &= 1.31607401295249 & T_{int,min} &= 0.14 & T_{ion} &= 10.0 \\T_{obs} &= 3600.0 & T_{snap} &= 600 & T_{solve} &= 600 \\amp_{f max} &= 1.034 & \epsilon_{w} &= 0.01 & \lambda &= 0.571033253333333 \\\lambda_{max} &= 0.85654988 & \lambda_{min} &= 0.285516626666667 & r_{facet base} &= 0.2 \\t_{ICAL,B} &= 3600.0 & t_{ICAL,G} &= 1.0 & t_{ICAL,I} &= 10.0 \\\end{aligned}$
Equations for SKA1_Mid DPrepA_Image:¶
$\displaystyle \begin{aligned}Rflop_{Correct} &= 8 N_{Ipatches} N_{beam} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.006362\,\text{PFLOP/s} \\ \\Rflop_{DFT} &= \frac{N_{beam} N_{majortotal} \left(32 N_{a}^{2} N_{source} + 128 N_{a}^{2} + 266 N_{a} N_{source}\right) \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)}}{N_{bl}} \\ &= 6.690211\,\text{PFLOP/s} \\ \\Rflop_{Degrid} &= 8 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,predict}{\left(B_{max}{\left(i \right)} \right)} N_{kernel2,predict}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.055705\,\text{PFLOP/s} \\ \\Rflop_{Degridding Kernel Update} &= \frac{5.0 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,predict}{\left(B_{max}{\left(i \right)} \right)} N_{gcf,used,predict}{\left(B_{max}{\left(i \right)} \right)} N_{kernel,AW,predict}^{2}{\left(B_{max}{\left(i \right)} \right)} \log{\left(N_{kernel,AW,predict}^{2}{\left(B_{max}{\left(i \right)} \right)} \right)}}{T_{kernel,predict}{\left(B_{max}{\left(i \right)} \right)}}}{\log{\left(2 \right)}} \\ &= 0.043445\,\text{PFLOP/s} \\ \\Rflop_{FFT} &= \frac{2.5 N_{beam} N_{f,FFT,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp} \log{\left(N_{pix,linear}^{2} \right)} \max\left(1, \frac{\Delta_{W max}{\left(B_{max} \right)}}{\Delta_{W stack}}\right)}{T_{snap} \log{\left(2 \right)}} \\ &= 0.027769\,\text{PFLOP/s} \\ \\Rflop_{Flag} &= 279 N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.055472\,\text{PFLOP/s} \\ \\Rflop_{Grid} &= 8 N_{beam} N_{facet}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)} N_{kernel2,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.055705\,\text{PFLOP/s} \\ \\Rflop_{Gridding Kernel Update} &= \frac{5.0 N_{beam} N_{facet}^{2} N_{majortotal} N_{mm} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,backward}{\left(B_{max}{\left(i \right)} \right)} N_{gcf,used,backward}{\left(B_{max}{\left(i \right)} \right)} N_{kernel,AW,backward}^{2}{\left(B_{max}{\left(i \right)} \right)} \log{\left(N_{kernel,AW,backward}^{2}{\left(B_{max}{\left(i \right)} \right)} \right)}}{T_{kernel,backward}{\left(B_{max}{\left(i \right)} \right)}}}{\log{\left(2 \right)}} \\ &= 0.043445\,\text{PFLOP/s} \\ \\Rflop_{IFFT} &= \frac{2.5 N_{beam} N_{f,FFT,predict} N_{facet,predict}^{2} N_{majortotal} N_{pix,linear,predict}^{2} N_{pp} \log{\left(N_{pix,linear,predict}^{2} \right)} \max\left(1, \frac{\Delta_{W max}{\left(B_{max} \right)}}{\Delta_{W stack}}\right)}{T_{snap} \log{\left(2 \right)}} \\ &= 0.138846\,\text{PFLOP/s} \\ \\Rflop_{Identify Component} &= \frac{N_{beam} N_{facet}^{2} N_{majortotal} N_{scales} N_{subbands} N_{tt} \left(2 N_{minor} N_{patch}^{2} + 2 N_{pix,linear}^{2}\right)}{T_{obs}} \\ &= 0.000209\,\text{PFLOP/s} \\ \\Rflop_{Image Spectral Fitting} &= \frac{N_{beam} N_{majortotal} N_{pix,linear,fov,total}^{2} N_{pp} N_{tt} \left(2.0 N_{f,FFT,backward} + 2.0 N_{f,FFT,predict}\right)}{T_{obs}} \\ &= 0.000207\,\text{PFLOP/s} \\ \\Rflop_{Phase Rotation} &= \frac{28 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 0.050103\,\text{PFLOP/s} \\ \\Rflop_{Phase Rotation Predict} &= \frac{28 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 0.050103\,\text{PFLOP/s} \\ \\Rflop_{Reprojection} &= \frac{50.0 N_{beam} N_{f,proj,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.000186\,\text{PFLOP/s} \\ \\Rflop_{Reprojection Predict} &= \frac{50.0 N_{beam} N_{f,proj,predict} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.000186\,\text{PFLOP/s} \\ \\Rflop_{Subtract Image Component} &= \frac{2 N_{beam} N_{majortotal} N_{minor} N_{patch}^{2} N_{pp} N_{scales} N_{subbands} N_{tt}}{T_{obs}} \\ &= 0.000041\,\text{PFLOP/s} \\ \\Rflop_{Subtract Visibility} &= 2 N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.000398\,\text{PFLOP/s} \\ \\Rflop_{Visibility Weighting} &= 16 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.000184\,\text{PFLOP/s} \\ \\ Rflop &= 7.218579\,\text{PFLOP/s}\end{aligned}$
$\displaystyle \begin{aligned}Rout_{Correct} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 2.385877\,\text{TB/s} \\ \\Rout_{DFT} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 2.385877\,\text{TB/s} \\ \\Rout_{Degrid} &= M_{vis} N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{N_{f,vis,predict}{\left(B_{max}{\left(i \right)} \right)}}{T_{coal,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.137678\,\text{TB/s} \\ \\Rout_{Degridding Kernel Update} &= 8 N_{beam} N_{facet,predict}^{2} N_{majortotal} N_{pp} Q_{gcf}^{3} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,predict}{\left(B_{max}{\left(i \right)} \right)} N_{gw,predict}^{3}{\left(B_{max}{\left(i \right)} \right)}}{T_{kernel,predict}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 116.720236\,\text{TB/s} \\ \\Rout_{FFT} &= \frac{M_{px} N_{beam} N_{f,FFT,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.238657\,\text{TB/s} \\ \\Rout_{Flag} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 2.385877\,\text{TB/s} \\ \\Rout_{Grid} &= \frac{M_{cpx} N_{beam} N_{facet}^{2} N_{majortotal} N_{pix,linear} N_{pp} \left(\frac{N_{pix,linear}}{2} + 1\right) \sum_{i=1}^{N_{bl}} N_{f,vis,backward}{\left(B_{max}{\left(i \right)} \right)}}{T_{snap}} \\ &= 848.087472\,\text{TB/s} \\ \\Rout_{Gridding Kernel Update} &= 8 N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} Q_{gcf}^{3} \sum_{i=1}^{N_{bl}} \frac{N_{f,gcf,backward}{\left(B_{max}{\left(i \right)} \right)} N_{gw,backward}^{3}{\left(B_{max}{\left(i \right)} \right)}}{T_{kernel,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 116.720236\,\text{TB/s} \\ \\Rout_{IFFT} &= \frac{M_{cpx} N_{beam} N_{f,FFT,predict} N_{facet,predict}^{2} N_{majortotal} N_{pix,linear,predict} N_{pp} \left(\frac{N_{pix,linear,predict}}{2} + 1\right)}{T_{snap}} \\ &= 1.193337\,\text{TB/s} \\ \\Rout_{Identify Component} &= \frac{M_{cpx} N_{beam} N_{facet}^{2} N_{majortotal} N_{minor} N_{subbands}}{T_{obs}} \\ &= 0.000000\,\text{TB/s} \\ \\Rout_{Image Spectral Fitting} &= \frac{M_{px} N_{beam} N_{majortotal} N_{pix,linear,fov,total}^{2} N_{pp} N_{tt}}{T_{obs}} \\ &= 0.003453\,\text{TB/s} \\ \\Rout_{Phase Rotation} &= M_{vis} N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} R_{vis,backward}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 