Performance vs Echoes

Before this page

Performance notes, for the paths timed below, and Cross-validation against Echoes, for the accuracy of the same runs.

Every number on this page is read from a JSON file written by a script of the repository, and can be measured again by anyone with the commands of the last section. Both codes run on the same machine, one thread each, timed by BenchmarkTools' @belapsed (the minimum of the samples); Echoes is called through its Python API, from Python modules that also run on their own.   means that MeanFieldHomogenization is faster; a ratio below 1 is in bold. The gap is the largest relative difference between the two results, so that each ratio compares two computations of the same number.

Kernels

Measured on AMD EPYC-Milan Processor, Julia 1.13.3, MeanFieldHomogenization 0.23.1, TensND 0.8.5, 2 thread; BenchmarkTools @belapsed minimum of 20 samples, one thread each; Echoes through PythonCall.

Hill tensor

The Julia back-end is the default one unless the case names another.

Case	Echoes				gap
Sphere a=b=c=1 / ISO	analytical	13.8 ns	72 ms	5200000×	6.6e-10
Prolate a=5,b=c=1 / ISO	analytical	85.7 ns	108 ms	1300000×	5.3e-10
Oblate a=b=5,c=1 / ISO	analytical	73.3 ns	67.1 ms	920000×	3.6e-10
Prolate a=3,b=c=1 / cubic	NUMINT3D	1.17 ms	98.3 ms	84.1×	9.8e-09
Prolate a=3,b=c=1 / cubic (Julia :decuhr)	NUMINT3D	1.17 ms	98.3 ms	84.3×	6.0e-06
Prolate a=3,b=c=1 / cubic (Julia :nestedquadgk)	NUMINT3D	224 µs	98 ms	440×	1.6e-09
Triaxial a=2,b=1,c=0.5 / triclinic	RESIDUES	3.67 ms	4.98 ms	1.36×	1.1e-07
Triaxial a=2,b=1,c=0.5 / triclinic	NUMINT3D	3.67 ms	89.8 ms	24.5×	2.5e-09
Triaxial a=2,b=1,c=0.5 / triclinic (Julia :decuhr)	RESIDUES	4.39 ms	4.98 ms	1.13×	1.1e-07
Triaxial a=2,b=1,c=0.5 / triclinic (Julia :decuhr)	NUMINT3D	4.39 ms	89.6 ms	20.4×	2.5e-09
Triaxial a=2,b=1,c=0.5 / triclinic (Julia :nestedquadgk)	RESIDUES	906 µs	4.97 ms	5.49×	1.1e-07
Triaxial a=2,b=1,c=0.5 / triclinic (Julia :nestedquadgk)	NUMINT3D	906 µs	89.5 ms	98.8×	2.5e-09
Rotated prolate a=5,b=c=1 / ISO (pi/4,pi/6,pi/3)	analytical	91.5 ns	108 ms	1200000×	3.0e-10
Rotated triaxial a=2,b=1,c=0.5 / triclinic (pi/4,pi/6,pi/3)	RESIDUES	2.19 ms	3.12 ms	1.43×	1.4e-12
Rotated triaxial a=2,b=1,c=0.5 / triclinic (pi/4,pi/6,pi/3)	NUMINT3D	2.19 ms	90.4 ms	41.4×	1.9e-10
Rotated triaxial a=2,b=1,c=0.5 / triclinic :decuhr (pi/4,pi/6,pi/3)	RESIDUES	2.99 ms	3.11 ms	1.04×	2.0e-10
Rotated triaxial a=2,b=1,c=0.5 / triclinic :decuhr (pi/4,pi/6,pi/3)	NUMINT3D	2.99 ms	90.6 ms	30.3×	5.5e-14

Crack compliance

Case	Echoes				gap
Penny / ISO	analytical	350 ns	57.3 ms	160000×	5.2e-10
Penny / cubic	NUMINT3D	2.5 ms	62.2 ms	24.9×	7.1e-09
Penny / triclinic	RESIDUES	1.71 ms	3.58 ms	2.1×	6.9e-10
Penny / triclinic	NUMINT3D	1.71 ms	105 ms	61.6×	6.9e-10

Conductivity

Case	Echoes				gap
Sphere a=b=c=1 / K iso	analytical	8.02 ns	4.84 µs	600×	0.0e+00
Prolate a=5,b=c=1 / K iso	analytical	165 ns	4.98 µs	30.2×	1.1e-16
Oblate a=b=5,c=1 / K iso	analytical	150 ns	4.94 µs	32.9×	1.8e-16
Triaxial a=2,b=1,c=0.5 / K aniso	RESIDUES	4.05 µs	8.58 µs	2.12×	2.2e-16
Triaxial a=2,b=1,c=0.5 / K aniso	NUMINT3D	4.05 µs	8.8 µs	2.17×	2.2e-16
Rotated triaxial a=2,b=1,c=0.5 / K aniso (pi/4,pi/6,pi/3)	RESIDUES	7.35 µs	9.13 µs	1.24×	1.0e-15
Rotated triaxial a=2,b=1,c=0.5 / K aniso (pi/4,pi/6,pi/3)	NUMINT3D	7.35 µs	9.17 µs	1.25×	1.0e-15

Derivative

ForwardDiff through hill_tensor on the Julia side, Echoes' hill_derivative on the other (a closed form for an isotropic medium).

