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SENR/NRPy: A Next-Generation, Dynamical SENR/NRPy: A Next-Generation, Dynamical Reference Metric Numerical Relativity CodeReference Metric Numerical Relativity Code
Zachariah B. EtienneZachariah B. EtienneIan RuchlinIan Ruchlin
in collaboration within collaboration with
Thomas W. BaumgarteThomas W. Baumgarte
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SENR/NRPy: Code OverviewSENR/NRPy: Code Overview
● NRPy: Python+sympy code generation for NRNRPy: Python+sympy code generation for NR● Similar to Kranc, but with no Mathematica!Similar to Kranc, but with no Mathematica!
– Equations Equations at your fingertipsat your fingertips, even on HPC systems!, even on HPC systems!– Input: Einstein notation + simple syntax Python codeInput: Einstein notation + simple syntax Python code– Output: Efficient, compiler-vectorizable C code (AVX)Output: Efficient, compiler-vectorizable C code (AVX)
● SENR: Simple, Efficient Numerical Relativity codeSENR: Simple, Efficient Numerical Relativity code● Contains NRPy wrappers, diagnostics, MoL, BCs for Contains NRPy wrappers, diagnostics, MoL, BCs for
solving BSSN equations in arbitrary coord systemssolving BSSN equations in arbitrary coord systems– Log-Spherical Polar, Cylindrical, Cartesian, Bispherical-likeLog-Spherical Polar, Cylindrical, Cartesian, Bispherical-like
https://tinyurl.com/senrcode
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SENR/NRPy: MotivationSENR/NRPy: Motivation
● SENR/NRPy = Simple, efficient, open (BSD-licensed, SENR/NRPy = Simple, efficient, open (BSD-licensed, Python-based) infrastructure for numerical relativity codes Python-based) infrastructure for numerical relativity codes and code generationand code generation
● Goal: When solving problem, choose the best coordinate Goal: When solving problem, choose the best coordinate grid for the task!grid for the task!● Black hole, neutron star: Log-Spherical polar coordsBlack hole, neutron star: Log-Spherical polar coords● Compact binary: Dynamical, Bispherical-like coordsCompact binary: Dynamical, Bispherical-like coords
● Better coordinate grids = Giant efficiency gain over AMR!Better coordinate grids = Giant efficiency gain over AMR!● At least At least ~160x decrease in # of gridpoints → use desktop for BHB~160x decrease in # of gridpoints → use desktop for BHB● Single grid patch = ~25x better scalability than AMR!Single grid patch = ~25x better scalability than AMR!
https://tinyurl.com/senrcode
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Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
BHBHBHBHdxdx dxdx
2 dx2 dx
4 dx4 dx
8 dx8 dx
16 dx, etc16 dx, etc
AMRAdaptive Mesh Refinement
(Most Popular Method in NR)
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BHBHBHBHdxdx dxdx
2 dx2 dx
4 dx4 dx
AMRAdaptive Mesh Refinement
(Most Popular Method in NR)
8 dx8 dx
16 dx, etc16 dx, etc
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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BHBHdxdx
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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BHBHBHBHdy=dx=drdy=dx=dr
Near-Spherical Near-Spherical ObjectObject
● Highest res needed Highest res needed in radial dirn, need in radial dirn, need ~1/3 points in ~1/3 points in angular directions angular directions Cost: Cost: Nr*Ntheta*Nphi ~ Nr*Ntheta*Nphi ~ 1/9 Nr1/9 Nr33
● Cartesian grid: Cartesian grid: need dx=dy=dz=dr. need dx=dy=dz=dr. Cost:Cost:Nx*Ny*Nz ~ NrNx*Ny*Nz ~ Nr33
● So far, spherical So far, spherical polar grid ~polar grid ~9x 9x more more efficient than efficient than CartesianCartesian
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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BHBHBHBHdy=dx=drdy=dx=dr
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
What about dr What about dr along diagonal?along diagonal?
