- Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls(arXiv)
Abstract : We prove the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls with bounded exterior condition in dimension d≥1. Our approach relies on a tree-based probabilistic representation based on a (2s)-stable branching processes for all s∈(0,1), and our existence results hold for sufficiently small exterior conditions and nonlinearity coefficients. In comparison with existing approaches, we consider a wide class of polynomial nonlinearities without imposing upper bounds on their maximal degree or number of terms. Numerical illustrations are provided in large dimensions.
2.Tweed and wireframe: accelerated relaxation algorithms for multigrid solution of ellipticPDEs on stretched structured grids (arXiv)
Abstract : Two new relaxation schemes are proposed for the smoothing step in the geometric multigrid solution of PDEs on 2D and 3D stretched structured grids. The new schemes are characterized by efficient line relaxation on branched sets of lines of alternating colour, where the lines are constructed to be everywhere orthogonal to the local direction of maximum grid clustering. Tweed relaxation is best suited for grid clustering near the boundaries of the computational domain, whereas wireframe relaxation is best suited for grid clustering near the centre of the computational domain. On strongly stretched grids of these types, multigrid leveraging these new smoothing schemes significantly outperforms multigrid based on other leading relaxation schemes, such as checkerboard and alternating-direction zebra relaxation, for the numerical solution of large linear systems arising from the discretization of elliptic PDEs
3. Nonlinear Neumann problems for fully nonlinear elliptic PDEs on a quadrant(arXiv)
Abstract : We consider the nonlinear Neumann problem for fully nonlinear elliptic PDEs on a quadrant. We establish a comparison theorem for viscosity sub and supersolutions of the nonlinear Neumann problem. The crucial argument in the proof of the comparison theorem is to build a C1,1 test function which takes care of the nonlinear Neumann boundary condition. A similar problem has been treated on a general n-dimensional orthant by Biswas, Ishii, Subhamay, and Wang [SIAM J. Control Optim. 55 (2017), pp. 365–396], where the functions (Hi in the main text) describing the boundary condition are required to be positively one-homogeneous, and the result in this paper removes the positive homogeneity in two-dimension. An existence result for solutions is also presented
4. A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics(arXiv)
Abstract : In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected dis- counted future dividend payments. In models which take multiple economic factors into ac- count, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differ- ential equations which can be identified with solutions of elliptic partial differential equations are approximated by means of deep neural networks.