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Exploiting Inexact Computations in Multilevel Monte Carlo and Other Sampling Methods

Josef Martínek, Erin Carson, Robert Scheichl

TL;DR

The paper addresses the high energy cost of uncertainty quantification using multilevel sampling by enabling safe, inexact computations on coarser levels. It develops a theoretical framework and an adaptive MPML algorithm that allocates minimum per-level precision without extra cost, validated on an elliptic PDE with lognormal coefficients. The results show substantial practical gains: memory savings up to $3.5\times$ and FLOP-speedups around $1.5\times$, with potential energy-efficiency benefits for large-scale HPC. The approach is extendable to other multilevel techniques and motivates future work on energy-aware solvers and parallel implementations. Overall, the work demonstrates a viable pathway to combine numerical efficiency with rigorous UQ accuracy in high-performance settings.

Abstract

Multilevel sampling methods, such as multilevel and multifidelity Monte Carlo, multilevel stochastic collocation, or delayed acceptance Markov chain Monte Carlo, have become standard uncertainty quantification (UQ) tools for a wide class of forward and inverse problems. The underlying idea is to achieve faster convergence by leveraging a hierarchy of models, such as partial differential equation (PDE) or stochastic differential equation (SDE) discretisations with increasing accuracy. By optimally redistributing work among the levels, multilevel methods can achieve significant performance improvement compared to single level methods working with one high-fidelity model. Intuitively, approximate solutions on coarser levels can tolerate large computational error without affecting the overall accuracy. We show how this can be used in high-performance computing applications to obtain a significant performance gain. As a use case, we analyse the computational error in the standard multilevel Monte Carlo method and formulate an adaptive algorithm which determines a minimum required computational accuracy on each level of discretisation. We show two examples of how the inexactness can be converted into actual gains using an elliptic PDE with lognormal random coefficients. Using a low precision sparse direct solver combined with iterative refinement results in a simulated gain in memory references of up to $3.5\times$ compared to the reference double precision solver; while using a MINRES iterative solver, a practical speedup of up to $1.5\times$ in terms of FLOPs is achieved. These results provide a step in the direction of energy-aware scientific computing, with significant potential for energy savings.

Exploiting Inexact Computations in Multilevel Monte Carlo and Other Sampling Methods

TL;DR

The paper addresses the high energy cost of uncertainty quantification using multilevel sampling by enabling safe, inexact computations on coarser levels. It develops a theoretical framework and an adaptive MPML algorithm that allocates minimum per-level precision without extra cost, validated on an elliptic PDE with lognormal coefficients. The results show substantial practical gains: memory savings up to and FLOP-speedups around , with potential energy-efficiency benefits for large-scale HPC. The approach is extendable to other multilevel techniques and motivates future work on energy-aware solvers and parallel implementations. Overall, the work demonstrates a viable pathway to combine numerical efficiency with rigorous UQ accuracy in high-performance settings.

Abstract

Multilevel sampling methods, such as multilevel and multifidelity Monte Carlo, multilevel stochastic collocation, or delayed acceptance Markov chain Monte Carlo, have become standard uncertainty quantification (UQ) tools for a wide class of forward and inverse problems. The underlying idea is to achieve faster convergence by leveraging a hierarchy of models, such as partial differential equation (PDE) or stochastic differential equation (SDE) discretisations with increasing accuracy. By optimally redistributing work among the levels, multilevel methods can achieve significant performance improvement compared to single level methods working with one high-fidelity model. Intuitively, approximate solutions on coarser levels can tolerate large computational error without affecting the overall accuracy. We show how this can be used in high-performance computing applications to obtain a significant performance gain. As a use case, we analyse the computational error in the standard multilevel Monte Carlo method and formulate an adaptive algorithm which determines a minimum required computational accuracy on each level of discretisation. We show two examples of how the inexactness can be converted into actual gains using an elliptic PDE with lognormal random coefficients. Using a low precision sparse direct solver combined with iterative refinement results in a simulated gain in memory references of up to compared to the reference double precision solver; while using a MINRES iterative solver, a practical speedup of up to in terms of FLOPs is achieved. These results provide a step in the direction of energy-aware scientific computing, with significant potential for energy savings.

Paper Structure

This paper contains 22 sections, 8 theorems, 43 equations, 9 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Inexact computations and energy-efficient computing
  3. Case study: multilevel Monte Carlo
  4. Main contributions and outline of the paper
  5. Inexact computations in linear solvers
  6. Floating point arithmetic
  7. Krylov subspace methods
  8. Iterative refinement
  9. Finite element methods for PDEs with random data
  10. An elliptic PDE with random data and its FE solution
  11. Approximate FE solutions using inexact solvers
  12. Inexact computations in multilevel Monte Carlo
  13. Standard multilevel Monte Carlo
  14. Computational error in multilevel Monte Carlo
  15. Adaptive MPML algorithm
  16. ...and 7 more sections

Key Result

Lemma 3.2

Let $u_h$ be the discrete solution of Problem pr_AVP_rnd and let $\widehat{u}_h$ be the approximation of $u_h$ defined above. Then where $C$ is independent of $h$, $u$, and $\omega$ and $\|{\cdot}\|_{2}$ denotes the Euclidean norm on $\mathbb{R}^n$.

Figures (9)

  • Figure 1: Computational error in the variance of the difference estimators $\mathop{\mathrm{\textrm{var}}}\nolimits[\Tilde{Y}_{l}]$ (left) and in the bias $|\mathbb{E}[\Tilde{Q}_{l}-Q]|$ (right), plotted against effective precision. Discretisation level $l$ corresponds to $h_l=1/8\times2^{-l}$. To achieve an effective precision $\delta_l$, relative residuals produced by MINRES are monitored.
  • Figure 2: Estimated MSE with approximate $95\%$ confidence intervals for the MPMC estimator when compared to the reference MLMC estimator for various target tolerances. As their linear solver, both estimators use MINRES; MPML uses an adaptively chosen stopping criterion.
  • Figure 3: Total cost gain in terms of FLOPs for various tolerances using adaptive MPML compared to the standard adaptive MLMC estimator. As their linear solver, both estimators use MINRES; MPML uses an adaptively chosen stopping criterion.
  • Figure 4: Average number of samples on each level for adaptive MLMC and adaptive MPML for $\text{TOL}^2=10^{-6}$ (with approximate $95\%$ confidence intervals).
  • Figure 5: MSE of the MPML algorithm for various values of $k_p$ compared to the reference MSE of MLMC (with approximate $95\%$ confidence intervals).
  • ...and 4 more figures

Theorems & Definitions (16)

  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Theorem 4.1: Computational error in MPML
  • proof
  • Corollary 4.2
  • Remark 4.3: Asymptotic MPML cost
  • Remark 4.4: Balancing the computational and model error
  • ...and 6 more