Parameter identification in PDEs by the solution of monotone inclusion problems

TL;DR

The paper addresses stable parameter identification for a semilinear parabolic PDE by formulating Lavrentiev regularization as a monotone inclusion $\mathcal{A}(u)+\partial(\lambda\mathcal{R}+\mu\mathcal{S})(u)\ni y^\delta$, where $\mathcal{R}$ is the total variation in time and $\mathcal{S}$ is a spatial $H^1$ semi-norm. It develops a globally convergent nested inertial primal-dual algorithm to solve the semi-discretized inclusion, with explicit proximal steps and a cocoercive forward operator $\mathcal{A}$. Theoretical results establish well-posedness of the regularized problem and convergence of the numerical method under standard monotone-operator assumptions, and numerical experiments in 1D demonstrate stable recovery of time-piecewise-constant sources and favorable convergence behavior compared to existing proximal schemes. This work provides a robust framework for PDE-constrained inverse problems that promotes temporal sparsity while preserving spatial smoothness, along with a scalable solver based on inertial forward-backward splitting.

Abstract

In this paper we consider the solution of monotone inverse problems using the particular example of a parameter identification problem for a semilinear parabolic PDE. For the regularized solution of this problem, we introduce a total variation based regularization method requiring the solution of a monotone inclusion problem. We show well-posedness in the sense of inverse problems of the resulting regularization scheme. In addition, we introduce and analyze a numerical algorithm for the solution of this inclusion problem using a nested inertial primal dual method. We demonstrate by means of numerical examples the convergence of both the numerical algorithm and the regularization method.

Figures (5)

• Figure 1: Solution of the PDE \ref{['eq:PDE']}. First row, left: right hand side $u = u_1^\dagger$ as defined in \ref{['eq:u1']}. First row, right: corresponding solution $y_1 := \mathcal{A}(u_1^\dagger)$. Second row, left: right hand side $u = u_2^\dagger$ as defined in \ref{['eq:u2']}. Second row, right: corresponding solution $y_2 := \mathcal{A}(u_2^\dagger)$.
• Figure 2: Regularised solution of \ref{['eq:ip']} by the solution of \ref{['eq:problem']}. First row, left: regularised solution $u_{\lambda,\mu}^\delta$ with noisy data $y^\delta = u_1^\dagger + n^\delta$ with $u_1^\dagger$ as defined in \ref{['eq:u1']}, noise level $\delta = 10^{-2}$, and regularisation parameters $\lambda = 10^{-4}$ and $\mu = 10^{-5}$. First row, right: resulting error $u_{\lambda,\mu}^\delta - u^\dagger$. Second row, left: regularised solution $u_{\lambda,\mu}^\delta$ with noisy data $y^\delta = u_2^\dagger + n^\delta$ with $u = u_2^\dagger$ as defined in \ref{['eq:u2']}, noise level $\delta = 10^{-2}$, and regularisation parameters $\lambda = 2\cdot 10^{-6}$ and $\mu = 10^{-5}$. Second row, right: resulting error $u_{\lambda,\mu}^\delta - u^\dagger$.
• Figure 3: Demonstration of the convergence behavior of Algorithm \ref{['alg:applied']}. Upper left: Size of the updates $\lVert u^{(\text{old})}-u\rVert_2$. Upper right Values of $\lVert \mathcal{A}(u)-y^\delta\rVert_2$ (red), the total variation regularization term $\lambda \mathcal{R}(u)$ (blue), and the $H^1$ regularization term $\mu\mathcal{S}(u)$ (green). Lower left: Value of the primal residual $r_1(u,v)$. Lower right: Value of the dual residual $r_2(u,v)$.
• Figure 4: Convergence of the regularization method. Left: Plot of the relative error $\lVert u-u_1^\dagger \rVert_2/\lVert u_1^\dagger\rVert_2$ (blue). The black dotted line indicates a rate of order $\sqrt{\delta}$. Right: Plot of the relative residual $\lVert \mathcal{A}(u)-y_1\rVert_2/\lVert y_1\rVert_2$ (blue). The black dotted line indicates a rate of order $\delta$.
• Figure 5: Comparison of accuracy versus computation times for the different algorithms and $N \in \{400,800,1200\}$ discretization points both for the $t$ and the $x$ variable. Each marker on the plot represents 100 steps with the respective algorithm. Algorithm 1 is shown in blue in all graphs, the fixed point iteration in green, and the inertial primal-dual forward-backward algorithm in red. Top row: Primal residuals for the different algorithms. Bottom row: Dual residuals for the different algorithms.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• proof
• Lemma 3
• proof
• Definition 1
• Lemma 4
• Lemma 5
• proof
• Theorem 6