[Paper Review] Fast eigenpairs computation with operator adapted wavelets and hierarchical subspace correction

Hehu Xie, Lei Zhang+1 more|arXiv (Cornell University)|Jun 2, 2018

Advanced Numerical Methods in Computational Mathematics50 references3 citations

TL;DR

This paper presents a fast, robust method for computing eigenpairs of multiscale elliptic PDEs with rough coefficients by combining operator-adapted wavelets (gamblets) with hierarchical subspace correction. The approach achieves near-linear complexity by using gamblets to create a stable, multiscale decomposition that enables efficient multigrid iterations, significantly accelerating convergence in eigenvalue solvers like LOBPCG while maintaining high accuracy even for challenging problems with extreme scale separation.

ABSTRACT

We present a method for the fast computation of the eigenpairs of a bijective positive symmetric linear operator $\mathcal{L}$. The method is based on a combination of operator adapted wavelets (gamblets) with hierarchical subspace correction.First, gamblets provide a raw but fast approximation of the eigensubspaces of $\mathcal{L}$ by block-diagonalizing $\mathcal{L}$ into sparse and well-conditioned blocks. Next, the hierarchical subspace correction method, computes the eigenpairs associated with the Galerkin restriction of $\mathcal{L}$ to a coarse (low dimensional) gamblet subspace, and then, corrects those eigenpairs by solving a hierarchy of linear problems in the finer gamblet subspaces (from coarse to fine, using multigrid iteration). The proposed algorithm is robust for the presence of multiple (a continuum of) scales and is shown to be of near-linear complexity when $\mathcal{L}$ is an (arbitrary local, e.g.~differential) operator mapping $\mathcal{H}^s_0(Ω)$ to $\mathcal{H}^{-s}(Ω)$ (e.g.~an elliptic PDE with rough coefficients).

Motivation & Objective

• To address the challenge of efficiently computing eigenpairs for elliptic PDEs with highly oscillatory or rough coefficients, where standard eigensolvers struggle due to scale coupling.
• To overcome the slow convergence of multilevel correction methods in multiscale settings by leveraging gamblets for robust, well-conditioned multiscale decomposition.
• To develop a method that achieves near-linear computational complexity for eigenvalue problems in $σ$-regularity spaces, even when the operator has non-separable multiple scales.
• To provide a preconditioning framework that enhances the performance of state-of-the-art eigensolvers such as LOBPCG and multilevel correction schemes.

Proposed method

• The method uses gamblets—operator-adapted wavelets that satisfy scale orthogonality, well-conditioned decomposition, and spatial localization—to construct a hierarchical multiscale basis for the operator $σ$.
• It applies hierarchical subspace correction by first solving the eigenvalue problem on a coarse gamblet subspace, then iteratively correcting the solution on finer subspaces using multigrid iterations.
• The eigenvalue problem is restricted to the Galerkin projection of $σ$ onto the coarse gamblet subspace, and corrections are computed via multigrid-accelerated linear solves in finer subspaces.
• The approach integrates gamblets with multigrid iteration to ensure fast and robust convergence of inner linear solves, crucial for eigenproblem solvers.
• It enables efficient preconditioning for LOBPCG by providing high-quality initial approximations and accelerating convergence through multiscale sparsity and conditioning.
• The method is designed to maintain near-linear complexity when $σ$ maps $σ^{s}_{0}(\Omega)$ to $σ^{-s}(\Omega)$, even for differential operators with rough coefficients.

Experimental results

Research questions

• RQ1 Can gamblets be used to construct a stable, multiscale decomposition that enables fast and robust eigenpair computation for operators with multiple scales?
• RQ2 Does combining hierarchical subspace correction with gamblet-based multigrid lead to near-linear complexity in solving multiscale eigenvalue problems?
• RQ3 How does the gamblet-based method compare to state-of-the-art eigensolvers like LOBPCG in terms of convergence speed and accuracy for problems with rough coefficients?
• RQ4 Can the gamblet decomposition serve as an effective preconditioner for iterative eigensolvers such as LOBPCG, and how does it improve convergence?
• RQ5 What is the performance of the hybrid method combining multilevel correction and LOBPCG on challenging problems like Anderson localization?

Key findings

• The gamblet-based multilevel correction scheme achieved an accuracy of $10^{-14}$ for the first 12 eigenvalues of the SPE10 problem without adaptive stopping criteria, demonstrating high robustness.
• The gamblet preconditioned LOBPCG required fewer outer iterations than standard preconditioners (e.g., ILU or geometric multigrid), indicating faster convergence.
• For the Anderson localization problem with $\varepsilon = 0.01$, the hybrid method—using gamblet-based multilevel correction to initialize LOBPCG—achieved rapid convergence with high accuracy.
• The method demonstrated near-linear computational complexity for elliptic PDEs with rough coefficients, even when the operator has a continuum of scales.
• Numerical results showed that the gamblet-based approach outperformed standard preconditioning techniques (e.g., ILU, geometric multigrid) in both convergence rate and robustness.
• The combination of gamblets with hierarchical subspace correction enabled fast, stable, and accurate computation of eigenpairs across multiple test cases, including high-contrast and multiscale problems.