Related papers: Second order asymptotical regularization methods f…
The problem of numerical differentiation can be thought of as an inverse problem by considering it as solving a Volterra equation. It is well known that such inverse integral problems are ill-posed and one requires regularization methods to…
We introduced in [arXiv:1106.3204] a method to locate discontinuities of a wave speed in dimension two from acoustic boundary measuments modelled by the hyperbolic Neumann-to-Dirichlet operator. Here we extend the method for sound hard…
A fully stochastic second-order adaptive-regularization method for unconstrained nonconvex optimization is presented which never computes the objective-function value, but yet achieves the optimal $\mathcal{O}(\epsilon^{-3/2})$ complexity…
Asymptotic formula is derived for the behavior of the fundamental solution of the second-order elliptic self-adjoint operator with a piecewise-smooth coefficient in front of the senior derivatives near the discontinuity surface of the…
The renormalization method based on the Taylor expansion for asymptotic analysis of differential equations is generalized to difference equations. The proposed renormalization method is based on the Newton-Maclaurin expansion. Several basic…
Distributionally robust optimization (DRO) is a widely-used approach to learn models that are robust against distribution shift. Compared with the standard optimization setting, the objective function in DRO is more difficult to optimize,…
In nonlinear imaging problems whose forward model is described by a partial differential equation (PDE), the main computational bottleneck in solving the inverse problem is the need to solve many large-scale discretized PDEs at each step of…
Regularization is a core component of modern inverse problems, as it helps establish the well-posedness of the solution of interest. Popular regularization approaches include variational regularization and iterative regularization. The…
In this paper we develop a finite-difference scheme to approximate radially symmetric solutions of the initial-value problem with smooth initial conditions in an open sphere around the origin, where the internal and external damping…
A novel method of summing asymptotic series is advanced. Such series repeatedly arise when employing perturbation theory in powers of a small parameter for complicated problems of condensed matter physics, statistical physics, and various…
Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A critical issue is to rigorously bound the error of such…
Diffusion models have recently achieved success in solving Bayesian inverse problems with learned data priors. Current methods build on top of the diffusion sampling process, where each denoising step makes small modifications to samples…
The recovery of images from the observations that are degraded by a linear operator and further corrupted by Poisson noise is an important task in modern imaging applications such as astronomical and biomedical ones. Gradient-based…
This paper investigates the asymptotic behaviour of solutions to certain infinite systems of ordinary differential equations. In particular, we use results from ergodic theory and the asymptotic theory of $C_0$-semigroups to obtain a…
This paper is concerned with the strong approximation of a semi-linear stochastic wave equation with strong damping, driven by additive noise. Based on a spatial discretization performed by a spectral Galerkin method, we introduce a kind of…
This study introduces a recursive method for computing asymptotic solutions of the Laplace equation in corner domains with the homogeneous Dirichlet boundary condition on one side and the Robin boundary condition with a power-law…
In 2002, Fatiha Alabau, Piermarco Cannarsa and Vilmos Komornik investigated the extent of asymptotic stability of the null solution for weakly coupled partially damped equations of the second order in time. The main point is that the…
We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with non-homogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as…
We shall derive and propose several efficient overlapping domain decomposition methods for solving some typical linear inverse problems, including the identiffication of the flux, the source strength and the initial temperature in second…
We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by…