Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hierarchical Super-Localized Orthogonal Decomposition Method

Jose C. Garay, Hannah Mohr, Daniel Peterseim, Christoph Zimmer

TL;DR

This work develops a superlocalized hierarchical, $a$-orthogonal basis to approximate the solution operator $\mathcal{A}^{-1}$ for elliptic PDEs with rough coefficients. By combining SLOD corrections with a hierarchical LOD-inspired construction, the method yields a sparse, block-structured operator whose per-level components can be solved independently, enabling mesh-independent conditioning and parallelizable online computation. The authors provide a four-stage compression framework with rigorous error bounds, and demonstrate through numerical experiments that HSLOD achieves optimal or near-optimal convergence rates even in high-contrast, channelized coefficients, while outperforming prior HLOD/gamblet-based approaches. The approach promises scalable, offline-online computational efficiency for multiscale diffusion problems and offers a pathway to extensions to elliptic optimal control problems with rough coefficients.

Abstract

We present the construction of a sparse-compressed operator that approximates the solution operator of elliptic PDEs with rough coefficients. To derive the compressed operator, we construct a hierarchical basis of an approximate solution space, with superlocalized basis functions that are quasi-orthogonal across hierarchy levels with respect to the inner product induced by the energy norm. The superlocalization is achieved through a novel variant of the Super-Localized Orthogonal Decomposition method that is built upon corrections of basis functions arising from the Localized Orthogonal Decomposition method. The hierarchical basis not only induces a sparse compression of the solution space but also enables an orthogonal multiresolution decomposition of the approximate solution operator, decoupling scales and solution contributions of each level of the hierarchy. With this decomposition, the solution of the PDE reduces to the solution of a set of independent linear systems per level with mesh-independent condition numbers that can be computed simultaneously. We present an accuracy study of the compressed solution operator as well as numerical results illustrating our theoretical findings and beyond, revealing that desired optimal error rates with well-behaved superlocalized basis functions can still be attained even in the challenging case of coefficients with high-contrast channels.

Hierarchical Super-Localized Orthogonal Decomposition Method

TL;DR

This work develops a superlocalized hierarchical, -orthogonal basis to approximate the solution operator for elliptic PDEs with rough coefficients. By combining SLOD corrections with a hierarchical LOD-inspired construction, the method yields a sparse, block-structured operator whose per-level components can be solved independently, enabling mesh-independent conditioning and parallelizable online computation. The authors provide a four-stage compression framework with rigorous error bounds, and demonstrate through numerical experiments that HSLOD achieves optimal or near-optimal convergence rates even in high-contrast, channelized coefficients, while outperforming prior HLOD/gamblet-based approaches. The approach promises scalable, offline-online computational efficiency for multiscale diffusion problems and offers a pathway to extensions to elliptic optimal control problems with rough coefficients.

Abstract

We present the construction of a sparse-compressed operator that approximates the solution operator of elliptic PDEs with rough coefficients. To derive the compressed operator, we construct a hierarchical basis of an approximate solution space, with superlocalized basis functions that are quasi-orthogonal across hierarchy levels with respect to the inner product induced by the energy norm. The superlocalization is achieved through a novel variant of the Super-Localized Orthogonal Decomposition method that is built upon corrections of basis functions arising from the Localized Orthogonal Decomposition method. The hierarchical basis not only induces a sparse compression of the solution space but also enables an orthogonal multiresolution decomposition of the approximate solution operator, decoupling scales and solution contributions of each level of the hierarchy. With this decomposition, the solution of the PDE reduces to the solution of a set of independent linear systems per level with mesh-independent condition numbers that can be computed simultaneously. We present an accuracy study of the compressed solution operator as well as numerical results illustrating our theoretical findings and beyond, revealing that desired optimal error rates with well-behaved superlocalized basis functions can still be attained even in the challenging case of coefficients with high-contrast channels.
Paper Structure (21 sections, 10 theorems, 115 equations, 6 figures, 3 tables)

This paper contains 21 sections, 10 theorems, 115 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Construction of hierarchical basis
  3. a-orthogonal bases of $\mathcal{V}_H$
  4. Toward basis functions with local support
  5. A practical hierarchical basis
  6. Stable localization of basis functions
  7. A stable SLOD basis for level $\ell$
  8. HSLOD basis functions at level $\ell$ leading to a Riesz stable basis
  9. Behavior of HSLOD basis functions at a given level and their associated stiffness matrix
  10. Operator compression and inversion
  11. First compression stage: rank reduction and sparsification by using the superlocalized basis
  12. Second compression stage: sparsification by discarding off- block- diagonal entries of $\hat{\mathbb{A}}_{H_L}$
  13. Third compression stage: sparsification by approximating inverses of diagonal blocks of $\check{\mathbb{A}}_{H_L}$
  14. Fourth compression stage: sparsification by discarding entries of $\mathbb{S}^{(k)}$ with absolute values below a prescribed tolerance
  15. Overall compression error of the approximate solution
  16. ...and 6 more sections

Key Result

Lemma 2.2

Let $\bar{\psi}_{g}$, $\psi_g$, and $\gamma_{\bar{\psi}_{g}}$ be defined as above. Then, the energy norm of the localization error has the bound where $\mathrm{diam}(\Omega)$ denotes the diameter of $\Omega$ and the constant $\alpha$ is given in (A-spectral-bound).

Figures (6)

  • Figure 3.1: Illustration of the two second-order patches $\omega^{(\ell,2)}_T$ (top left) and $\widetilde{\omega}^{(\ell,2)}_K$ on the mesh $\mathcal{T}_\ell$ for some $\ell>0$. The patches are centered around some mesh elements $T\in\mathcal{T}_{\ell-1}$ and $K\in\mathcal{T}_\ell$, respectively. Additionally, the coarser mesh $\mathcal{T}_{\ell-1}$ is depicted with bold lines.
  • Figure 5.1: Piecewise-constant coefficient (left), coefficient with high-contrast channels (middle) and piecewise-constant right-hand side $f\in L^2(\Omega)$ with respect to the mesh with mesh size $2^{-5}$ (right).
  • Figure 5.2: Sparsity pattern of the complete stiffness matrix for varying patch orders $m$. Utilizing a logarithmic gray-scale for color representation, darker shades indicate larger magnitudes of the corresponding entries, while zero entries are depicted in white.
  • Figure 5.3: Sparsity pattern and number of non-zeros of the inverse of the stiffness matrix after the different compression stages. Left: inverse of block-diagonal matrix; middle: CG approximation of each inverted block after seven iterations; right: discarding entries of the CG approximation with absolute value smaller than $10^{-5}$. The patch order is set to $m=2$.
  • Figure 5.4: Plot of the relative energy errors $\|u_h - u_{h,L}^m \|_a/\|u_h\|_a$ of the HSLOD and the HLOD in dependence of the mesh size $H_L$ for a piecewise-constant coefficient $\mathbf{A}$. Left: errors for a smooth right-hand side, right: errors for a piecewise-constant right-hand side.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Remark 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 25 more