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Stochastic BDDC algorithms

Xuemin Tu, Jinjin Zhang

TL;DR

This paper addresses fast solvers for sampling-based SPDEs with random diffusion by developing stochastic BDDC preconditioners. It employs offline construction of PC surrogates through local Karhunen-Loève expansions and Polynomial Chaos, enabling fast per-sample online preconditioning via a stochastic BDDC framework that avoids forming the full global Schur complement. Theoretical results show that the stochastic preconditioners achieve conditioning comparable to exact BDDC and outperform mean-value baselines as the stochastic dimensions and PC degrees grow. Numerical experiments confirm strong performance and scalability across different sampling methods (SG/SC) and sample sizes, including cases with inexact Schur complements. The work provides a practical route to robust, scalable solvers for SPDEs with high-contrast or high-dimensional uncertainty, and suggests extensions to multi-level Monte Carlo and adaptive primal constraints.

Abstract

Stochastic balancing domain decomposition by constraints (BDDC) algorithms are developed and analyzed for the sampling of the solutions of linear stochastic elliptic equations with random coefficients. Different from the deterministic BDDC algorithms, the stochastic BDDC algorithms have online and offline stages. At the offline stage, the Polynomial Chaos (PC) expansions of different components of the BDDC algorithms are constructed based on the subdomain local parametrization of the stochastic coefficients. During the online stage, the sample-dependent BDDC algorithm can be implemented with a small cost. Under some assumptions, the condition number of the stochastic BDDC preconditioned operator is estimated. Numerical experiments confirm the theory and show that the stochastic BDDC algorithm outperforms the BDDC preconditioner constructed using the mean value of the stochastic coefficients.

Stochastic BDDC algorithms

TL;DR

This paper addresses fast solvers for sampling-based SPDEs with random diffusion by developing stochastic BDDC preconditioners. It employs offline construction of PC surrogates through local Karhunen-Loève expansions and Polynomial Chaos, enabling fast per-sample online preconditioning via a stochastic BDDC framework that avoids forming the full global Schur complement. Theoretical results show that the stochastic preconditioners achieve conditioning comparable to exact BDDC and outperform mean-value baselines as the stochastic dimensions and PC degrees grow. Numerical experiments confirm strong performance and scalability across different sampling methods (SG/SC) and sample sizes, including cases with inexact Schur complements. The work provides a practical route to robust, scalable solvers for SPDEs with high-contrast or high-dimensional uncertainty, and suggests extensions to multi-level Monte Carlo and adaptive primal constraints.

Abstract

Stochastic balancing domain decomposition by constraints (BDDC) algorithms are developed and analyzed for the sampling of the solutions of linear stochastic elliptic equations with random coefficients. Different from the deterministic BDDC algorithms, the stochastic BDDC algorithms have online and offline stages. At the offline stage, the Polynomial Chaos (PC) expansions of different components of the BDDC algorithms are constructed based on the subdomain local parametrization of the stochastic coefficients. During the online stage, the sample-dependent BDDC algorithm can be implemented with a small cost. Under some assumptions, the condition number of the stochastic BDDC preconditioned operator is estimated. Numerical experiments confirm the theory and show that the stochastic BDDC algorithm outperforms the BDDC preconditioner constructed using the mean value of the stochastic coefficients.

Paper Structure

This paper contains 10 sections, 9 theorems, 132 equations, 2 figures, 13 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A stochastic elliptic equation and a finite element discretization
  3. Domain decomposition and deterministic BDDC algorithms
  4. Stochastic BDDC algorithms
  5. Subdomain local KL expansions and PC approximations
  6. Stochastic BDDC Preconditioner
  7. SG
  8. SC
  9. Analysis of the stochastic BDDC algorithm
  10. Numerical Experiments

Key Result

Lemma 5.2

We have Recall $A^{(i)}_{L}$ is any subdomain local matrices constructed using $a_{ML}$ and $A^{{(i)}}_{L,PC_d}$ is its PC approximation, defined in (equation:pc) and based on $a_{M_{KL}}^{(i)}$.

Figures (2)

  • Figure 1: The CG iterations for $100$ samples with $N_s=256$, $\frac{H}{h}=8$, $N_{KL}=1$, $d=4$, $l=1$, $\sigma^2=0.5$..
  • Figure 2: The $\|S_{\Pi}-S_{\Pi,PC_d}\|_F$ for the mean of $100$ samples with $N_s=256$, $\frac{H}{h}=8$, $l=0.1$, $\sigma^2=0.5$. The top figure is for $N_{KL}=2,3,4$ and $d=4$ (corresponding to the last row in Table \ref{['Tab:exL1-1']}); the bottom figure is for $d=3,4,5$ and $N_{KL}=3$ (corresponding to the last row in Table \ref{['Tab:exLp-3']}).

Theorems & Definitions (20)

  • Remark 4.1
  • Remark 4.2
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • proof
  • Lemma 5.5
  • proof
  • ...and 10 more