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A polynomial dimension-dependence analysis of Bramble--Pasciak--Xu preconditioners

Boou Jiang, Jongho Park, Jinchao Xu

TL;DR

This work establishes that BPX preconditioners for high-dimensional finite element discretizations have condition numbers that depend only polynomially on the spatial dimension, provided a geometry- and patch-convex triangulation framework. It achieves this by deriving dimension-aware estimates for core FE tools (regularity, Bramble–Hilbert, trace, and inverse inequalities), and by introducing an averaged Scott--Zhang interpolation whose error bounds scale polynomially with dimension. A multilevel norm equivalence theorem is proved, and a BPX preconditioner is constructed with explicit polynomial bounds on its dimensional dependence; viewing BPX as a parallel subspace correction method yields sharper, dimension-dependent spectral estimates. The results have potential implications for quantum algorithms, where polynomial dimension dependence can translate into exponential speedups over classical approaches for PDE solvers.

Abstract

We investigate the dimension dependence of Bramble--Pasciak--Xu (BPX) preconditioners for high-dimensional partial differential equations and establish that the condition numbers of BPX-preconditioned systems grow only polynomially with the spatial dimension. Our analysis requires a careful derivation of the dimension dependence of several fundamental tools in the theory of finite element methods, including the elliptic regularity, Bramble--Hilbert lemma, trace inequalities, and inverse inequalities. We further introduce a new quasi-interpolation operator into finite element spaces, a variant of the classical Scott--Zhang interpolation, whose associated constants scale polynomially with the dimension. Building on these ingredients, we prove a multilevel norm equivalence theorem and derive a BPX preconditioner with explicit polynomial bounds on its dimensional dependence. This result has notable implications for emerging quantum computing methodologies: recent studies indicate that polynomial dependence of BPX preconditioners on dimension can yield exponential speedups for quantum-algorithmic approaches over their classical counterparts.

A polynomial dimension-dependence analysis of Bramble--Pasciak--Xu preconditioners

TL;DR

This work establishes that BPX preconditioners for high-dimensional finite element discretizations have condition numbers that depend only polynomially on the spatial dimension, provided a geometry- and patch-convex triangulation framework. It achieves this by deriving dimension-aware estimates for core FE tools (regularity, Bramble–Hilbert, trace, and inverse inequalities), and by introducing an averaged Scott--Zhang interpolation whose error bounds scale polynomially with dimension. A multilevel norm equivalence theorem is proved, and a BPX preconditioner is constructed with explicit polynomial bounds on its dimensional dependence; viewing BPX as a parallel subspace correction method yields sharper, dimension-dependent spectral estimates. The results have potential implications for quantum algorithms, where polynomial dimension dependence can translate into exponential speedups over classical approaches for PDE solvers.

Abstract

We investigate the dimension dependence of Bramble--Pasciak--Xu (BPX) preconditioners for high-dimensional partial differential equations and establish that the condition numbers of BPX-preconditioned systems grow only polynomially with the spatial dimension. Our analysis requires a careful derivation of the dimension dependence of several fundamental tools in the theory of finite element methods, including the elliptic regularity, Bramble--Hilbert lemma, trace inequalities, and inverse inequalities. We further introduce a new quasi-interpolation operator into finite element spaces, a variant of the classical Scott--Zhang interpolation, whose associated constants scale polynomially with the dimension. Building on these ingredients, we prove a multilevel norm equivalence theorem and derive a BPX preconditioner with explicit polynomial bounds on its dimensional dependence. This result has notable implications for emerging quantum computing methodologies: recent studies indicate that polynomial dependence of BPX preconditioners on dimension can yield exponential speedups for quantum-algorithmic approaches over their classical counterparts.

Paper Structure

This paper contains 24 sections, 28 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Simplices and triangulations
  4. Finite element spaces
  5. Triangulations in high dimensions
  6. Shape-regularity and quasi-uniformity
  7. Freudenthal triangulations
  8. Fundamental tools
  9. Nodal basis functions
  10. Regularity on convex domains
  11. Bramble--Hilbert lemma
  12. Trace inequalities
  13. Inverse inequalities
  14. Averaged Scott--Zhang interpolation
  15. Definitions
  16. ...and 9 more sections

Key Result

Lemma 3.2

Let $\mathcal{T}_h$ be a $\rho$-shape-regular triangulation of $\Omega \subset \mathbb{R}^d$, and let $\tau \in \mathcal{T}_h$. For a given $(d-1)$-face $F \in \mathcal{F}(\tau)$, let $h_F$ denote the distance from the vertex of $\tau$ opposite to $F$ to the hyperplane containing $F$. Then we have

Theorems & Definitions (53)

  • Definition 3.1: shape-regularity
  • Lemma 3.2
  • Proof 1
  • Proposition 3.3
  • Proof 2
  • Remark 3.4
  • Definition 3.5: quasi-uniformity
  • Lemma 4.1
  • Proof 3
  • Lemma 4.2: elliptic regularity on convex domains
  • ...and 43 more