Inverse problems on low-dimensional manifolds

Abstract

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability results in this general setting, also in the case when only a finite discretization of the measurements is available. Then, a Landweber-type reconstruction algorithm from a finite number of measurements is proposed, for which we prove global convergence, thanks to a new criterion for finding a suitable initial guess. These general results are then applied to several examples, including two classical nonlinear ill-posed inverse boundary value problems. The first is Calderón’s inverse conductivity problem, for which we prove a Lipschitz stability estimate from a finite number of measurements for piece-wise constant conductivities with discontinuities on an unknown triangle. A similar stability result is then obtained for Gel’fand-Calderón’s problem for the Schrödinger equation, in the case of piece-wise constant potentials with discontinuities on a finite number of non-intersecting balls.

Stable determination of an anisotropic inclusion in the Schrödinger equation from local Cauchy data

“…Our idea is to first estimate the L 2 norm of (γ (1) − γ (2) ) on W 1 , namely δ 2 , by means of δ * 1 and then to estimate the L 2 norm of (q (2) − q (1) ) on W 1 , namely δ2 , in terms of δ 2 .…”

“…This result was subsequently extended by Di Cristo and Rondi [23] for the inverse scattering problem and by Sincich [41] for the corrosion detection problem. Recently, in [2], Alberti et al have extended these ideas by proving that for coefficients belonging to finite dimensional manifolds, uniqueness and stability are guaranteed. In this direction, Lipschitz stability estimates have been proved for real and complex finite dimensional isotropic coefficients ([8, 6, 17]), for a special type of anisotropic conductivities ( [28,25]), for polyhedral inclusions in a conductive medium ([19, 13, 18]), for the non local operator ( [39]) and for the elasticity case ( [24]).…”

Discretization of parameter identification in PDEs using neural networks

“…Combining this with (21) and (17), we can replace boundedness of ( f N ) L ∞ (Ω) by boundedness of f N L ∞ (Ω y ) , and use the weaker regularizer…”

“…some examples for Lipschitz continuous and coercive activation functions are: ReLU, Leaky ReLU (coercive on R), softplus etc. Assume further that the exact f † can be expressed exactly via a NN, possibly with infinitely many hyperparameters, say f † ∈ C ∞ with f † (0) = 0, similar to (21) we have…”

“…In our case, X N = (Z × V) K × C N , so X N is a linear space in case of a linear activation function σ albeit not necessarily finite dimensional (for approximation on manifolds, see e.g. [21]). It yields the kth iterate…”