Related papers: Estimates on the stability constant for the trunca…
Numerical issues for the 3D travel time tomography problem with non-overdetemined data are considered. Truncated Fourier series with respect to a special orthonormal basis of functions depending on the source position is used. In addition,…
We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are givenby the far field patterns $u^\infity(\hat{x},k)$ for all…
A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all…
We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton's method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function.…
We prove a H\"{o}lder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function $v$ on $\mathbb{R}^d$ from its Fourier transform $\mathcal{F} v$ given on $[-r,r]^d$. This estimate relies…
We introduce a method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we…
We consider the inverse shape and parameter problem for detecting corrosion from partial boundary measurements. This problem models the non-destructive testing for a partially buried object from electrostatic measurements on the accessible…
We shall prove pointwise estimates for the decreasing rearrangement of $Tf$, where $T$ covers a wide range of interesting operators in Harmonic Analysis such as operators satisfying a Fefferman-Stein inequality, the Bochner-Riesz operator,…
In this paper, we study the inverse random source scattering problem for the biharmonic Schrodinger equation in two and three dimensions. The driven source is assumed to be a generalized microlocally isotropic Gaussian random function whose…
In this paper a deterministic sparse Fourier transform algorithm is presented which breaks the quadratic-in-sparsity runtime bottleneck for a large class of periodic functions exhibiting structured frequency support. These functions…
We consider two formally determined inverse problems for the wave equation in more than one space dimension. Motivated by the fixed angle inverse scattering problem, we show that a compactly supported potential is uniquely determined by the…
In this paper we establish a global Carleman estimate for the fourth order Schr\"odinger equation posed on a $1-d$ finite domain. The Carleman estimate is used to prove the Lipschitz stability for an inverse problem consisting in retrieving…
A fundamental small-noise sensitivity analysis of spike localization in the presence of adversarial perturbations and an arbitrary point spread function (PSF) is presented. The analysis leverages the local Lipschitz property of the inverse…
We give new formulas for finding a compactly supported function $v$ on $\mathbb{R}^d$, $d\geq 1$, from its Fourier transform $\mathcal{F} v$ given within the ball $B_r$. For the one-dimensional case, these formulas are based on the theory…
Robustness and stability of image-reconstruction algorithms have recently come under scrutiny. Their importance to medical imaging cannot be overstated. We review the known results for the topical variational regularization strategies…
We prove weighted $L^2$ and refined $L^p$ decoupling estimates for functions whose Fourier transforms are supported in a small neighborhood of the unit sphere or the truncated paraboloid with an additional lower-dimensional frequency…
We study the inverse problem of recovering the spatial support of parameter variations in a system of partial differential equations (PDEs) from boundary measurements. A reconstruction method is developed based on the monotonicity…
We consider first-order methods with constant step size for minimizing locally Lipschitz coercive functions that are tame in an o-minimal structure on the real field. We prove that if the method is approximated by subgradient trajectories,…
The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This result also holds for frames that are possibly continuous…
Mathematically rigorous inversion method is developed to recover compactly supported potentials from the fixed-energy scattering data in three dimensions. Error estimates are given for the solution. An algorithm for inversion of noisy…