Showing papers by "Hung V. Tran published in 2021"

Hamilton-Jacobi Equations with Semilinear Costs and State Constraints, with Applications to Large Deviations in Games

Abstract: We characterize solutions of a class of time-homogeneous optimal control problems with semilinear running costs and state constraints as maximal viscosity subsolutions to Hamilton-Jacobi equations ...

Coagulation‐Fragmentation Equations with Multiplicative Coagulation Kernel and Constant Fragmentation Kernel

Abstract: We study a critical case of Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation. Our results include wellposedness, regularity and long-time behaviors of viscosity solutions to the Hamilton-Jacobi equation in certain regimes, which have implications to wellposedness and long-time behaviors of \emph{mass-conserving} solutions to the Coagulation-Fragmentation equation.

A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications

Abstract: We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented.

The large time profile for Hamilton–Jacobi–Bellman equations

Abstract: Here, we study the large-time limit of viscosity solutions of the Cauchy problem for second-order Hamilton–Jacobi–Bellman equations with convex Hamiltonians in the torus. This large-time limit solves the corresponding stationary problem, sometimes called the ergodic problem. This problem, however, has multiple viscosity solutions and, thus, a key question is which of these solutions is selected by the limit. Here, we provide a representation for the viscosity solution to the Cauchy problem in terms of generalized holonomic measures. Then, we use this representation to characterize the large-time limit in terms of the initial data and generalized Mather measures. In addition, we establish various results on generalized Mather measures and duality theorems that are of independent interest.

Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method

Abstract: We propose a globally convergent numerical method, called the convexification, to numerically compute the viscosity solution to first-order Hamilton-Jacobi equations through the vanishing viscosity process where the viscosity parameter is a fixed small number. By convexification, we mean that we employ a suitable Carleman weight function to convexify the cost functional defined directly from the form of the Hamilton-Jacobi equation under consideration. The strict convexity of this functional is rigorously proved using a new Carleman estimate. We also prove that the unique minimizer of the this strictly convex functional can be reached by the gradient descent method. Moreover, we show that the minimizer well approximates the viscosity solution of the Hamilton-Jacobi equation as the noise contained in the boundary data tends to zero. Some interesting numerical illustrations are presented.

On a class of divergence form linear parabolic equations with degenerate coefficients

Abstract: We study a class of linear parabolic equations in divergence form with degenerate coefficients on the upper half space. Specifically, the equations are considered in $(-\infty, T) \times \mathbb{R}^d_+$, where $\mathbb{R}^d_+ = \{x \in \mathbb{R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given, and the diffusion matrices are the product of $x_d$ and bounded uniformly elliptic matrices, which are degenerate at $\{x_d=0\}$. As such, our class of equations resembles well the corresponding class of degenerate viscous Hamilton-Jacobi equations. We obtain wellposedness results and regularity type estimates in some appropriate weighted Sobolev spaces for the solutions.

Large time behavior for a Hamilton–Jacobi equation in a critical coagulation-fragmentation model

Abstract: We study the large time behavior of the sublinear viscosity solution to a singular Hamilton-Jacobi equation that appears in a critical Coagulation-Fragmentation model with multiplicative coagulation and constant fragmentation kernels. Our results include complete characterizations of stationary solutions and optimal conditions to guarantee large time convergence. In particular, we obtain convergence results under certain natural conditions on the initial data, and a nonconvergence result when such conditions fail.

Level-set forced mean curvature flow with the Neumann boundary condition

Abstract: Here, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz. We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations. Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz.

Degenerate linear parabolic equations in divergence form on the upper half space

Abstract: We study a class of second-order degenerate linear parabolic equations in divergence form in $(-\infty, T) \times \mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial \mathbb R^d_+$, where $\mathbb R^d_+ = \{x \in \mathbb R^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The coefficient matrices of the equations are the product of $\mu(x_d)$ and bounded uniformly elliptic matrices, where $\mu(x_d)$ behaves like $x_d^\alpha$ for some given $\alpha \in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain. Under a partially VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.

Optimal Convergence Rates for Elliptic Homogenization Problems in Nondivergence-Form: Analysis and Numerical Illustrations

Abstract: We study optimal convergence rates in the periodic homogenization of linear elliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary co...