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Preconditioned normal equations for solving discretised partial differential equations

Lorenzo Lazzarino, Yuji Nakatsukasa, Umberto Zerbinati

TL;DR

The paper addresses solving non-symmetric square linear systems $A\mathbf{x}=\mathbf{b}$ from PDE discretizations by leveraging the normal equations $B\mathbf{x}=A^{\!T}\mathbf{b}$ with $B=A^{\!T}A$ and applying CGNE/LSQR. It introduces the concept of normal preconditioning, designing preconditioners from discretizations of a related "normal" PDE using a Riesz map $T$ so that the preconditioned problem $(P^{\!T}TP)^{-1}A^{\!T}TA\mathbf{x}=(P^{\!T}TP)^{-1}A^{\!T}T\mathbf{b}$ has favorable spectral properties governed by the $T$-weighted singular values of $AP^{-1}$. The authors show that a wide class of normal preconditioners exists and that, in advection–diffusion settings, CGNE with such preconditioners can achieve fast convergence and, in some regimes, mesh-independent performance when $ u$ is small; they provide a functional-analytic framework for these results and discuss backward stability with iterative refinement. The work suggests practical pathways to robust solvers for non-symmetric discretizations, with future directions including LSQ-FEM connections and extensions to other PDEs such as Oseen or Helmholtz equations.

Abstract

This paper explores preconditioning the normal equation for non-symmetric square linear systems arising from PDE discretization, focusing on methods like CGNE and LSQR. The concept of ``normal'' preconditioning is introduced and a strategy to construct preconditioners studying the associated ``normal'' PDE is presented. Numerical experiments on convection-diffusion problems demonstrate the effectiveness of this approach in achieving fast and stable convergence.

Preconditioned normal equations for solving discretised partial differential equations

TL;DR

The paper addresses solving non-symmetric square linear systems from PDE discretizations by leveraging the normal equations with and applying CGNE/LSQR. It introduces the concept of normal preconditioning, designing preconditioners from discretizations of a related "normal" PDE using a Riesz map so that the preconditioned problem has favorable spectral properties governed by the -weighted singular values of . The authors show that a wide class of normal preconditioners exists and that, in advection–diffusion settings, CGNE with such preconditioners can achieve fast convergence and, in some regimes, mesh-independent performance when is small; they provide a functional-analytic framework for these results and discuss backward stability with iterative refinement. The work suggests practical pathways to robust solvers for non-symmetric discretizations, with future directions including LSQ-FEM connections and extensions to other PDEs such as Oseen or Helmholtz equations.

Abstract

This paper explores preconditioning the normal equation for non-symmetric square linear systems arising from PDE discretization, focusing on methods like CGNE and LSQR. The concept of ``normal'' preconditioning is introduced and a strategy to construct preconditioners studying the associated ``normal'' PDE is presented. Numerical experiments on convection-diffusion problems demonstrate the effectiveness of this approach in achieving fast and stable convergence.

Paper Structure

This paper contains 5 sections, 1 theorem, 43 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Preconditioning the normal equations
  3. Normal preconditioning
  4. Functional Analytic Perspective
  5. Conclusions

Key Result

Theorem 4.1

The relative error at the $k$-th iteration of the CG method applied to the preconditioned equation is such that where $x^*$ is the exact solution of eq:PrecnormalPDE, $P_k$ is the space of polynomials $p$ of degree less than or equal to $k$ and unitary lowest order coefficient, where for a matrix $M$ we denote by $M^\circ$ the $T$-transpose of $M$, i.e. the matrix such that for any vectors $u,v

Figures (3)

  • Figure 1: The discrete solution $u_h$ of the reaction diffusion equation \ref{['eq:weakForm']}, with $\mathbf{\beta}=(1,0)$, for different value of $\nu$ at the finest mesh size $512\times 512$, together with $exp(-\lvert\nabla\cdot \beta u_h\lvert^2)$.
  • Figure 2: The discrete solution $u_h$ of the reaction diffusion equation \ref{['eq:weakForm']}, with $\sqrt{2}\mathbf{\beta}=(1,1)$, for different value of $\nu$ at the finest mesh size $512\times 512$, together with $exp(-\lvert\nabla\cdot \beta u_h\lvert^2)$.
  • Figure 3: Comparison of convergence history for the CGNE method preconditioned by the inversion via PETSc GAMG of \ref{['eq:discreteReactionDiffusion2']}, for different values of $\nu$ and different mesh sizes. The wind is fixed to $\left\lVert\beta\right\rVert\beta = (1,1)$ and as right-hand side we consider the function $f(x,y) \equiv 1$. The CGNE method was terminated when the absolute residual was less than $10^{-5}$.

Theorems & Definitions (11)

  • Example 1: Normal preconditioning
  • Example 2: Upwind convection diffusion
  • Theorem 4.1
  • Proof 1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Example 3: Preconditioning via anisotropic diffusion
  • Remark 4.5
  • Example 4: Preconditioning via reaction diffusion
  • ...and 1 more