Case	Echoes				gap
Sphere a=b=c=1 / ISO | κ=3k	ISO	557 ns	8.47 µs	15.2×	4.8e-16
Sphere a=b=c=1 / ISO | η=2μ	ISO	557 ns	8.47 µs	15.2×	9.8e-17
Prolate a=5,b=c=1 / ISO | κ=3k	ISO	1.25 µs	8.82 µs	7.04×	2.9e-16
Prolate a=5,b=c=1 / ISO | η=2μ	ISO	1.25 µs	8.79 µs	7.03×	6.2e-16
Oblate a=b=5,c=1 / ISO | κ=3k	ISO	1.23 µs	8.64 µs	7.05×	1.5e-16
Oblate a=b=5,c=1 / ISO | η=2μ	ISO	1.23 µs	8.55 µs	6.98×	4.8e-16
Prolate a=3,b=c=1 / cubic-as-ORTHO | C11	NUMINT3D	2.93 ms	108 ms	36.7×	5.1e-05
Prolate a=3,b=c=1 / cubic-as-ORTHO | C22	NUMINT3D	2.95 ms	111 ms	37.5×	1.8e-06
Prolate a=3,b=c=1 / cubic-as-ORTHO | C33	NUMINT3D	2.93 ms	112 ms	38.2×	2.1e-05
Prolate a=3,b=c=1 / cubic-as-ORTHO | C12	NUMINT3D	2.91 ms	123 ms	42.3×	1.3e-05
Prolate a=3,b=c=1 / cubic-as-ORTHO | C13	NUMINT3D	2.95 ms	121 ms	40.8×	1.3e-04
Prolate a=3,b=c=1 / cubic-as-ORTHO | C23	NUMINT3D	2.91 ms	128 ms	43.9×	9.3e-06
Prolate a=3,b=c=1 / cubic-as-ORTHO | μ₂₃	NUMINT3D	2.97 ms	153 ms	51.5×	5.2e-06
Prolate a=3,b=c=1 / cubic-as-ORTHO | μ₁₃	NUMINT3D	2.94 ms	133 ms	45.3×	3.7e-05
Prolate a=3,b=c=1 / cubic-as-ORTHO | μ₁₂	NUMINT3D	2.94 ms	130 ms	44.3×	1.0e-05

Schemes

Whole homogenizations of the same RVEs, with the same tolerances on both sides: numerical Hill tensors to a relative (absolute ), fixed points to a relative . The aging viscoelastic cases discretize a Maxwell matrix on 11 instants.

Measured on AMD EPYC-Milan Processor, Julia 1.13.2, MeanFieldHomogenization 0.20.1, TensND 0.8.3, 1 thread; BenchmarkTools @belapsed minimum, one thread; Hill tensors to rtol 1e-6, atol 1e-8; fixed points to rtol 1e-8.

Case	Scheme				gap
isotropic matrix, spheres	mori tanaka	791 ns	169 ms	210000×	1.9e-13
isotropic matrix, spheres	self consistent	9.8 µs	1.35 s	140000×	1.8e-13
isotropic matrix, spheres	differential	127 µs	85.7 s	680000×	5.5e-05
isotropic matrix, aligned prolate spheroids 5:1:1	mori tanaka	942 ns	225 ms	240000×	2.1e-12
isotropic matrix, aligned prolate spheroids 5:1:1	self consistent	75.7 µs	1.88 s	25000×	3.4e-13
isotropic matrix, oblate pores 1:1:0.1, isotropic average	mori tanaka	916 ns	165 ms	180000×	1.7e-13
isotropic matrix, oblate pores 1:1:0.1, isotropic average	self consistent	36.7 µs	6.65 s	180000×	3.0e-12
anisotropic matrix, two rotated triaxial families	dilute	1.13 ms	226 ms	200×	1.4e-11
anisotropic matrix, two rotated triaxial families	mori tanaka	1.17 ms	316 ms	270×	1.4e-11
conduction, isotropic matrix, spheres	mori tanaka	792 ns	35.8 µs	45.2×	1.4e-16
conduction, isotropic matrix, spheres	self consistent	22.4 µs	185 µs	8.26×	2.6e-16
conduction, anisotropic matrix, rotated triaxial ellipsoids	dilute	8.44 µs	48.4 µs	5.73×	9.7e-17
conduction, anisotropic matrix, rotated triaxial ellipsoids	mori tanaka	9.9 µs	56.1 µs	5.67×	3.5e-16
conduction, anisotropic matrix, rotated triaxial ellipsoids	self consistent	161 µs	288 µs	1.79×	4.6e-16
ALV: Maxwell matrix, elastic spheres	mori tanaka	107 µs	1.03 ms	9.66×	2.9e-16
ALV: Maxwell matrix, elastic spheres	self consistent	1.18 ms	6.58 ms	5.59×	2.1e-16
ALV: Maxwell matrix, oblate soft inclusions 1:1:0.1, isotropic average	mori tanaka	180 µs	1.05 ms	5.85×	7.9e-16
ALV: Maxwell matrix, oblate soft inclusions 1:1:0.1, isotropic average	self consistent	1.92 ms	6.69 ms	3.48×	6.7e-16

