Related papers: Identification of the diffusion parameter in nonlo…
We construct a novel estimator for the diffusion coefficient of the limiting homogenized equation, when observing the slow dynamics of a multiscale model, in the case when the slow dynamics are of bounded variation. Previous research…
We formulate the problem of material identification as a problem of optimal control in which the deformation of the specimen is the state variable and the unknown material law is the control variable. We assume that the material obeys…
We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad…
For a model convection-diffusion problem, we obtain new error estimates for a general upwinding finite element discretization based on bubble modification of the test space. The key analysis tool is based on finding representations of the…
In this paper, we propose a new adaptation of the D-iteration algorithm to numerically solve the differential equations. This problem can be reinterpreted in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the…
We investigate the error of the (semidiscrete) Galerkin method applied to a semilinear subdiffusion equation in the presence of a nonsmooth initial data. The diffusion coefficient is allowed to depend on time. It is well-known that in such…
We present a Bayesian methodology for infinite as well as finite dimensional parameter identification for partial differential equation models. The Bayesian framework provides a rigorous mathematical framework for incorporating prior…
We present a new stabilization technique for multiscale convection diffusion problems. Stabilization for these problems has been a challenging task, especially for the case with high Peclet numbers. Our method is based on a constraint…
We present an optimization-based coupling method for local and nonlocal continuum models. Our approach couches the coupling of the models into a control problem where the states are the solutions of the nonlocal and local equations, the…
In this paper, we consider a free boundary problem with a nonlocal diffusion kernel function $k(x)$. Due to the long distance exchange effect of nonlocal diffusion, the free boundary can expand discontinuously, which makes the problem…
In this paper we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed due to the strong smoothing property of…
We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the \$n\$th time level…
Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and…
We provide a new perspective on the study of parameterized optimization problems. Our approach combines methods for post-optimal sensitivity analysis and ordinary differential equations to quantify the uncertainty in the minimizer due to…
A singularly perturbed convection-diffusion problem,posed on the unit square in $\mathbb{R}^2$, is studied; its solution has both exponential and characteristic boundary layers. The problem is solved numerically using the local…
Identification of nonlinear dynamical systems is crucial across various fields, facilitating tasks such as control, prediction, optimization, and fault detection. Many applications require methods capable of handling complex systems while…
Consider a set of discounted optimal stopping problems for a one-parameter family of objective functions and a fixed diffusion process, started at a fixed point. A standard problem in stochastic control/optimal stopping is to solve for the…
This paper deals with a stochastic optimal feedback control problem for the controlled stochastic partial differential equations. More precisely, we establish the existence of stochastic optimal feedback control for the controlled…
Traditionally, systems governed by linear Partial Differential Equations (PDEs) are spatially discretized to exploit their algebraic structure and reduce the computational effort for controlling them. Due to beneficial insights of the PDEs,…
We consider numerical methods for linear parabolic equations in one spatial dimension having piecewise constant diffusion coefficients defined by a one parameter family of interface conditions at the discontinuity. We construct immersed…