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A parallel solver for random input problems via Karhunen-Loève expansion and diagonalized coarse grid correction

Dou Dai, Qiuqi Li, Huailing Song

TL;DR

The paper tackles slow convergence of parallel-in-time parareal in stochastic settings by introducing a hybrid KLE-CGC approach that first reduces input dimensionality via Karhunen-Loève expansion and then builds a gPC-driven surrogate to produce high-quality initial guesses for coarse-grid corrections. It proves that the method retains the standard parareal convergence rate while significantly reducing iteration counts, validated through three nonlinear PDEs with random inputs. The diagonalization-based CGC enables fully parallel coarse-grid corrections, and the KL-gPC surrogate further improves initial-value accuracy, leading to substantial parallel efficiency gains. The results demonstrate robust performance across advection-diffusion, Burgers, and Allen-Cahn equations, including energy-dissipative behavior in the Allen-Cahn case, highlighting the method's practical impact for rapid, high-accuracy, parametric PDE simulations.

Abstract

This paper is dedicated to enhancing the computational efficiency of traditional parallel-in-time methods for solving stochastic initial-value problems. The standard parareal algorithm often suffers from slow convergence when applied to problems with stochastic inputs, primarily due to the poor quality of the initial guess. To address this issue, we propose a hybrid parallel algorithm, termed KLE-CGC, which integrates the Karhunen-Loève (KL) expansion with the coarse grid correction (CGC). The method first employs the KL expansion to achieve a low-dimensional parameterization of high-dimensional stochastic parameter fields. Subsequently, a generalized Polynomial Chaos (gPC) spectral surrogate model is constructed to enable rapid prediction of the solution field. Utilizing this prediction as the initial value significantly improves the initial accuracy for the parareal iterations. A rigorous convergence analysis is provided, establishing that the proposed framework retains the same theoretical convergence rate as the standard parareal algorithm. Numerical experiments demonstrate that KLE-CGC maintains the same convergence order as the original algorithm while substantially reducing the number of iterations and improving parallel scalability.

A parallel solver for random input problems via Karhunen-Loève expansion and diagonalized coarse grid correction

TL;DR

The paper tackles slow convergence of parallel-in-time parareal in stochastic settings by introducing a hybrid KLE-CGC approach that first reduces input dimensionality via Karhunen-Loève expansion and then builds a gPC-driven surrogate to produce high-quality initial guesses for coarse-grid corrections. It proves that the method retains the standard parareal convergence rate while significantly reducing iteration counts, validated through three nonlinear PDEs with random inputs. The diagonalization-based CGC enables fully parallel coarse-grid corrections, and the KL-gPC surrogate further improves initial-value accuracy, leading to substantial parallel efficiency gains. The results demonstrate robust performance across advection-diffusion, Burgers, and Allen-Cahn equations, including energy-dissipative behavior in the Allen-Cahn case, highlighting the method's practical impact for rapid, high-accuracy, parametric PDE simulations.

Abstract

This paper is dedicated to enhancing the computational efficiency of traditional parallel-in-time methods for solving stochastic initial-value problems. The standard parareal algorithm often suffers from slow convergence when applied to problems with stochastic inputs, primarily due to the poor quality of the initial guess. To address this issue, we propose a hybrid parallel algorithm, termed KLE-CGC, which integrates the Karhunen-Loève (KL) expansion with the coarse grid correction (CGC). The method first employs the KL expansion to achieve a low-dimensional parameterization of high-dimensional stochastic parameter fields. Subsequently, a generalized Polynomial Chaos (gPC) spectral surrogate model is constructed to enable rapid prediction of the solution field. Utilizing this prediction as the initial value significantly improves the initial accuracy for the parareal iterations. A rigorous convergence analysis is provided, establishing that the proposed framework retains the same theoretical convergence rate as the standard parareal algorithm. Numerical experiments demonstrate that KLE-CGC maintains the same convergence order as the original algorithm while substantially reducing the number of iterations and improving parallel scalability.

Paper Structure

This paper contains 16 sections, 6 theorems, 80 equations, 4 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Karhunen-Loève expansion
  4. Generalized polynomial chaos formulation
  5. Karhunen-Loève expansion with parallel coarse-grid correction method
  6. The parareal algorithm
  7. Diagonalization-based parallel CGC
  8. The implementation of the KLE-PCGC method
  9. Convergence analysis
  10. The linear systems
  11. The nonlinear systems
  12. Numerical results
  13. Advection-diffusion equation
  14. Burgers Equation
  15. Allen-Cahn Equation
  16. ...and 1 more sections

Key Result

Lemma 4.1

Let $\mathcal{F}$ and $\mathcal{G}$ be two one-step numerical methods with stability functions $\mathcal{R}_f(z)$ and $\mathcal{R}_g(z)$, which are, respectively, applied to the ODE system eq:ODE and eq:WODE with small step size $\Delta t$ and large step size $\Delta T$. Then, the error $\mathbf{e}^ where $\|\boldsymbol{\cdot}\|$ is an arbitrary norm and the matrices $\mathbf{G}$ and $\mathbf{F}$

Figures (4)

  • Figure 5.1: Comparison of the mean error corresponding to the parareal, parallel CGC and the KLE-PCGC algorithm.
  • Figure 5.2: The mean reference solution $u_N(\boldsymbol{\xi})$ (left), the KLE-PCGC solution $\mathrm{U}_N^k(\boldsymbol{\xi})$ (middle), and the Parareal solution $u_N^k(\boldsymbol{\xi})$ (right) at the final time $t=T$.
  • Figure 5.3: Comparison of the mean error between the parareal algorithm and the KLE-CGC algorithm (left) and the mean reference solution $u_N$, KLE-CGC solution $\mathrm{U}_N^k$, and parareal solution $u_N^k$ at the final time $t=T$ (right).
  • Figure 5.4: Comparison of the mean error (left) and the evolution of the mean energy $E(t)$ (right) between the parareal algorithm and the KLE-CGC algorithm.

Theorems & Definitions (13)

  • Remark 1
  • Lemma 4.1: general result deduced from M.J
  • Theorem 4.1
  • proof
  • Remark 2
  • Theorem 4.2
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • ...and 3 more