● Cube diagonal = Cube diagonal = √√3*sidelength → to 3*sidelength → to get dr resolution in get dr resolution in all directions, need all directions, need to reduce dx,dy,dz to reduce dx,dy,dz by by √√33
● Since cost in Since cost in memory ~1/dxmemory ~1/dx33, , “fitting the round “fitting the round peg in a square peg in a square hole” increases hole” increases cost by another cost by another factor of factor of ((√√3)3)33~5.2x!~5.2x!
Near-Spherical Near-Spherical ObjectObject
● Highest res needed Highest res needed in radial dirn, need in radial dirn, need ~1/3 points in ~1/3 points in angular directions angular directions Cost: Cost: Nr*Ntheta*Nphi ~ Nr*Ntheta*Nphi ~ 1/9 Nr1/9 Nr33
● Cartesian grid: Cartesian grid: need dx=dy=dz=dr. need dx=dy=dz=dr. Cost:Cost:Nx*Ny*Nz ~ NrNx*Ny*Nz ~ Nr33
● So far, spherical So far, spherical polar grid ~polar grid ~9x 9x more more efficient than efficient than CartesianCartesian
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BHBHBHBHdy=dx=drdy=dx=dr
Inefficiencies so far: Inefficiencies so far: ~47x~47x
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
What about dr What about dr along diagonal?along diagonal?
● Cube diagonal = Cube diagonal = √√3*sidelength → to 3*sidelength → to get dr resolution in get dr resolution in all directions, need all directions, need to reduce dx,dy,dz to reduce dx,dy,dz by by √√33
● Since cost in Since cost in memory ~1/dxmemory ~1/dx33, , “fitting the round “fitting the round peg in a square peg in a square hole” increases hole” increases cost by another cost by another factor of factor of ((√√3)3)33~5.2x!~5.2x!
Near-Spherical Near-Spherical ObjectObject
● Highest res needed Highest res needed in radial dirn, need in radial dirn, need ~1/3 points in ~1/3 points in angular directions angular directions Cost: Cost: Nr*Ntheta*Nphi ~ Nr*Ntheta*Nphi ~ 1/9 Nr1/9 Nr33
● Cartesian grid: Cartesian grid: need dx=dy=dz=dr. need dx=dy=dz=dr. Cost:Cost:Nx*Ny*Nz ~ NrNx*Ny*Nz ~ Nr33
● So far, spherical So far, spherical polar grid ~polar grid ~9x 9x more more efficient than efficient than CartesianCartesian
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BHBHBHBH AMR Box side-AMR Box side-length = 2length = 2
AMR Box AMR Box Boundary is a Boundary is a
Cube...Cube...
● … … but fields fall off but fields fall off radially!radially!
● →→region outside region outside orange circle is orange circle is over-resolved by 2xover-resolved by 2x
● Total volume of Total volume of over-resolved over-resolved region = 8-4/3 pi ~ region = 8-4/3 pi ~ 3.8 = about half the 3.8 = about half the cube!cube!
● Bispherical Bispherical coordinate system: coordinate system: Gain another ~1.7xGain another ~1.7x
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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BHBHBHBH AMR Box side-AMR Box side-length = 2length = 2
AMR Box AMR Box Boundary is a Boundary is a
Cube...Cube...
● … … but fields fall off but fields fall off radially!radially!
● →→region outside region outside orange circle is orange circle is over-resolved by 2xover-resolved by 2x
● Total volume of Total volume of over-resolved over-resolved region = 8-4/3 pi ~ region = 8-4/3 pi ~ 3.8 = about half the 3.8 = about half the cube!cube!
● Bispherical Bispherical coordinate system: coordinate system: Gain another ~1.7xGain another ~1.7x
Inefficiencies so far: Inefficiencies so far: ~80x~80x
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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BHBHBHBHdxdx dxdx
2 dx2 dx
4 dx4 dx
AMRAdaptive Mesh Refinement
(Most Popular Method in NR)
8 dx8 dx
AMRAMR
● Information must be interpolated Information must be interpolated across refinement boundaries.across refinement boundaries.
