Related papers: Optimisation approach for the Monge-Ampere equatio…
A least-squares method for solving the hyperbolic Monge-Amp\`ere equation with transport boundary condition is introduced. The method relies on an iterative procedure for the gradient of the solution, the so-called mapping. By formulating…
We study the Dirichlet problem for the complex Monge-Amp\`ere operator on a B-regular domain $\Omega$, allowing boundary data that is singular or unbounded. We introduce the concept of pluri-quasibounded functions on $\Omega$ and $\partial…
We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.
The classical Minkowski problem for convex bodies has deeply influenced the development of differential geometry. During the past several decades, abundant mathematical theories have been developed for studying the solutions of the…
A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at…
Monge--Amp\`ere equation plays an important part in Analysis. For example, it is instrumental in mass transport problems. On the other hand, the Bellman function technique appeared recently as a way to consider certain Harmonic Analysis…
We collect examples of boundary-value problems of Dirichlet and Dirichlet-Neumann type which we found instructive when designing and analysing numerical methods for fully nonlinear elliptic partial differential equations. In particular, our…
We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method is based on a numerical resolution of the corresponding…
In this work we present a numerical method for the Optimal Mass Transportation problem. Optimal Mass Transportation (OT) is an active research field in mathematics.It has recently led to significant theoretical results as well as…
We shall consider the regularity problem of solutions for complex Monge-Ampere equations. First we prove interior $C^2$ estimates of solutions in a bounded domain for complex Monge-Ampere equation with assumption of certain $L^p$ bound for…
In this paper, we propose a novel neural network framework, the Legendre-Kolmogorov-Arnold Network (Legendre-KAN) method, designed to solve fully nonlinear Monge-Amp\`ere equations with Dirichlet boundary conditions. The architecture…
We show, that under natural assumptions, solutions of Dirichlet problems for uniformly elliptic divergence form operator can be approximated pointwise by solutions of some versions of Robin problems. The proof is based on stochastic…
We introduce a monotone (degenerate elliptic) discretization of the Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is consistent provided the solution hessian condition number is uniformly bounded. Our approach…
By developing an integral approach, we present a new method for the interior regularity of strictly convex solution of the Monge-Amp\`{e}re equation $\det D^2 u = 1$.
We propose a novel multimesh rational approximation scheme for the numerical solution of the (homogeneous) Dirichlet problem for the spectral fractional Laplacian. The scheme combines a rational approximation of the function $\lambda…
We consider the numerical approximation of surfaces of prescribed Gaussian curvature via the solution of a fully nonlinear partial differential equation of Monge-Amp\`ere type. These surfaces need not be continuous up to the boundary of the…
In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampere type equations under sharp hypotheses. In particular we assume that the matrix function in the…
We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative…
We consider nonsmooth optimal control problems subject to a linear elliptic partial differential equation with homogeneous Dirichlet boundary conditions. It is well-known that local solutions satisfy the celebrated Pontryagin maximum…
We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More…