An optional GPU backend recorded an 11% speedup over native CPU solvers in the larger of two numerical benchmarks reported in an arXiv preprint on risk-averse optimization in random materials. The result comes from a computational workflow for modeling and optimization, not from a physical-material trial.
The same comparison showed that Ginkgo/OpenMP on CPUs incurred around a 15% performance loss versus the same algorithm in M++/MPI. The figures therefore describe these tested software paths and problems, rather than a universal GPU advantage.
The model behind the numbers
At its core, the work asks how to choose controls for a three-dimensional system described by partial differential equations—equations that represent how the modeled system varies across space—when the material properties are random and risk matters. The framework uses finite elements, multilevel estimators and multilevel stochastic gradient descent.
Random inputs are generated as Gaussian random fields using an SPDE approach. Dirichlet–Neumann Averaging is used to mitigate artifacts at the boundary when the modeled domain is truncated. In the numerical PDE tests, the random coefficient is sampled with a truncated Karhunen–Loève expansion.
The paper also describes sampling extensions for non-stationary and anisotropic fields, along with multiphase, level-cut and deep Gaussian processing models. They are presented as options within the random-material sampling framework.
The CPU comparison
For its CPU-scaling experiment, the study fixed the risk-aversion parameter at 40 and varied CPU count from 64 to 2,048. Each configuration ran for approximately four hours of wall-clock time.
More CPU resources were associated with more computed samples. At 512 CPUs or above, the configurations also used additional levels in the multilevel Monte Carlo hierarchy. Reported gradient-estimator error estimates decreased, and all configurations converged to the same solution.
Within the investigated range, no saturation was observed. The estimated convergence rate was 0.26 when measured against CPU count and 0.31 against wall-clock time. Both values are estimates, and the supplied analysis reports no uncertainty intervals.
What the GPU test measured
M++ was integrated with Ginkgo as an optional GPU-accelerated linear-algebra backend. The interface supports CUDA, HIP and SYCL, with data translation between the packages’ indexing formats.
The first SPDE/DNA benchmark used 58 cells, 256 samples, 2,048 linear solves and eight boundary conditions in the 3D average. A second used 68 cells, 64 samples and 64 MPI ranks. These were comparatively small benchmark configurations.
On those tests, the interface and targeted backends showed strong scaling. One GPU performed similarly to two GPUs, while two GPUs limited oversubscription to 32-fold. That pattern suggested incomplete GPU utilization.
Preliminary tests of the state and adjoint calculations found comparable performance on GPUs between M++’s geometrical multigrid-preconditioned conjugate-gradient method and Ginkgo’s algebraic multigrid preconditioner. The configuration and interface were still being optimized.
A result tied to its test design
The manuscript is an arXiv preprint, version 1, dated 26 Aug 2026. Its evidence is computational, covering random-field sampling, three-dimensional PDE-constrained optimization and CPU/GPU benchmark comparisons. The reported results therefore concern the tested algorithms and hardware configurations.
The CPU result—no saturation in the investigated range—does not establish what would happen beyond that range. The GPU comparison does not show that an accelerator will be faster for every problem size or solver choice. Performance is bounded by the reported solver, hardware, problem-size and oversubscription configurations.
The reported computing period used 684,717 CPU hours and 133 GPU hours, with no breakdown by experiment provided in the supplied analysis. The authors acknowledge HoreKa computing time through NHR@KIT, backed jointly by federal and Baden-Württemberg support, with partial DFG funding for HoreKa.
The paper states that all collected performance data are available and cites a Zenodo record. The supplied description does not further detail the data contents or access conditions.
Paper data and sources
Original title: Risk-averse Optimization in Random Materials: Algorithmic Advances and HPC Acceleration
Authors: Niklas Baumgarten, Marcel Koch, David Schneiderhan et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text