The differential scheme is not the same algorithm on both sides: Echoes takes 1000 explicit first-order steps, MeanFieldHomogenization integrates adaptively to a relative , hence a gap of a few , the error of the first-order steps.

The three-scale strength model: the same route, two stopping rules

The model of Quasi-brittle strength of cement paste and mortar computes a strength and its derivative with respect to the shear modulus of one family of hydrate needles. Both codes take the same route, described on the page of the model: the self-consistent foam is first converged with the needles averaged isotropically, whose Hill tensors are closed forms, and the 20 transversely isotropic families start from that estimate. They stop differently. Echoes takes a single step over the families and obtains the derivative there, with numerical Hill tensors to . MeanFieldHomogenization iterates the families to convergence, the derivative included through dual numbers, every Hill tensor to . The second row starts the families from the hydrate stiffness instead, the direct route kept to validate the first (Validating the angular discretization); both rows reach the same strength.

Case	Scheme				gap
three-scale strength model (20 needle families), value and derivative	SC + MT + MT, from the isotropic foam	2.7 s	2.29 s	0.85×	6.0e-04
three-scale strength model (20 needle families), value and derivative	SC + MT + MT, direct	4.51 s	2.29 s	0.51×	6.0e-04

What the numbers say

• Closed forms against Echoes' calls. On an isotropic medium both codes evaluate a closed form, and a Hill tensor costs tens of nanoseconds in Julia against tens of milliseconds through Echoes' Python API; the isotropic schemes inherit the factor.

• Same algorithm, comparable speed. Residues against residues on a triclinic medium, and the anisotropic conductivity, differ by a factor 2 to 2.5. The one case where Echoes is faster, :decuhr on the rotated triclinic ellipsoid against Echoes' residues, is a cubature against residues.

• The cubatures are the fastest general back-ends. :nestedquadgk agrees with NUMINT3D to and is 80 to 400 times faster.

• Derivatives by automatic differentiation cost no more than closed forms. ForwardDiff through hill_tensor beats Echoes' closed-form derivative on the isotropic cases.

• Aging viscoelasticity: 3.6 to 10 times faster on 11 instants, the self-consistent schemes in the cheapest algebra of the reference kernel.

Reproducing

The Julia scripts call the Echoes side through PythonCall, from Python modules next to them. Install Echoes in a Python environment of your choice, which must also have sympy (pip install sympy), activate it, then start Julia from the same shell: scripts/bench_echoes/echoes_python.jl, included by every script before PythonCall loads, takes the active python3 (or the interpreter named by ECHOES_PYTHON), checks that it imports echoes and sympy, and tells PythonCall to use it and CondaPkg to manage nothing. The explicit alternative is JULIA_CONDAPKG_BACKEND=Null and JULIA_PYTHONCALL_EXE=<path to that python>. CondaPkg then touches nothing, so the numpy version Echoes was built against (often numpy 1.x, numpy < 2) is kept; never let CondaPkg manage an environment where Echoes is installed, or pin that numpy in its CondaPkg.toml. Details in scripts/bench_echoes/README.md. From the root of the repository, alone on the machine:

julia --project=scripts/bench_echoes scripts/bench_echoes/benchmark.jl
julia --project=scripts/bench_echoes scripts/bench_echoes/benchmark_schemes_speed.jl

They write scripts/bench_echoes/figures/benchmark_kernels.json and benchmark_schemes_speed.json, which this page reads. The Echoes side runs on its own as well, with the cases defined once in these modules:

python3 -I scripts/bench_echoes/kernels_echoes.py
python3 -I scripts/bench_echoes/schemes_speed_echoes.py

with python3 the interpreter in which import echoes works.

Internal (Julia-only) regression benchmarking is separate: scripts/bench/ holds a gated suite with a committed baseline and a bitwise checksum gate, described in its own README.md.

Where to go next

Roadmap lists what remains to be implemented.