Related papers: A Shape Design Approximation for Degenerate Partia…
In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results…
We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear…
We provide a broad overview on qualitative versus quantitative regularity estimates in the theory of degenerate parabolic pdes. The former relates to DiBenedetto's revolutionary method of intrinsic scaling, while the latter is achieved by…
In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when "the gradient is small". Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations…
Shape optimization models with one or more shapes are considered in this chapter. Of particular interest for applications are problems in which where a so-called shape functional is constrained by a partial differential equation (PDE)…
The article addresses the convergence of implicit and semi-implicit, fully discrete approximations of a class of nonlinear parabolic evolution problems. Such schemes are popular in the numerical solution of evolutions defined with the…
Ordinary Differential Equations are a simple but powerful framework for modeling complex systems. Parameter estimation from times series can be done by Nonlinear Least Squares (or other classical approaches), but this can give…
In this paper, authors shall introduce a finite element method by using a weakly defined gradient operator over discontinuous functions with heterogeneous properties. The use of weak gradients and their approximations results in a new…
We introduce a novel numerical approach for a class of stochastic dynamic programs which arise as discretizations of backward stochastic differential equations or semi-linear partial differential equations. Solving such dynamic programs…
This manuscript is devoted to the study of a class of nonlinear non-instantaneous impulsive first order abstract retarded type functional differential equations in an arbitrary separable Hilbert space H. A new set of sufficient conditions…
We establish the boundedness of time derivatives of solutions to parabolic $p$-Laplace equations. Our approach relies on the Bernstein technique combined with a suitable approximation method. As a consequence, we obtain an optimal…
Estimating correspondences between deformed shape instances is a long-standing problem in computer graphics; numerous applications, from texture transfer to statistical modelling, rely on recovering an accurate correspondence map. Many…
Motivated by applications to probability and mathematical finance, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable…
An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
We study the parabolic equation \begin{align} \notag &u_t(t,x)=a^{ij}(t)u_{x^ix^j}(t,x)+f(t,x), \quad (t,x) \in [0,T] \times \mathbf{R}^d \\ &u(0,x)=u_0(x) \label{main eqn} \end{align} with the full degeneracy of the leading coefficients,…
We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to…
We consider a $2\times 2$ system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse…
We give a simplified and direct proof of the Kato square root estimate for parabolic operators with elliptic part in divergence form and coefficients possibly depending on space and time in a merely measurable way. The argument relies on…
We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic.…