Related papers: Introduction to inverse problems for hyperbolic PD…
In this paper, a quadratic optimal control problem is considered for second-order parabolic PDEs with homogeneous Dirichlet boundary conditions, in which the "point" control function (depending only on time) constitutes a source term. These…
In the present paper we consider an inverse source problem for time-fractional mixed parabolic-hyperbolic equation with the Caputo derivative. In case, when hyperbolic part of the considered mixed type equation is wave equation, the…
These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental…
This paper is concerned with a class of partial differential equations, which are the linear combinations, with constant coefficients, of the classical flows of the KdV hierarchy. A boundary value problem with inhomogeneous boundary…
We discuss possibilities of application of Numerical Analysis methods to proving computability, in the sense of the TTE approach, of solution operators of boundary-value problems for systems of PDEs. We prove computability of the solution…
The comparative study of two globally convergent numerical methods for acoustic tomography is carried out in two dimensions. These are the boundary control method and the quasi-reversibility method. The novelty is that in the latter a…
In this paper we prove stability estimates of logarithmic type for an inverse problem consisting in the determination of unknown portions of the boundary of a domain in $\mathbb{R}^n$, from a knowledge, in a finite time observation, of…
The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the…
We formulate the immersed-boundary method (IBM) as an inverse problem. A control variable is introduced on the boundary of a larger domain that encompasses the target domain. The optimal control is the one that minimizes the mismatch…
We investigate inverse boundary problems associated with a time-dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. We establish in several generic…
Quantum systems are exceedingly difficult to engineer because they are sensitive to various types of noises. In particular, time-dependent noises are frequently encountered in experiments but how to overcome them remains a challenging…
A theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial…
We propose and study several inverse boundary problems associated with a quasilinear hyperbolic equation of the form ${c(x)^{-2}}\partial_t^2u=\Delta_g(u+F(x, u))+G(x, u)$ on a compact Riemannian manifold $(M, g)$ with boundary. We show…
Convex optimization encompasses a wide range of optimization problems that contain many efficiently solvable subclasses. Interior point methods are currently the state-of-the-art approach for solving such problems, particularly effective…
We investigate the qualitative properties of the weak solutions to the boundary value problems for the hyperbolic fourth-order linear equations with constant coefficients in the plane bounded domain convex with respect to characteristics.…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…
Solving parabolic optimal control problems can be inherently challenging in the field of science and engineering, especially with constraints on the nonsmooth distributed control. Motivated by the extensive applicability of the alternating…
Solving inverse partial differential equation (PDE) problems is a fundamental topic in scientific research due to its broad significance across a wide range of real-world applications. Inverse PDE problems arise across medical imaging,…
Solution of Helmholtz equation with impedance boundary condition on finite interval is equivalently reformulated as steady state of initial boundary value problem for first order hyperbolic system of partial differential equations.…
Leading eigenvalue problems for large scale matrices arise in many applications. Coordinate-wise descent methods are considered in this work for such problems based on a reformulation of the leading eigenvalue problem as a non-convex…