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Risk-averse Optimization in Random Materials: Algorithmic Advances and HPC Acceleration
Niklas Baumgarten, Marcel Koch, David Schneiderhan, Tim Schrader, Robin Weiß
TL;DR
The paper targets computationally costly risk-averse optimization for three-dimensional PDEs with random materials. It combines entropic-risk optimization, random-material sampling, and M++–Ginkgo hardware integration, finding improved estimator and solution quality with additional CPU resources, while identifying multilevel heterogeneity as a challenge for parallelization.
Problem
Three-dimensional stochastic risk-averse optimization is expensive, and unfavorable samples can disproportionately increase costs while random material structure affects system behavior.
Method
The paper combines entropic-risk PDE-constrained optimization, random-material sampling including deep Gaussian processing, and an M++–Ginkgo interface for accelerated linear algebra.
Results
Additional CPU resources reduce gradient-estimator errors, enable more samples and levels, improve solution quality, and show no saturation within the investigated scaling range.
Takeaways & Limitations
Sampling-based risk-averse optimization and GPU-compatible linear algebra are promising for high-dimensional random materials and three-dimensional PDE computations.
Takeaways & Limitations
The multilevel structure creates parallelization challenges because different levels require solving problems of vastly different sizes.
Abstract
from arXiv · showhide
We summarize our advances in the algorithmic development and hardware utilization for risk-averse optimization problems in random materials. This includes risk-averse optimization using the entropic risk measure, as well as recently developed sampling techniques for random materials, that are interoperable with the optimization framework. Furthermore, we discuss recent progress in the efficient utilization of modern hybrid hardware architectures for these methods to solve three-dimensional partial differential equations.
1 Introduction
The paper addresses risk-averse optimization of uncertain engineered systems whose material structure is incompletely known. It combines stochastic material sampling, entropic risk optimization, and GPU-oriented software integration for costly three-dimensional PDE computations.
- Engineered-system optimization must account for uncertainty in the spatial structure of materials and minimize a risk measure of state-target discrepancy.
- The approach uses the entropic risk measure with three-dimensional PDE constraints solved by finite element methods.
- Random-field models represent natural and synthetic materials while capturing microstructural features that influence structural integrity, longevity, fatigue, and macroscopic properties.
- Three-dimensional stochastic optimization is computationally expensive, and unfavorable samples can disproportionately amplify risk-averse objective costs.
- The paper covers risk-averse optimization, random-material sampling, and an M++–Ginkgo interface, with initial evaluations on modern hardware architectures.
2 Risk-averse Optimization
The paper formulates entropic-risk optimal control for three-dimensional elliptic PDEs with random material coefficients and solves it using multilevel stochastic gradient methods. Experiments show that increasing CPU resources supports more samples and levels, reduces estimator errors, and improves gradient-based solution quality without observed saturation in the tested range.
- Risk-averse formulation: The entropic risk parameter θ interpolates continuously between expectation-based and smooth worst-case optimization.
- Risk-averse formulation: The problem seeks an optimal control for a target state under a three-dimensional elliptic PDE with random material coefficients and control constraints.
- Random-material model: The random coefficient is sampled with a truncated Karhunen–Loève expansion using correlation length ρ=0.15 and a sinusoidal three-dimensional target state.
- Computational challenges: Three-dimensional finite-element PDE solves are computationally demanding, and optimization multiplies this cost across iterations and sampled realizations.
- Computational challenges: As θ increases, more samples are required to capture rare unfavorable outcomes that dominate the entropic-risk objective.
- Numerical experiments: For θ=40, increasing CPUs from P=64 to P=2048 enables more samples and, for P≥512, additional discretization levels.
- Algorithm: MLMC adaptively selects samples and discretization levels, while adjoint solves provide sample-wise gradient estimates for the iteration scheme.
- Numerical experiments: More resources decrease gradient-estimator RMS error, permit more optimization iterations, and yield smaller final gradient norms without observed CPU-scaling saturation.
3 SPDE-Based Sampling of Random Materials
The paper develops Gaussian random field sampling methods for random materials, extending them to non-stationary, anisotropic, and multiphase structures. These sampled fields support material constructions ranging from two-phase porous media to ordered and competitive multiphase realizations.
- Gaussian random fields: Matérn-covariance Gaussian random fields provide an alternative to Karhunen–Loève expansions for modeling random input data.Their parameters control field smoothness, correlation length, and variance scaling.
