Inverse problems on low-dimensional manifolds

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