● Interpolation → grids must overlap Interpolation → grids must overlap ● Overlap regions (grey) can take up Overlap regions (grey) can take up
50% of overall computational domain!50% of overall computational domain!
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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BHBHBHBHdxdx dxdx
2 dx2 dx
4 dx4 dx
AMRAdaptive Mesh Refinement
(Most Popular Method in NR)
8 dx8 dx
AMRAMR
● Information must be interpolated Information must be interpolated across refinement boundaries.across refinement boundaries.
● Interpolation → grids must overlap Interpolation → grids must overlap ● Overlap regions (grey) can take up Overlap regions (grey) can take up
50% of overall computational domain!50% of overall computational domain!
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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BHBHBHBH
AMR Box AMR Box Boundary is a Boundary is a
Cube...Cube...
● … … but fields fall off but fields fall off radially!radially!
● →→region outside region outside orange circle is orange circle is over-resolved by 2xover-resolved by 2x
● Total volume of Total volume of over-resolved over-resolved region = 8-4/3 pi ~ region = 8-4/3 pi ~ 3.8 = about half the 3.8 = about half the cube!cube!
● Bispherical Bispherical coordinate system: coordinate system: Gain another ~1.7xGain another ~1.7x
High-order finite High-order finite difference with difference with
AMRAMR
● → → Enormous Enormous number of ghost number of ghost zones at refinement zones at refinement boundaries!boundaries!
● Ghost zones can Ghost zones can take up 50% of take up 50% of overall overall computational computational domain!domain!
● Bispherical Bispherical coordinate system: coordinate system: Gain another ~2xGain another ~2x
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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BHBHBHBH
AMR Box AMR Box Boundary is a Boundary is a
Cube...Cube...
● … … but fields fall off but fields fall off radially!radially!
● →→region outside region outside orange circle is orange circle is over-resolved by 2xover-resolved by 2x
● Total volume of Total volume of over-resolved over-resolved region = 8-4/3 pi ~ region = 8-4/3 pi ~ 3.8 = about half the 3.8 = about half the cube!cube!
● Bispherical Bispherical coordinate system: coordinate system: Gain another ~1.7xGain another ~1.7x
AMR Inefficiencies: AMR Inefficiencies: ~160x~160x
High-order finite High-order finite difference with difference with
AMRAMR
● → → Enormous Enormous number of ghost number of ghost zones at refinement zones at refinement boundaries!boundaries!
● Ghost zones can Ghost zones can take up 50% of take up 50% of overall overall computational computational domain!domain!
● Bispherical Bispherical coordinate system: coordinate system: Gain another ~2xGain another ~2x
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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AMR Inefficiencies: AMR Inefficiencies: ~160x~160x
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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AMR Inefficiencies: AMR Inefficiencies: ~160x (estimated)~160x (estimated)
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
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AMR Inefficiencies: AMR Inefficiencies: ~160x (estimated)~160x (estimated)
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
Inefficiency Inefficiency MeasurementMeasurement
● Single black holeSingle black hole● Moderate spin:Moderate spin:
● a/M = 0.5a/M = 0.5● Set up AMR Set up AMR
(Carpet) grid, (Carpet) grid, measure H measure H constraint violationconstraint violation
● Adjust SENR grids:Adjust SENR grids:● H_SENR < H_AMR H_SENR < H_AMR
at all pointsat all points
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AMR Inefficiencies: AMR Inefficiencies: ~160x (estimated)~160x (estimated)
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
