Related papers: A direct method for reconstructing inclusions and …
Compressive sensing is a methodology for the reconstruction of sparse or compressible signals using far fewer samples than required by the Nyquist criterion. However, many of the results in compressive sensing concern random sampling…
In this work, we study the perception problem for sampled surfaces (possibly with boundary) using tools from computational topology, specifically, how to identify their underlying topology starting from point-cloud samples in space, such as…
We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear…
The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes…
We address the problem of reconstructing a multi-band signal from its sub-Nyquist point-wise samples. To date, all reconstruction methods proposed for this class of signals assumed knowledge of the band locations. In this paper, we develop…
The problem of identifying the set of Dirichlet-to-Neumann (DtN) maps arising from conductivities on a smooth domain, among operators acting on functions on the boundary, is a challenging issue in the mathematical analysis of the Calder\'on…
In this paper, we propose a new spectral decomposition method to simulate waves propagating in complicated waveguides. For the numerical solutions of waveguide scattering problems, an important task is to approximate the…
In this paper, we propose a general methodology for sampling from un-normalized densities defined on Riemannian manifolds, with a particular focus on multi-modal targets that remain challenging for existing sampling methods. Inspired by the…
We consider the problem of reconstructing an embedding of a compact connected Riemannian manifold in a Euclidean space up to an almost isometry, given the information on intrinsic distances between points from its ``sufficiently large''…
Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…
Electrical impedance tomography aims at reconstructing the conductivity inside a physical body from boundary measurements of current and voltage at a finite number of contact electrodes. In many practical applications, the shape of the…
This paper introduces a new approach for solving electrical impedance tomography (EIT) problems using deep neural networks. The mathematical problem of EIT is to invert the electrical conductivity from the Dirichlet-to-Neumann (DtN) map.…
We explore the geometric implications of introducing a spectral cut-off on Riemannian manifolds. This is naturally phrased in the framework of non-commutative geometry, where we work with spectral triples that are \emph{truncated} by…
We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of…
This paper concerns with iterative schemes for the perfect reconstruction of functions belonging to multiresolution spaces on bounded manifolds from nonuniform sampling. The schemes have optimal complexity in the sense that the…
The reconstruction of unsteady flow fields from limited measurements is a challenging and crucial task for many engineering applications. Machine learning models are gaining popularity for solving this problem due to their ability to learn…
We propose a globally convergent computational technique for the nonlinear inverse problem of reconstructing the zero-order coefficient in a parabolic equation using partial boundary data. This technique is called the "reduced dimensional…
We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids.…
Accurate determination of the complex effective permittivity is fundamental to optical material engineering, but it remains a critical metrology challenge for heterogeneous systems. In polymer blends and optical composites, scattering and…
This paper is devoted to the inverse problem of recovering simultaneously a potential and a point source in a Shr\"odinger equation from the associated nonlinear Dirichlet to Neumann map. The uniqueness of the inversion is proved and…