19,907 research outputs found
Numerical approximation of BSDEs using local polynomial drivers and branching processes
We propose a new numerical scheme for Backward Stochastic Differential
Equations based on branching processes. We approximate an arbitrary (Lipschitz)
driver by local polynomials and then use a Picard iteration scheme. Each step
of the Picard iteration can be solved by using a representation in terms of
branching diffusion systems, thus avoiding the need for a fine time
discretization. In contrast to the previous literature on the numerical
resolution of BSDEs based on branching processes, we prove the convergence of
our numerical scheme without limitation on the time horizon. Numerical
simulations are provided to illustrate the performance of the algorithm.Comment: 28 page
A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations
In this paper, we develop regularized discrete least squares collocation and
finite volume methods for solving two-dimensional nonlinear time-dependent
partial differential equations on irregular domains. The solution is
approximated using tensor product cubic spline basis functions defined on a
background rectangular (interpolation) mesh, which leads to high spatial
accuracy and straightforward implementation, and establishes a solid base for
extending the computational framework to three-dimensional problems. A
semi-implicit time-stepping method is employed to transform the nonlinear
partial differential equation into a linear boundary value problem. A key
finding of our study is that the newly proposed mesh-free finite volume method
based on circular control volumes reduces to the collocation method as the
radius limits to zero. Both methods produce a large constrained least-squares
problem that must be solved at each time step in the advancement of the
solution. We have found that regularization yields a relatively
well-conditioned system that can be solved accurately using QR factorization.
An extensive numerical investigation is performed to illustrate the
effectiveness of the present methods, including the application of the new
method to a coupled system of time-fractional partial differential equations
having different fractional indices in different (irregularly shaped) regions
of the solution domain
Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion
We present a class of spline finite element methods for time-domain wave
propagation which are particularly amenable to explicit time-stepping. The
proposed methods utilize a discontinuous Galerkin discretization to enforce
continuity of the solution field across geometric patches in a multi-patch
setting, which yields a mass matrix with convenient block diagonal structure.
Over each patch, we show how to accurately and efficiently invert mass matrices
in the presence of curved geometries by using a weight-adjusted approximation
of the mass matrix inverse. This approximation restores a tensor product
structure while retaining provable high order accuracy and semi-discrete energy
stability. We also estimate the maximum stable timestep for spline-based finite
elements and show that the use of spline spaces result in less stringent CFL
restrictions than equivalent piecewise continuous or discontinuous finite
element spaces. Finally, we explore the use of optimal knot vectors based on L2
n-widths. We show how the use of optimal knot vectors can improve both
approximation properties and the maximum stable timestep, and present a simple
heuristic method for approximating optimal knot positions. Numerical
experiments confirm the accuracy and stability of the proposed methods
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