3,971 research outputs found
Multilevel Monte Carlo for Random Degenerate Scalar Convection Diffusion Equation
We consider the numerical solution of scalar, nonlinear degenerate
convection-diffusion problems with random diffusion coefficient and with random
flux functions. Building on recent results on the existence, uniqueness and
continuous dependence of weak solutions on data in the deterministic case, we
develop a definition of random entropy solution. We establish existence,
uniqueness, measurability and integrability results for these random entropy
solutions, generalizing \cite{Mishr478,MishSch10a} to possibly degenerate
hyperbolic-parabolic problems with random data. We next address the numerical
approximation of random entropy solutions, specifically the approximation of
the deterministic first and second order statistics. To this end, we consider
explicit and implicit time discretization and Finite Difference methods in
space, and single as well as Multi-Level Monte-Carlo methods to sample the
statistics. We establish convergence rate estimates with respect to the
discretization parameters, as well as with respect to the overall work,
indicating substantial gains in efficiency are afforded under realistic
regularity assumptions by the use of the Multi-Level Monte-Carlo method.
Numerical experiments are presented which confirm the theoretical convergence
estimates.Comment: 24 Page
An approximation scheme for semilinear parabolic PDEs with convex and coercive Hamiltonians
We propose an approximation scheme for a class of semilinear parabolic
equations that are convex and coercive in their gradients. Such equations arise
often in pricing and portfolio management in incomplete markets and, more
broadly, are directly connected to the representation of solutions to backward
stochastic differential equations. The proposed scheme is based on splitting
the equation in two parts, the first corresponding to a linear parabolic
equation and the second to a Hamilton-Jacobi equation. The solutions of these
two equations are approximated using, respectively, the Feynman-Kac and the
Hopf-Lax formulae. We establish the convergence of the scheme and determine the
convergence rate, combining Krylov's shaking coefficients technique and
Barles-Jakobsen's optimal switching approximation.Comment: 24 page
Exponential Runge-Kutta methods for stiff kinetic equations
We introduce a class of exponential Runge-Kutta integration methods for
kinetic equations. The methods are based on a decomposition of the collision
operator into an equilibrium and a non equilibrium part and are exact for
relaxation operators of BGK type. For Boltzmann type kinetic equations they
work uniformly for a wide range of relaxation times and avoid the solution of
nonlinear systems of equations even in stiff regimes. We give sufficient
conditions in order that such methods are unconditionally asymptotically stable
and asymptotic preserving. Such stability properties are essential to guarantee
the correct asymptotic behavior for small relaxation times. The methods also
offer favorable properties such as nonnegativity of the solution and entropy
inequality. For this reason, as we will show, the methods are suitable both for
deterministic as well as probabilistic numerical techniques
A 3D radiative transfer framework: I. non-local operator splitting and continuum scattering problems
We describe a highly flexible framework to solve 3D radiation transfer
problems in scattering dominated environments based on a long characteristics
piece-wise parabolic formal solution and an operator splitting method. We find
that the linear systems are efficiently solved with iterative solvers such as
Gauss-Seidel and Jordan techniques. We use a sphere-in-a-box test model to
compare the 3D results to 1D solutions in order to assess the accuracy of the
method. We have implemented the method for static media, however, it can be
used to solve problems in the Eulerian-frame for media with low velocity
fields.Comment: A&A, in press. 14 pages, 19 figures. Full resolution figures
available at ftp://phoenix.hs.uni-hamburg.de/preprints/3DRT_paper1.pdf HTML
version (low res figures) at
http://hobbes.hs.uni-hamburg.de/~yeti/PAPERS/3drt_paper1/index.htm
Regularity theory for nonlinear systems of SPDEs
We consider systems of stochastic evolutionary equations of the type
where is a non-linear
operator, for instance the -Laplacian with and grows linearly.
We extend known results about the deterministic problem to the stochastic
situation. First we verify the natural regularity: where
. If we have Uhlenbeck-structure then
is finite for all
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
- …