Infinite-Dimensional Inverse Problems with Finite Measurements

Abstract

We present a general framework to study uniqueness, stability and reconstruction for infinite-dimensional inverse problems when only a finite-dimensional approximation of the measurements is available. For a large class of inverse problems satisfying Lipschitz stability we show that the same estimate holds even with a finite number of measurements. We also derive a globally convergent reconstruction algorithm based on the Landweber iteration. This theory applies to nonlinear ill-posed problems such as electrical impedance tomography (EIT), inverse scattering and quantitative photoacoustic tomography (QPAT), under the assumption that the unknown belongs to a finite-dimensional subspace. In particular, we derive Lipschitz stability estimates for EIT with a matrix approximation of the Neumann-to-Dirichlet map; for the inverse scattering problem with measurements of the scattering amplitude at a finite number of directions on $S^2 \times S^2$; and for QPAT with a low-pass filter of the internal energy.

Appendix: A Local Convergence of Landweber Iteration

For the sake of completeness, we describe the convergence result used in Section 3.1. We present a simplified version of [66, Theorem 3.2] that is sufficient for our scopes.

Let X and Y be Hilbert spaces, $A\subseteq X$ be an open set, $K\subseteq A$ be a compact set and $F:A\rightarrow Y$ be such that

• 1. $F\in C^1(A,Y)$ and $F':A\rightarrow {\mathcal {L}}_c(X,Y)$ is Lipschitz continuous, namely

  $$\begin{aligned}&\Vert F'(x_1)-F'(x_2)\Vert _{X\rightarrow Y}\leqq L\Vert x_1-x_2\Vert _X,\qquad x_1,x_2\in A,\\&\Vert F'(x)\Vert _{X\rightarrow Y}\leqq \hat{L},\qquad x\in A, \end{aligned}$$

  for some $L,\hat{L}>0$;

• 2. $F^{-1}$ is Lipschitz continuous, namely

  $$\begin{aligned} \Vert x_1-x_2\Vert _X \leqq C\Vert F(x_1)-F(x_2)\Vert _Y,\qquad x_1,x_2\in A, \end{aligned}$$

  for some $C>0$.

Proposition 2

There exist $\rho ,\mu >0$ and $c\in (0,1)$ such that the following is true. Take $x^{\dagger }\in K$ and let $y=F(x^{\dagger })$. If $x_0\in K$ satisfies

$$\begin{aligned} \Vert x^{\dagger }-x_0\Vert <\rho , \end{aligned}$$

then the iterates $(x_k)$ of the Landweber iteration

$$\begin{aligned} x_{k+1}=x_{k}-\mu F'(x_{k})^{*}\left( F(x_{k})-y\right) ,\qquad k\in {\mathbb {N}}, \end{aligned}$$

converge to $x^{\dagger }$ and satisfy

$$\begin{aligned} \Vert x^{\dagger }-x_k\Vert \leqq \rho c^k,\qquad k\in {\mathbb {N}}. \end{aligned}$$

Remark The constants $\rho$, c and $\mu$ are given explicitly in [66] as functions of the a priori data. For instance, the constant $\rho$, which measures how close to $x^{\dagger }$ the initial guess $x_0$ needs to be, may be chosen as

$$\begin{aligned} \rho =\frac{1}{2L^2\hat{L}^2 C^4}, \end{aligned}$$

and the step-size $\mu$ needs to satisfy

$$\begin{aligned} \mu<\frac{1}{\hat{L}^{2}}\quad \text {and} \quad \mu \bigl (1-\mu \hat{L}^{2}\bigr )<C. \end{aligned}$$

The statement provided in [66] requires F to be weakly sequentially closed, but a close look to the proof indicates that this assumption may be dropped, since in this case the existence of the minimizer $x^{\dagger }$ is granted.