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Reduced basis techniques for stochastic problems

Sébastien Boyaval, Claude Le Bris, Tony Lelièvre, Yvon Maday, Ngoc Cuong Nguyen, Anthony T. Patera

arXiv:1004.0357v1math.NAcs.NA

TL;DR

Reduced-basis methods target repeated evaluations in stochastic problems, trading costly offline work for cheaper online computation. The paper reviews two approaches, including a reported 50-fold reduction in computation time for one random-coefficient problem.

  • Problem

    Many-query stochastic analyses need repeated evaluations, while reduced-basis efficiency trades costly offline preprocessing for cheaper online evaluations.

    Source: §1 · p. 2, §2.1 · p. 6

  • Method

    The paper reviews reduced-basis discretization for random-coefficient PDEs and a statistical variant using correlated ensembles and variance reduction for parametrized SDEs.

    Source: §1 · pp. 2–3

  • Results

    For δ = 0.2 and K = 45, reports total computational time fifty times as short as direct finite-element computations; reports good variance reduction for large online parameter variations.

    Source: §3.4 · p. 23, §4.3 · p. 28

  • Takeaways & Limitations

    The review illustrates reduced-basis methods in two stochastic settings: random-coefficient PDE discretization and variance reduction for parametrized SDEs.

    Source: §1 · pp. 2–3

    Reduced-basis efficiency depends on a many-query workload, because the expensive offline step is worthwhile only when many input values must be computed.

    Source: §2.1 · p. 6

Abstract

from arXiv · show

We report here on the recent application of a now classical general reduction technique, the Reduced-Basis approach initiated in [C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A. T. Patera, and G. Turinici. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods. Journal of Fluids Engineering, 124(1):7080, 2002.], to the specific context of differential equations with random coefficients. After an elementary presentation of the approach, we review two contributions of the authors: [S. Boyaval, C. Le Bris, Y. Maday, N.C. Nguyen, and A.T. Patera. A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin co-efficient. Computer Methods in Applied Mechanics and Engineering, 198(4144):3187-3206, 2009.], which presents the application of the RB approach for the discretization of a simple second order elliptic equation supplied with a random boundary condition, and [S. Boyaval and T. Leli\`evre, A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm with T. Leli\`evre, Commun. Math. Sci. 8, special Issue "Mathematical Issue on Complex Fluids" P. Zhang ed., to appear, 2010, ARXIV preprint arXiv:0906.3600], which uses a RB type approach to reduce the variance in the Monte-Carlo simulation of a stochastic differential equation. We conclude the review with some general comments and also discuss possible tracks for further research in the direction.

1 Introduction

The review applies certified reduced-basis methods to stochastic problems through two routes: a random-coefficient PDE and a variance-reduction method for stochastic differential equations [BL09]. It frames these applications within the reduced-basis trade-off between greater offline cost and cheaper repeated evaluations.

1 Introduction

  • Reduced-basis approximation and a posteriori error estimation target rapid, reliable input-output evaluation for systems with random parameters, including PDEs with random coefficients and stochastic differential equations.The many-query statistical-analysis setting offers an opportunity to represent a solution manifold with a low-dimensional space, trading increased offline cost for reduced online cost.Source: §1 · p. 2
  • Recent deterministic reduced-basis methods emphasized parameter sampling, rigorous output-error estimation, and separating offline from online computation.These developments supported application to broader parameter domains with greater certainty and error control.Source: §1 · p. 2
  • For random-coefficient PDEs, the first route maps realizations of the random solution field to deterministic parametrized PDE solutions and applies classical reduced-basis methods.Statistical information can then be obtained, for example, through Monte Carlo approximations; both the input and solution random fields must be approximated.Source: §1 · pp. 2–3
  • For stochastic differential equations, the second route reinterprets reduced-basis ideas statistically, using correlated ensembles, variance reduction, and fine/coarse ensembles for offline/online stages.The approach generates companion variables used as control variates to reduce variance in Monte Carlo quantities.Source: §1 · p. 3

2 An Initiation to Reduced-Basis Techniques

This section introduces Reduced Basis techniques, explaining how snapshot-based offline basis construction supports fast online approximations and error estimation for repeated parameterized PDE computations. It also reviews theoretical guarantees and extensions, providing background for later applications to random coefficients and variance reduction.