Inefficiency Inefficiency MeasurementMeasurement
● Single black holeSingle black hole● Moderate spin:Moderate spin:
● a/M = 0.5a/M = 0.5● Set up AMR Set up AMR
(Carpet) grid, (Carpet) grid, measure H measure H constraint violationconstraint violation
● Adjust SENR grids:Adjust SENR grids:● H_SENR < H_AMR H_SENR < H_AMR
at all pointsat all points
AMR Grid:AMR Grid:● 10 GB10 GB
SENR's Log-SENR's Log-Spherical Spherical Polar Grid:Polar Grid:
● 40 MB40 MB● (un-optimized grid (un-optimized grid
structure, another structure, another 4-10x drop 4-10x drop possible)possible)
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AMR Inefficiencies: AMR Inefficiencies: ~160x (estimated)~160x (estimated)
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
Inefficiency Inefficiency MeasurementMeasurement
● Single black holeSingle black hole● Moderate spin:Moderate spin:
● a/M = 0.5a/M = 0.5● Set up AMR Set up AMR
(Carpet) grid, (Carpet) grid, measure H measure H constraint violationconstraint violation
● Adjust SENR grids:Adjust SENR grids:● H_SENR < H_AMR H_SENR < H_AMR
at all pointsat all points
AMR Grid:AMR Grid:● 10 GB10 GB
SENR's Log-SENR's Log-Spherical Spherical Polar Grid:Polar Grid:
● 40 MB40 MB● (un-optimized grid (un-optimized grid
structure, another structure, another 4-10x drop 4-10x drop possible)possible)
AMR Inefficiencies: AMR Inefficiencies: 250x (measured)250x (measured)
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AMR Inefficiencies: AMR Inefficiencies: ~160x (estimated)~160x (estimated)
Enormous Inefficiencies Exist in Enormous Inefficiencies Exist in Numerical Relativity (NR) SimulationsNumerical Relativity (NR) Simulations
Inefficiency Inefficiency MeasurementMeasurement
● Single black holeSingle black hole● Moderate spin:Moderate spin:
● a/M = 0.5a/M = 0.5● Set up AMR Set up AMR
(Carpet) grid, (Carpet) grid, measure H measure H constraint violationconstraint violation
● Adjust SENR grids:Adjust SENR grids:● H_SENR < H_AMR H_SENR < H_AMR
at all pointsat all points
AMR Grid:AMR Grid:● 10 GB10 GB
SENR's Log-SENR's Log-Spherical Spherical Polar Grid:Polar Grid:
● 40 MB40 MB● (un-optimized grid (un-optimized grid
structure, another structure, another 4-10x drop 4-10x drop possible)possible)
AMR Inefficiencies: AMR Inefficiencies: 250x (measured)250x (measured)
Stable long-term BH evolutions!Stable long-term BH evolutions!But does it converge?But does it converge?
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SENR Results: SENR Results: Exponential convergence of Exponential convergence of
numerical errors!numerical errors!
Simulating Simulating black black hole without hole without
excisionexcision: :
Numerical errors Numerical errors converge to zero converge to zero
exponentiallyexponentially with with increased increased
polynomial polynomial approximation order!approximation order!
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SENR/NRPy: SummarySENR/NRPy: Summary● Open Source, Open DevelopmentOpen Source, Open Development → → Greater AdoptionGreater Adoption
● http://tinyurl.com/senrcodehttp://tinyurl.com/senrcode● Algorithmic SimplicityAlgorithmic Simplicity → → More Science FasterMore Science Faster
● Easier to debug & extendEasier to debug & extend● Build on tried & true algorithmsBuild on tried & true algorithms
– BSSN in Spherical Polar Coords techniques pioneered by T. BSSN in Spherical Polar Coords techniques pioneered by T. Baumgarte et alBaumgarte et al
● SENR: Extend ideas to support arbitrary, SENR: Extend ideas to support arbitrary, dynamicaldynamical coords coords● Memory Efficiency Is Key FocusMemory Efficiency Is Key Focus: : Unlock the DesktopUnlock the Desktop
● Get public involved → ~10,000x more GW throughput!Get public involved → ~10,000x more GW throughput!
● Bottom line: Bottom line: Maximize science with minimal human & computational Maximize science with minimal human & computational resourcesresources