Probabilistic ODE Solvers with Runge-Kutta Means

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct this family such that posterior means match the outputs of the Runge-Kutta family exactly, thus inheriting their proven good properties. Remaining degrees of freedom not identified by the match to Runge-Kutta are chosen such that the posterior probability measure fits the observed structure of the ODE. Our results shed light on the structure of Runge-Kutta solvers from a new direction, provide a richer, probabilistic output, have low computational cost, and raise new research questions.Comment: 18 pages (9 page conference paper, plus supplements); appears in Advances in Neural Information Processing Systems (NIPS), 201

Fourth Order Algorithms for Solving the Multivariable Langevin Equation and the Kramers Equation

We develop a fourth order simulation algorithm for solving the stochastic Langevin equation. The method consists of identifying solvable operators in the Fokker-Planck equation, factorizing the evolution operator for small time steps to fourth order and implementing the factorization process numerically. A key contribution of this work is to show how certain double commutators in the factorization process can be simulated in practice. The method is general, applicable to the multivariable case, and systematic, with known procedures for doing fourth order factorizations. The fourth order convergence of the resulting algorithm allowed very large time steps to be used. In simulating the Brownian dynamics of 121 Yukawa particles in two dimensions, the converged result of a first order algorithm can be obtained by using time steps 50 times as large. To further demostrate the versatility of our method, we derive two new classes of fourth order algorithms for solving the simpler Kramers equation without requiring the derivative of the force. The convergence of many fourth order algorithms for solving this equation are compared.Comment: 19 pages, 2 figure

A Fourier interpolation method for numerical solution of FBSDEs: Global convergence, stability, and higher order discretizations

The implementation of the convolution method for the numerical solution of backward stochastic differential equations (BSDEs) introduced in [19] uses a uniform space grid. Locally, this approach produces a truncation error, a space discretization error, and an additional extrapolation error. Even if the extrapolation error is convergent in time, the resulting absolute error may be high at the boundaries of the uniform space grid. In order to solve this problem, we propose a tree-like grid for the space discretization which suppresses the extrapolation error leading to a globally convergent numerical solution for the (F)BSDE. On this alternative grid the conditional expectations involved in the BSDE time discretization are computed using Fourier analysis and the fast Fourier transform (FFT) algorithm as in the initial implementation. The method is then extended to higher-order time discretizations of FBSDEs. Numerical results demonstrating convergence are also presented.Comment: 28 pages, 8 figures; Previously titled 'Global convergence and stability of a convolution method for numerical solution of BSDEs' (1410.8595v1

A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows

The potential of the hybridized discontinuous Galerkin (HDG) method has been recognized for the computation of stationary flows. Extending the method to time-dependent problems can, e.g., be done by backward difference formulae (BDF) or diagonally implicit Runge-Kutta (DIRK) methods. In this work, we investigate the use of embedded DIRK methods in an HDG solver, including the use of adaptive time-step control. Numerical results demonstrate the performance of the method for both linear and nonlinear (systems of) time-dependent convection-diffusion equations

Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs

We introduce a new algebraic framework based on a modification (called exotic) of aromatic Butcher-series for the systematic study of the accuracy of numerical integrators for the invariant measure of a class of ergodic stochastic differential equations (SDEs) with additive noise. The proposed analysis covers Runge-Kutta type schemes including the cases of partitioned methods and postprocessed methods. We also show that the introduced exotic aromatic B-series satisfy an isometric equivariance property.Comment: 33 page