Preconditioned pseudo-time continuation for parameterized inverse problems

TL;DR

This work tackles PDE-constrained inverse problems with uncertain auxiliary parameters by employing a pseudo-time continuation framework to track the inversion parameter ${\boldsymbol m}$ as the auxiliary parameters vary along a controlled path. A predictor-corrector scheme is used to evolve the solution, with Hessian solves accelerated by two adaptive quasi-Newton preconditioners that leverage the continuation structure: a parametric update accounting for $\boldsymbol{\theta}$ changes and a block update that exploits PCG history. The proposed preconditioners are initialized (often via a low-rank approximation of the data-misfit Hessian preconditioned by the regularization) and updated at each time step to maintain efficiency, enabling mesh-independent performance in large-scale problems. Numerical experiments on diffusion coefficient estimation and ice-sheet bedrock inversion demonstrate substantial reductions in Hessian-solve cost and robustness to perturbations, highlighting the method’s potential for efficient uncertainty quantification in complex PDE inverse problems.

Abstract

We consider parameterized variational inverse problems that are constrained by partial differential equations (PDEs). We seek to efficiently compute the solution of the inverse problem when auxiliary model parameters, which appear in the governing PDE, are varied. Computing the solution of the inverse problem for different auxiliary parameter values is crucial for uncertainty quantification. This, however, is computationally challenging since it requires solving many optimization problems for different realizations of the auxiliary parameters. We leverage pseudo-time continuation and solve an initial value problem to evolve the optimal solution along an auxiliary parameter path. This article introduces the use of an adaptive quasi-Newton Hessian preconditioner to accelerate the computation. Our proposed preconditioner exploits properties of the pseudo-time continuation process to achieve reliable and efficient computation. We elaborate our proposed framework and elucidate its properties for two nonlinear inverse problems.

Figures (7)

• Figure 1: Newton's method iterations and pseudo-time continuation steps generated by minimizing $J(m,4.0)$ when initialized with the minimizer of $J(m,1.0)$. The iterations for Newton's method and time steps for pseudo-time continuation approach are marked by crosses and stars, respectively. The horizontal line indicates the minimizer of $J(m,4.0)$.
• Figure 1: Left: log diffusion coefficient used to generate synthetic data; right: the state solution with observation locations denoted by red dots.
• Figure 2: Comparison of Forward and modified Euler time-stepping for three different $\boldsymbol{\theta}$ perturbation magnitudes $\alpha=1.0$ (left), $\alpha=1.25$ (center), and $\alpha=1.5$ (right).
• Figure 3: Comparison of Forward and modified Euler for various PCG tolerances.
• Figure 4: The cost of the pseudo-time continuation procedure for various preconditioner rank parameters. The computational cost in number of PDE solves is shown in the left panel and the number of vectors stored is shown in the right panel. Each curve corresponds to a rank $r_\text{init}$ used to initialize the preconditioner via a low-rank approximation of the regularization-preconditioned data-misfit Hessian.

Theorems & Definitions (1)

• Example 2.1