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Quasi-Monte Carlo for Bayesian design of experiment problems governed by parametric PDEs

Vesa Kaarnioja, Claudia Schillings

TL;DR

This work advances Bayesian optimal experimental design for PDE-governed inverse problems by establishing parametric regularity of the expected information gain integrand and rigorously analyzing quasi-Monte Carlo approaches. It develops both full-tensor and sparse-tensor QMC frameworks, deriving dimension-robust convergence rates under suitable regularity and weight assumptions, and demonstrates substantial practical gains through an elliptic PDE test with unknown diffusion coefficients. The combination of inner-outer regularity analysis and CFL-style tail control enables efficient nested integration for $EIG$, with sparse-tensor approaches recovering near-optimal rates in numerical experiments. Overall, the methodology offers a scalable, high-dimensional Bayesian design tool with strong theoretical guarantees and demonstrated applicability to complex PDE-based systems.

Abstract

This paper contributes to the study of optimal experimental design for Bayesian inverse problems governed by partial differential equations (PDEs). We derive estimates for the parametric regularity of multivariate double integration problems over high-dimensional parameter and data domains arising in Bayesian optimal design problems. We provide a detailed analysis for these double integration problems using two approaches: a full tensor product and a sparse tensor product combination of quasi-Monte Carlo (QMC) cubature rules over the parameter and data domains. Specifically, we show that the latter approach significantly improves the convergence rate, exhibiting performance comparable to that of QMC integration of a single high-dimensional integral. Furthermore, we numerically verify the predicted convergence rates for an elliptic PDE problem with an unknown diffusion coefficient in two spatial dimensions, offering empirical evidence supporting the theoretical results and highlighting practical applicability.

Quasi-Monte Carlo for Bayesian design of experiment problems governed by parametric PDEs

TL;DR

This work advances Bayesian optimal experimental design for PDE-governed inverse problems by establishing parametric regularity of the expected information gain integrand and rigorously analyzing quasi-Monte Carlo approaches. It develops both full-tensor and sparse-tensor QMC frameworks, deriving dimension-robust convergence rates under suitable regularity and weight assumptions, and demonstrates substantial practical gains through an elliptic PDE test with unknown diffusion coefficients. The combination of inner-outer regularity analysis and CFL-style tail control enables efficient nested integration for , with sparse-tensor approaches recovering near-optimal rates in numerical experiments. Overall, the methodology offers a scalable, high-dimensional Bayesian design tool with strong theoretical guarantees and demonstrated applicability to complex PDE-based systems.

Abstract

This paper contributes to the study of optimal experimental design for Bayesian inverse problems governed by partial differential equations (PDEs). We derive estimates for the parametric regularity of multivariate double integration problems over high-dimensional parameter and data domains arising in Bayesian optimal design problems. We provide a detailed analysis for these double integration problems using two approaches: a full tensor product and a sparse tensor product combination of quasi-Monte Carlo (QMC) cubature rules over the parameter and data domains. Specifically, we show that the latter approach significantly improves the convergence rate, exhibiting performance comparable to that of QMC integration of a single high-dimensional integral. Furthermore, we numerically verify the predicted convergence rates for an elliptic PDE problem with an unknown diffusion coefficient in two spatial dimensions, offering empirical evidence supporting the theoretical results and highlighting practical applicability.
Paper Structure (11 sections, 4 theorems, 50 equations)

This paper contains 11 sections, 4 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Literature overview
  3. Outline of the paper
  4. Notations and preliminaries
  5. Problem setting
  6. Quasi-Monte Carlo integration
  7. Model problem
  8. Decomposing the high-dimensional integral
  9. Parametric regularity
  10. Parametric regularity of the inner integrand
  11. Mixed regularity

Key Result

lemma thmcounterlemma

Let $F$ belong to the weighted unanchored Sobolev space over $[0,1]^s$ with weights $\boldsymbol{\gamma}=(\gamma_{\mathfrak u})_{\mathfrak u\subseteq\{1:s\}}$. An $s$-dimensional randomly shifted lattice rule with $n=2^m$ points, $m\geq 0$, can be constructed by a CBC algorithm such that, for $R$ in where R.M.S. error$\,:=\sqrt{\mathbb{E}_{\boldsymbol{\Delta}}[|I_{s}(F)-\overline{Q}_{n,R}(F)|^2]}$

Theorems & Definitions (6)

  • lemma thmcounterlemma: cf. kuonuyenssurvey
  • lemma thmcounterlemma
  • proof
  • remark thmcounterremark
  • lemma thmcounterlemma
  • lemma thmcounterlemma