Source-linked AI summary
Optimized Multilevel Sampling Methods under Resource Constraints
Niklas Baumgarten
TL;DR
Limited memory and processing power constrain numerical precision and complicate scaling for PDE-based UQ and OC. The paper develops budgeted MLMC and MLSGD methods that allocate distributed computational resources across multilevel samples and coupled PDE solves. The methods operate within these constraints and, for MLSGD, achieve comparable accuracy 18× faster than standard batched SGD while reducing error fivefold at equal cost.
Problem
Limited memory and processing power restrict the precision of numerical results for PDE-based uncertainty quantification and optimal control.
Method
The paper develops budgeted MLMC for full-field estimates and MLSGD for optimal control, using resource-aware knapsack allocation and MLMC gradient estimates.
Results
MLSGD achieves comparable accuracy 18× faster than standard batched SGD and reduces error by a factor of 5 at the same cost.
Takeaways & Limitations
Multilevel sampling methods can support full-field UQ and PDE-constrained optimal control under fixed computational budgets.
Takeaways & Limitations
The optimal-control setting considers a time-independent elliptic diffusion equation with log-normal coefficients and homogeneous Dirichlet boundary conditions.
Abstract
from arXiv · showhide
We present recent developments in multilevel sampling methods under resource constraints. Over the past 15 years, multilevel methods have become widely used for uncertainty quantification. However, scaling them to high-dimensional problems and high-performance computing (HPC) environments remains challenging. In this work, we discuss two algorithms designed to address these issues: the budgeted Multilevel Monte Carlo (MLMC) method and the Multilevel Stochastic Gradient Descent (MLSGD) method. We demonstrate their effectiveness on HPC systems under consideration of the available computational resources for applications in forward uncertainty quantification (UQ) and optimal control (OC) under uncertainty.
1 Introduction
Resource constraints make high-dimensional UQ and OC computationally challenging, motivating two multilevel algorithms that optimize memory and CPU-time usage. The budgeted MLMC method targets full-field PDE estimates, while MLSGD finds optimal controls under uncertainty.
- Limited memory and processing power restrict numerical precision for PDE-based uncertainty quantification and optimal control.
- The budgeted Multilevel Monte Carlo method computes full-field estimates of PDE systems under uncertainty within imposed computational constraints.Such estimates include mean fields for applications including weather forecasting, climate simulation, and materials science.
- The Multilevel Stochastic Gradient Descent method finds optimal controls for PDE systems under uncertainty using full-field MLMC estimates to approximate optimization gradients.
- Both algorithms use distributed-memory multiindex finite-element meshes and currying to approximate solutions of multiple coupled PDEs.The coupled stages include Gaussian Random Field sampling, subsurface diffusion, and mass transport, with each stage supplying the next stage’s input.
- The algorithms are implemented in M++ and applied to UQ and OC within an HPC project spanning several scientific domains.
- The approach formulates resource allocation as knapsack problems that minimize error under fixed CPU-time and memory budgets.Distributed dynamic programming allocates resources across resolution levels, samples, and coupled PDEs.
2 Problem Statements
The paper formulates forward uncertainty quantification and PDE-constrained optimal control under uncertainty, then addresses their computational constraints with multilevel estimators and resource-aware optimization. The problems involve coupled uncertain PDE systems, approximate numerical solutions, and allocation of limited CPU time and memory across levels, samples, and coupled equations.
- Forward Uncertainty Quantification: Forward UQ estimates statistical moments such as the mean field of a PDE solution with randomly distributed input data.The solution depends on probability outcomes, spatial location, and possibly time.
- Forward Uncertainty Quantification: The forward application computes mass transport through a composition of Gaussian Random Field sampling, subsurface diffusion, and hyperbolic transport.Each stage’s output supplies the input for the next stage.
- Optimal Control under Uncertainty: The optimal-control problem seeks a control whose uncertain PDE state is as close as possible in expectation to a target state.The control problem uses an uncertain PDE operator and a cost factor.
- Optimal Control under Uncertainty: Batched stochastic gradient descent updates the control iteratively, while MLSGD replaces its Monte Carlo gradient estimate with a multilevel Monte Carlo estimate.The gradient is approximated by solving the adjoint system for samples combined into an estimator.
- Resource Constraints: Both problem classes require minimizing approximation error under fixed CPU-time and memory budgets.The resource-allocation problem distributes resources across resolution levels, samples, and coupled PDEs using distributed dynamic programming and knapsack formulations.
- Resource Constraints: The multilevel methods use distributed multiindex finite-element meshes that combine sample and domain parallelization across discretization levels.The data structure favors sample parallelization on lower levels and domain parallelization on higher levels, while limiting persistent memory to domain-parallel components.
3 Numerical Experiments
The experiments evaluate budget-constrained multilevel methods for full-field estimation and stochastic optimization on HPC systems. MLSGD outperforms batched SGD and improves solution quality as resources increase, although parallel efficiency diminishes.
- Experimental setting: The experiments ran on HoreKa using 64–1,024 processing units, while preliminary three-dimensional tests reached 16,384 units.The reported section focuses on two-dimensional problems.
- Resource constraints: The achievable error is bounded above by the CPU-time budget and below by the memory constraint.These constraints motivate the associated multilevel knapsack problems.
- Full-field MLMC: Full-field estimation adds no extra memory or CPU-time cost compared with scalar quantities of interest under the tested configuration.At P = 256 CPUs, full-field updates remained nearly identical in memory usage and CPU time to the scalar case while respecting imposed limits.
- MLSGD versus SGD: MLSGD achieves comparable accuracy 18× faster than batched SGD and reduces error by a factor of 5 at the same cost.The comparison used P = 64 CPUs and a one-hour computational budget.
- MLSGD versus SGD: MLSGD attains a convergence rate of δ≈0.5 versus δ≈0.37 for SGD.The result is consistent with the theoretical predictions cited by the authors.
- Resource scaling: Increasing resources improves MLSGD solution quality, but the smaller improvement from P = 256 to P = 1,024 indicates diminishing parallel efficiency.At P = 1,024, the method computes more samples and adds an extra level; the serial fraction λ_p is identified as the cause.
4 Outlook
The outlook identifies automated performance benchmarking and GPU implementation as directions for extending the multilevel methods. The authors plan to use Ginkgo for GPU-oriented high-performance computing.
- Benchmarking: The orange line in Figure 8 establishes a baseline for evaluating future MLSGD developments and M++ performance improvements.Future measurements can be automated through the cited software-engineering workflow.
- GPU extension: The planned GPU extension will use Ginkgo, a linear algebra framework for high-performance computing.Initial work includes designing simple test cases and transferring the implementation toward GPUs.