2 An Initiation to Reduced-Basis Techniques

  • The section deliberately presents the RB approach elementarily, leaving technicalities aside and directing familiar readers to the later application sections or pedagogic introductions.Source: §2 · p. 3

2.1 Outline of the Reduced Basis approach

  • RB targets many-query settings: it builds a small snapshot space offline, then computes reduced Galerkin approximations online and uses output-error estimators for greedy selection and certification.The reduced space is motivated by the expectation that the solution manifold can be approximated well in far fewer dimensions than the ambient discretization; efficiency depends on the number of queries justifying the offline cost.Source: §2.1 · pp. 5–6

2.2 Some more details on a simple case

  • An affine parameter dependence of the bilinear form is crucial to RB speedup, with offline cost scaling as N × O(N^k) and online cost as J × O(N^3), independent of the ambient dimension N.The parameter dependence must be identified or constructed in affine form, or approximated appropriately.Source: §2.2 · pp. 7–8

2.2.1 A posteriori estimator.

  • The output-error estimator is based on a residual-based bound for the state error, and its sharpness depends on the quality of the coercivity lower bound αLB(µ).The stated construction assumes a lower bound for the coercivity constant over the parameter range.Source: §2.2.1 · pp. 9–10

2.2.2 Offline stage and greedy algorithm.

  • The described greedy procedure starts by choosing µ1 randomly, computing uN(µ1), and setting the initial reduced space to Span(uN(µ1)).Source: §2.2.2 · p. 10
  • The trial sample Λtrial is a delicate choice: random samples may miss the accuracy target, and no systematic way to construct a good initial sample or quantify its quality is known.The RB approach therefore checks and can improve the sample a posteriori; adaptive training samples may improve it offline [HO08b].Source: §2.2.2 · pp. 10–11

2.2.3 Online stage: fast computations including a posteriori estimators.

  • For J ≫ 1 online queries without basis enrichment, RB computation costs N × O(N^k) + (J + O(|Λtrial|)) × O(N^3), versus J × O(N^k) directly.The authors describe the resulting savings as tremendous as J tends to infinity. In practice, the reduced system matrix may be nearly singular, so they recommend an orthonormal basis for stable computation.Source: §2.2.3 · pp. 12–13

2.3.1 Some elements of theory.

  • Theoretical analysis gives exponential accuracy in reduced-basis dimension N for the simple problem, assuming A1 is non-negative.The result is an a priori bound; the stated passage does not provide its numerical constants.Source: §2.3.1 · p. 13
  • The practical greedy space differs from the space used in the exponential-accuracy result, and numerical experiments suggest it may be less effective.Kolmogorov width and approximation theory offer tools for analyzing the greedy algorithm’s construction of spaces approximating the solution family.Source: §2.3.1 · pp. 13–14
  • Extensions cover affine variational formulations, non-coercive problems using inf-sup stability, and non-compliant problems using an adjoint reduced basis.For non-compliant problems, a primal–dual error bound is quadratic in the residual and may be much smaller than a simpler linear bound.Source: §2.3.2 · pp. 15–16
  • Non-affine parameter dependence can be handled with suitable affine approximations, including interpolation at sequentially selected magic points.Nonlinear elliptic extensions have been reported, but require efficient a posteriori estimators and lack a general theory; semi-discretized parabolic problems use RB in parameters and POD in time.Source: §2.3.2 · pp. 16–17

3 RB Approach for Boundary Value Problems with Stochastic Coefficients

This section reviews the first reduced-basis application to stochastic coefficients, adapting the method to a random-Robin heat-conduction problem and its statistical outputs.

3.1 Setting of the problem

  • The heat-sink model randomizes the convective Robin coefficient and targets E(S) and Var(S); Monte Carlo requires repeated PDE solves, creating a many-query workload.Source: §3.1 · pp. 18–19

3.2 Discretization of the problem

  • The Robin coefficient is KL-truncated into an affine parameterization; positivity at each truncation order preserves well-posedness, and online error estimation accounts for truncation.Source: §3.2 · pp. 19–20

3.3 Reduced-Basis ingredients

  • The global bound combines independent RB and KL-truncation errors; statistical estimators require M parameter queries and cannot drive offline construction, while truncation bounds can justify reducing K.Source: §3.3 · pp. 20–21