- SPDE-based sampling: SPDE-based sampling generates Gaussian random fields from stationary solutions driven by Gaussian white noise.The operator determines correlation structure, while the exponent controls smoothness and can support isotropic or non-stationary fields.
- SPDE-based sampling: Finite-element discretization on bounded domains, together with Dirichlet–Neumann Averaging, mitigates boundary artifacts from SPDE truncation.The resulting PDE solves use linear finite elements, multigrid preconditioning, and Krylov solvers, with GPU acceleration available.
- Material construction: Thresholding a Gaussian random field produces two-phase porous media with prescribed constituent volume fractions.Each point is assigned to either an inclusion phase or a matrix phase.
- Material construction: Anisotropic extensions use a symmetric positive-definite tensor to control correlation lengths along rotated principal directions.The same framework extends ordered multiphase materials by assigning volume fractions through 𝝓 = (𝜙1, . . . , 𝜙H).
- Material construction: Deep Gaussian processing creates non-stationary fields by conditioning a new realization on an initial field, while competitive sampling allows every phase to touch every other phase.The paper illustrates these structures for multiphase materials, including three- and ten-phase realizations.
4 HPC Acceleration
The paper accelerates FEM-based PDE solves by integrating M++ with Ginkgo, reconciling overlapping MPI data with globally indexed distributed linear algebra and supporting GPU backends. Initial benchmarks show strong scaling, a 15% CPU translation loss, and an 11% GPU speedup for the larger problem.
- 4 HPC Acceleration: FEM discretizations reduce the PDE, adjoint, and sampling solves to sparse linear systems whose repeated solution dominates optimization and uncertainty-quantification cost.The finite element coefficient vector is solved against a matrix incorporating material coefficients and a load vector capturing external forces.
- 4 HPC Acceleration: M++ can offload distributed linear algebra to CUDA, HIP, and SYCL through Ginkgo’s configurable solvers and preconditioners.The GinkgoSolver remains usable for PDEs in M++ by selecting an executor backend and configuring Ginkgo in JSON or code.
- 4.1 Interface design: The M++–Ginkgo interface maps overlapping, locally indexed vectors and matrices to globally indexed distributed vectors and CSR matrices.Owned and non-owned nodal points are separated using stored ownership information to establish unified global indexing.
- 4 HPC Acceleration: The interface addresses heterogeneous MPI and GPU memory models, including configurations where multiple MPI ranks share one GPU and GPU scheduling handles oversubscription.A single GPU may handle the workload of multiple CPU ranks, while alternative CPU/GPU distribution schemes remain future work.
- 4.2 Performance Evaluation and Maintenance: Strong scaling held for the comparatively small benchmarks, while one GPU performed similarly to two GPUs, suggesting incomplete GPU utilization.The benchmarks used 58 cells with 256 samples and 68 cells with 64 samples on up to 64 MPI ranks.
- 4.2 Performance Evaluation and Maintenance: 15% CPU performance loss from the translation layer was outweighed by an 11% GPU speedup over native CPU solvers for the larger problem.The reported CPU comparison used Jacobi-preconditioned conjugate gradients through OpenMP versus the same algorithm in M++ via MPI.
- 4.2 Performance Evaluation and Maintenance: Preliminary tests found comparable performance between M++ geometrical multigrid-preconditioned CG and Ginkgo algebraic multigrid on GPUs.Ongoing work targets repartitioning multiple MPI ranks onto a GPU-assigned rank and reducing intermediate interface computation.
5 Outlook and Conclusion
The paper reports advances in risk-averse optimization, random-material sampling, and an M++–Ginkgo interface for accelerated linear algebra. Future work will expand GPU use and applications to hyperbolic PDEs, while multilevel problems motivate hybrid and asynchronous CPU/GPU approaches.
- 5 Outlook and Conclusion: The work combines risk-averse optimization, SPDE and DNA random-material sampling, and an M++–Ginkgo interface for accelerated linear algebra.Experiments used NHR@KIT’s HoreKa systems.
- 5 Outlook and Conclusion: The authors plan to expand GPU capabilities and develop applications involving hyperbolic PDE systems with discontinuous Galerkin discretizations.
- 5 Outlook and Conclusion: Multilevel problems contain vastly different problem sizes, motivating hybrid and asynchronous approaches that could assign lower levels to GPUs and higher levels to CPUs.The multilevel structure reduces algorithmic complexity but challenges native M++ parallelization and the Ginkgo interface.