0.137678\,\text{TB/s} \\ \\Rout_{Phase Rotation Predict} &= \frac{M_{vis} N_{beam} N_{facet}^{2} N_{majortotal} N_{pp} \operatorname{sign}{\left(N_{facet} - 1 \right)} \sum_{i=1}^{N_{bl}} N_{f,vis}{\left(B_{max}{\left(i \right)} \right)}}{T_{int,used}} \\ &= 21.472897\,\text{TB/s} \\ \\Rout_{Reprojection} &= \frac{M_{px} N_{beam} N_{f,proj,backward} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.029832\,\text{TB/s} \\ \\Rout_{Reprojection Predict} &= \frac{M_{px} N_{beam} N_{f,proj,predict} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{pp}}{T_{snap}} \\ &= 0.029832\,\text{TB/s} \\ \\Rout_{Subtract Image Component} &= \frac{M_{px} N_{beam} N_{facet}^{2} N_{majortotal} N_{pix,linear}^{2} N_{subbands} N_{tt}}{T_{obs}} \\ &= 0.009944\,\text{TB/s} \\ \\Rout_{Subtract Visibility} &= M_{vis} N_{beam} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} R_{vis}{\left(B_{max}{\left(i \right)},1 \right)} \\ &= 2.385877\,\text{TB/s} \\ \\Rout_{Visibility Weighting} &= M_{vis} N_{beam} N_{f,min,gran} N_{facet}^{2} N_{majortotal} N_{pp} \sum_{i=1}^{N_{bl}} \frac{1}{T_{coal,backward}{\left(B_{max}{\left(i \right)} \right)}} \\ &= 0.010757\,\text{TB/s} \\ \\ Rout &= 1114.335518\,\text{TB/s}\end{aligned}$
Intermediate equations¶
$\displaystyle \begin{aligned}\Delta_{W max} &= \Delta_{W max} \\N_{f,FFT,backward} &= N_{f,out} \\N_{f,FFT,predict} &= \min\left(N_{f,max}, N_{f,out} N_{tt}\right) \\N_{f,gcf,backward}(b) &= \max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \\N_{f,gcf,predict}(b) &= \max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \\N_{f,proj,backward} &= N_{tt} \\N_{f,proj,predict} &= N_{tt} \\N_{f,vis}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{B_{max} Q_{bw} \Theta_{fov,total}} \right)}}\right)\right) \\N_{f,vis,backward}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right) \\N_{f,vis,predict}(b) &= \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov,predict} b} \right)}}\right)\right) \\N_{gcf,used,backward}(b) &= Q_{gcf}^{2} \cdot \left(1 - \left(1 - \frac{1}{Q_{gcf}^{2}}\right)^{\frac{T_{ion} \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right)}{\max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)}}\right) \\N_{gcf,used,predict}(b) &= Q_{gcf}^{2} \cdot \left(1 - \left(1 - \frac{1}{Q_{gcf}^{2}}\right)^{\frac{T_{ion} \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov,predict} b} \right)}}\right)\right)}{\max\left(N_{f,min}, \min\left(N_{f,max}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1.0 + \frac{\epsilon_{f approx}}{Q_{kernel} \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5}} \right)}}\right)\right) \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)}}\right) \\N_{gw,backward}(b) &= 1.0 \Theta_{fov} \sqrt{\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}} \\N_{gw,predict}(b) &= 1.0 \Theta_{fov,predict} \sqrt{\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}} \\N_{kernel,AW,backward}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5} \\N_{kernel,AW,predict}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{0.5} \\N_{kernel2,backward}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov}^{2} \left(\Theta_{fov}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{1.0} \\N_{kernel2,predict}(b) &= \left(N_{aa}^{2} + 