3.4 Numerical results

  • At K=20, global Monte Carlo output error decays exponentially as the reduced-basis size N grows from 1 to 14.Figure 2 displays RB-approximation and KL-truncation error bounds across basis sizes and truncation orders.Source: §3.4 · pp. 21, 23, Figure 2 · p. 22
  • At fixed K, truncation error soon dominates as N grows, producing a plateau beyond Ncrit(K), which increases with K.The reported global error combines RB-approximation and KL-truncation contributions.Source: §3.3 · p. 20, §3.4 · p. 23
  • At δ=0.2, the method treated K=45 parameters in total computational time fifty times shorter than direct finite-element computations.Source: §3.4 · p. 23

4 Variance Reduction using an RB approach

The paper adapts reduced-basis ideas to Monte Carlo estimation for parameterized stochastic differential equations, using reduced-basis approximations as control variates. Its offline/online algorithms target repeated expectation estimates, and the reported tests show substantial variance reduction.

4.1 Setting of the problem

  • Repeated expectation estimates E(Zλ) across parameter values are the target; Monte Carlo error depends on variance, while the optimal control variate is impractical because it requires the unknown expectation.The alternative PDE formulation is also impractical because solving it is at least as difficult as evaluating E(Zλ).Source: §4.1 · pp. 23, 25

4.2 Two algorithms for variance reduction by the RB approach

  • Offline, the algorithms select parameter snapshots and compute accurate expectations or PDE approximations; online, they choose a variance-minimizing linear combination of the resulting control variates.Algorithm 1 uses expectation-based approximations; Algorithm 2 uses approximations to the backward Kolmogorov solution, and its online linear solve scales as N^3 rather than N^2 for Algorithm 1.Source: §4.2 · pp. 26–27

4.3 Reduced-Basis ingredients

  • Both algorithms use offline/online stages, snapshots, and an inexpensive empirical-variance estimator; Algorithm 1 also extends to parameterized random variables when correlated samples can be generated.Using the same Brownian motions for outputs and control variates is a practical ingredient.Source: §4.2 · p. 26, §4.3 · p. 28
  • At least 10^2 and typically 10^4 variance reduction versus plain Monte Carlo; Algorithm 2 is more costly but more robust to broad parameter variation.Algorithm 2 appears less general and may be impractical for high-dimensional state spaces.Source: §4.3 · p. 28

4.4 Numerical Results

  • For FENE with b = 16, Algorithm 1 reduces variance by about 4 orders of magnitude; Algorithm 2 also reduces it, apparently less, while seeming slightly more robust on extrapolative tests.The figure plots the minimum, mean, and maximum variance-reduction statistic against reduced-basis size N for both algorithms.Source: §4.4 · pp. 29–30, Figure 3 · p. 30
  • Although preliminary, the tests suggest repeated parameterized Monte Carlo estimates are a promising use of reduced bases; even coarse estimates may serve as starting points for refined approximations.The paper mentions variance reduction and preconditioning based on a coarse-grained reduced-basis model as possible next steps.Source: §4.4 · p. 30

5 Perspectives

The perspectives identify open methodological needs for stochastic reduced-basis methods and outline possible extensions. They focus on better error estimation, affine treatment of random inputs, Bayesian sampling, and a connection to functional quantization.

5.1 A posteriori estimation in the stochastic context

  • Stochastic reduced-basis methods still need accurate, fast error estimators; confidence intervals are reliable only asymptotically, and non-asymptotic bounds or alternative variance estimates merit study.The authors also suggest error bounds relative to a reference computed with many realizations.Source: §5.1 · p. 31

5.2 Affine decompositions and the stochastic context

  • Reduced-basis efficiency currently relies on affine parametrization; for random fields with small correlation lengths, Karhunen-Loeve truncations may converge too slowly, motivating other affine representations.Possible approaches include projection onto suitable bases or empirical interpolation.Source: §5.2 · pp. 31–32

5.3 Application to Bayesian statistics

  • Bayesian posterior sampling over many observation parameters could use Algorithm 1: build a reduced basis from selected parameter values offline, then reduce sampling variance for other values online.The proposed setting generates correlated samples across parameter values by weighting prior samples with the likelihood.Source: §5.3 · pp. 32–33

5.4 Relation to functional quantization

  • Functional quantization shares the offline/online structure of reduced-basis methods; the authors propose quantizing parameter-specific SDE solutions and combining precomputed solutions for new parameters.The cited quantization approach first quantizes Brownian motion and then solves ordinary differential equations.Source: §5.4 · p. 33
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