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The inverse problem of reconstructing the initial state in quasilinear parabolic equations from time averages is investigated. Under suitable regularity assumptions on the quasilinear structure and a superlinear growth condition near zero…
A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the…
In this work we analyze the inverse problem of recovering the space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very…
In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model…
In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs.…
In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without…
For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge-Kutta methods such as the Radau IIA methods…
A convexification-based numerical method for a Coefficient Inverse Problem for a parabolic PDE is presented. The key element of this method is the presence of the so-called Carleman Weight Function in the numerical scheme. Convergence…
This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we study the inverse problem of determining the…
The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE…
We consider the problem of reconstructing a function $f\in L^2(\mathbb{R})$ given phase-less samples of its Gabor transform, which is defined by $$\mathcal{G} f(x,\omega) := 2^{\frac14} \int_{\mathbb{R}} f(t) e^{-\pi (t-x)^2} e^{-2\pi i y…
This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity $a(x)$ and the nonlinear reaction term $f(u)$ in a reaction-diffusion equation from overposed data. These measurements can consist of:…
In this paper, we consider the inverse boundary problems of recovering the time-dependent nonlinearity and damping term for a semilinear wave equation on a Riemannian manifold. The Carleman estimate and the construction of Gaussian beams…
The growth-fragmentation equation arises in many different contexts, ranging from cell division, protein polymerization, biopolymers, neurosciences etc. Direct observation of temporal dynamics being often difficult, it is of main interest…
In this paper we develop an adaptive procedure for the numerical solution of semilinear parabolic problems, with possible singular perturbations. Our approach combines a linearization technique using Newton's method with an adaptive…
In this paper we develop a reconstruction algorithm for the solution of an inverse boundary value problem dealing with a semilinear elliptic partial differential equation of interest in cardiac electrophysiology. The goal is the detection…
In this paper, two numerical schemes for a nonlinear integral equation of Fredholm type with weakly singular kernel are proposed. These numerical methods combine sinc-collocation and sinc-convolution approximations with Newton and steepest…
In this article we derive a priori error estimates for the $hp$-version of the mortar finite element method for parabolic initial-boundary value problems. Both semidiscrete and fully discrete methods are analysed in $L^2$- and $H^1$-norms.…
We derive a posteriori error estimates in the $L_\infty((0,T];L_\infty(\Omega))$ norm for approximations of solutions to linear para bolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat…
We consider parametrized problems driven by spatially nonlocal integral operators with parameter-dependent kernels. In particular, kernels with varying nonlocal interaction radius $\delta > 0$ and fractional Laplace kernels, parametrized by…