Related papers: An adjoint control method for initial condition id…
This paper is concerned with a compositional approach for constructing abstractions of interconnected discrete-time stochastic control systems. The abstraction framework is based on new notions of so-called stochastic simulation functions,…
The forward problem here is the Cauchy problem for a 1D hyperbolic PDE with a variable coefficient in the principal part of the operator. That coefficient is the spatially distributed dielectric constant. The inverse problem consists of the…
In this paper we consider the inverse problem of identifying the initial data in a fractionally damped wave equation from time trace measurements on a surface, as relevant in photoacoustic or thermoacoustic tomography. We derive and analyze…
This paper analyzes the discretization of a Neumann boundary control problem with a stochastic parabolic equation, where an additive noise occurs in the Neumann boundary condition. The convergence is established for general filtrations, and…
We study the Cauchy problem for the Hamilton-Jacobi equation with a semiconcave initial condition. We prove an inequality between two types of weak solutions emanating from such an initial condition (the variational and the viscosity…
This article begins with a brief introduction to numerical relativity aimed at readers who have a background in applied mathematics but not necessarily in general relativity. I then introduce and summarise my work on the problem of treating…
We consider an inverse problem of determining coefficient matrices in an $N$-system of second-order elliptic equations in a bounded two dimensional domain by a set of Cauchy data on arbitrary subboundary. The main result of the article is…
In this paper, we consider the approximate controllability of partial differential equations with time derivatives of non-integer order via boundary control. We first show the unique existence of the solution under smooth boundary…
The paper concerned with higher order asymptotic expansion of solutions to the Cauchy problem of abstract hyperbolic equations of the form $u''+Au+u'=0$ in a Hilbert space, where $A$ is a nonnegative selfadjoint operator. The result says…
We use the adjoint methods to study the static Hamilton-Jacobi equations and to prove the speed of convergence for those equations. The main new ideas are to introduce adjoint equations corresponding to the formal linearizations of…
We study for the first time the Cauchy problem for semilinear fractional elliptic equation. This paper is concerned with the Gaussian white noise model for the initial Cauchy data. We establish the ill-posedness of the problem. Then, under…
We study Cauchy problems of fractional differential equations in both space and time variables by expressing the solution in terms of ``stochastic composition" of the solutions to two simpler problems. These Cauchy sub-problems respectively…
In this work we develop a new numerical approach for recovering a spatially dependent source component in a standard parabolic equation from partial interior measurements. We establish novel conditional Lipschitz stability and H\"{o}lder…
We establish relationships between the classical moments problems which are problems of a construction of a measure supported on a real line, on a half-line or on an interval from prescribed set of moments with the Boundary control approach…
This paper investigates the simultaneous identification of a spatially dependent potential and the initial condition in a subdiffusion model based on two terminal observations. The existence, uniqueness, and conditional stability of the…
We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations…
I development a Conjugate Gradient Method for solving a partial differential system with multiply controls. Some numerical results are depicted. Also, I present an explication of why the control over a partial differential equations system…
We develop a novel iterative direct sampling method (IDSM) for solving linear or nonlinear elliptic inverse problems with partial Cauchy data. It integrates three innovations: a data completion scheme to reconstruct missing boundary…
We study the Cauchy problem for a general homogeneous linear partial differential equation in two complex variables with constant coefficients and with divergent initial data. We state necessary and sufficient conditions for the summability…
In the recent developments of regularization theory for inverse and ill-posed problems, a variational quasi-reversibility (QR) method has been designed to solve a class of time-reversed quasi-linear parabolic problems. Known as a PDE-based…