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DirichletProblem Class Reference

#include <DirichletProblem.h>

Inheritance diagram for DirichletProblem:

DirichletProblemExample List of all members.

Detailed Description

This class solves the Dirichlet problem for a EuclideanRegion G (in any dimension): for a given function h defined on the boundary of G we seek a function f satisfying

Brownian motion can be used to compute f(x) as follows: let X be a Brownian motion starting at the point x in G. If the function f satisfies then is a martingale and hence , for each stopping time . In particular if is the first exit time from G (equivalently the hitting time for the boundary of G), then since f=h on the boundary of G. Consequently we can compute f(x) as

where is the first exit time from G.

Definition at line 64 of file DirichletProblem.h.

Public Member Functions

virtual Real boundaryFunction (const RealVector &u)=0
 DirichletProblem (EuclideanRegion *D, int T, Real dt=0.01)
Real solution (const RealVector &x, int nPath=50000, bool reportHits=false)

Protected Attributes

int dim
EuclideanRegionG
VectorBrownianMotionX
StoppingTimetau


Constructor & Destructor Documentation

DirichletProblem::DirichletProblem EuclideanRegion   D,
int    T,
Real    dt = 0.01
 

The number T of time steps is fixed in advance to enable a Quasi Monte Carlo computation where this number determines the dimension of the Sobol generator driving the Brownian motion. This number has to be large enough to ensure that the boundary is hit. The solution routine can report what percentage of paths hit the boundary. If this is too small increase T.

Parameters:
D the region on which the problem is solved.
T time steps alloted to hit the boundary.
dt size of time steps.


Member Function Documentation

virtual Real DirichletProblem::boundaryFunction const RealVector   u [pure virtual]
 

The boundary function h.

Implemented in DirichletProblemExample.

Real DirichletProblem::solution const RealVector   x,
int    nPath = 50000,
bool    reportHits = false
 

The solution , each value computed from nPath paths of X.

Parameters:
nPath number of paths launched for the boundary.
reportHits report what percentage of paths hit the boundary.


Member Data Documentation

int DirichletProblem::dim [protected]
 

Definition at line 68 of file DirichletProblem.h.

EuclideanRegion* DirichletProblem::G [protected]
 

Definition at line 69 of file DirichletProblem.h.

VectorBrownianMotion* DirichletProblem::X [protected]
 

Definition at line 70 of file DirichletProblem.h.

StoppingTime* DirichletProblem::tau [protected]
 

Definition at line 71 of file DirichletProblem.h.


The documentation for this class was generated from the following file:
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