1.0 \Theta_{fov,predict}^{2} \left(\Theta_{fov,predict}^{2} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{2} + \frac{0.636619772367581 \Theta_{fov,predict} \min\left(\Delta_{W stack}, \Delta_{W max}{\left(b \right)}\right)^{1.5}}{\epsilon_{w}}\right)\right)^{1.0} \\R_{vis}(b,bcount) &= \frac{bcount \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{B_{max} Q_{bw} \Theta_{fov,total}} \right)}}\right)\right)}{T_{int,used}} \\R_{vis,backward}(b,bcount) &= \frac{bcount \min\left(N_{f,max}, \max\left(N_{f,out}, \frac{\log{\left(\frac{\lambda_{max}}{\lambda_{min}} \right)}}{\log{\left(1 + \frac{1.5 \lambda}{Q_{bw} \Theta_{fov} b} \right)}}\right)\right)}{\min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right)} \\T_{coal,backward}(b) &= \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right) \\T_{coal,predict}(b) &= \min\left(T_{ion}, T_{int,used} \max\left(1.0, \min\left(\frac{T_{snap}}{T_{int,used}}, \left\lfloor{\frac{\epsilon_{f approx} \lambda}{\Omega_{E} T_{int,used} \Theta_{fov} b}}\right\rfloor\right)\right)\right) \\T_{kernel,backward}(b) &= T_{ion} \\T_{kernel,predict}(b) &= T_{ion} \\\end{aligned}$
$\displaystyle \begin{aligned}N_{facet,predict} &= N_{facet} \\N_{pix,linear} &= \frac{3.25100497089045 B_{max} Q_{fov} Q_{pix} Q_{subband} \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\N_{pix,linear,fov,total} &= \frac{3.25100497089045 B_{max} Q_{fov} Q_{pix} Q_{subband}}{D_{s}} \\N_{pix,linear,predict} &= \frac{3.25100497089045 B_{max} Q_{fov} Q_{pix} Q_{subband} \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\Q_{bw} &= \frac{0.600124986981879}{\sqrt{1 - \frac{1}{amp_{f max}}}} \\T_{int,used} &= \max\left(T_{int,min}, \min\left(T_{snap}, \frac{1.0046082220589 D_{s} \sqrt{1 - \frac{1}{amp_{f max}}}}{B_{max} \Omega_{E} Q_{\mathcal{fov,I}} \sqrt{Q_{subband}}}\right)\right) \\\Theta_{fov} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\\Theta_{fov,predict} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda \left(r_{facet base} \operatorname{sign}{\left(N_{facet} - 1 \right)} + 1\right)}{D_{s} N_{facet}} \\\Theta_{fov,total} &= \frac{2.43825372816784 Q_{fov} \sqrt{Q_{subband}} \lambda}{D_{s}} \\\epsilon_{f approx} &= 2.44948974278318 \sqrt{1 - \frac{1}{amp_{f max}}} \\\end{aligned}$
Constants¶
$\displaystyle \begin{aligned}B_{max} &= 150000 & D_{s} &= 13.5 & \Delta_{W stack} &= 479.425367253958 \\M_{cpx} &= 16.0 & M_{px} &= 8.0 & M_{vis} &= 12 \\N_{Ipatches} &= 1 & N_{a} &= 197 & N_{aa} &= 9 \\N_{beam} &= 1 & N_{bl} &= 19306.0 & N_{f,max} &= 65536 \\N_{f,min} &= 40 & N_{f,min,gran} &= 800 & N_{f,out} &= 40 \\N_{facet} &= 3 & N_{majortotal} &= 11 & N_{minor} &= 1000 \\N_{mm} &= 4 & N_{patch} &= 4097 & N_{pp} &= 2 \\N_{scales} &= 10 & N_{source} &= 1000 & N_{subbands} &= 4 \\N_{tt} &= 5 & \Omega_{E} &= 7.292115 \cdot 10^{-5} & Q_{fov} &= 1.0 \\Q_{\mathcal{fov,I}} &= 2.7 & Q_{gcf} &= 8.0 & Q_{kernel} &= 10.0 \\Q_{pix} &= 2.5 & Q_{subband} &= 1.31607401295249 & T_{int,min} &= 0.14 \\T_{ion} &= 10.0 & T_{obs} &= 3600.0 & T_{snap} &= 600 \\amp_{f max} &= 1.034 & \epsilon_{w} &= 0.01 & \lambda &= 0.571033253333333 \\\lambda_{max} &= 0.85654988 & \lambda_{min} &= 0.285516626666667 & r_{facet base} &= 0.2 \\\end{aligned}$
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