• Corpus ID: 246015420

Numerical evaluation of ODE solutions by Monte Carlo enumeration of Butcher series

  title={Numerical evaluation of ODE solutions by Monte Carlo enumeration of Butcher series},
  author={Guillaume Penent and Nicolas Privault},
We present an algorithm for the numerical solution of ordinary differential equations by random enumeration of the Butcher trees used in the implementation of the RungeKutta method. Our Monte Carlo scheme allows for the direct numerical evaluation of an ODE solution at any given time within a certain interval, without iteration through multiple time steps. In particular, this approach does not involve a discretization step size, and it does not require the truncation of Taylor series. 
1 Citations
A fully nonlinear Feynman-Kac formula with derivatives of arbitrary orders
We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations with arbitrary gradient nonlinearities. This algorithm extends the classical Feynman-Kac


Trees and numerical methods for ordinary differential equations
  • J. Butcher
  • Computer Science
    Numerical Algorithms
  • 2009
Trees have a central role in the theory of Runge–Kutta methods and they also have applications to more general methods, involving multiple values and multiple stages.
Coefficients for the study of Runge-Kutta integration processes
  • J. Butcher
  • Mathematics
    Journal of the Australian Mathematical Society
  • 1963
We consider a set of η first order simultaneous differential equations in the dependent variables y1, y2, …, yn and the independent variable x ⋮ No loss of gernerality results from taking the
Bayesian Solution of Ordinary Differential Equations
In the numerical solution of ordinary differential equations, a function y(x) is to be reconstructed from knowledge of the functional form of its derivative: dy/dx = f (x, y), together with an
Branching diffusion representation of semilinear PDEs and Monte Carlo approximation
We provide a representation result of parabolic semi-linear PD-Es, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced
Existence and probabilistic representation of the solutions of semilinear parabolic PDEs with fractional Laplacians
We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on Rd with polynomial nonlinearities in (u,∇u), using a tree-based probabilistic representation. This
A Feynman-Kac Type Theorem for ODEs: Solutions of Second Order ODEs as Modes of Diffusions
A Feynman-Kac type result is proved that the solution to a system of second order ordinary differential equations is the mode of a diffusion, defined through the Onsager-Machlup formalism.
Butcher series: A story of rooted trees and numerical methods for evolution equations
Butcher series appear when Runge-Kutta methods for ordinary differential equations are expanded in power series of the step size parameter. Each term in a Butcher series consists of a weighted
Nonexplosion of a class of semilinear equations via branching particle representations
We consider a branching particle system where an individual particle gives birth to a random number of offspring at the place where it dies. The probability distribution of the number of offspring is
depends on the dimension d and power /, where G is the infinitesimal generator of a linear nonnegative contraction semigroup on the space B(Rd) of bounded measurable functions on Rd and c is a
Numerical Methods for Ordinary Differential Equations
and in each case one should label the axes and curves via xlabel, ylabel and legend. One may now experiment with different time spans, initial conditions, and parameters (just a for now). As a second