Monte Carlo Methods
in
Partial Differential Equations
Random Walk Methods
• Based on diffusion algorithms
• Applicable to elliptic and parabolic eqn’s
• Variations:
– Random walks on grids
– Brownian motion simulation
– Walk on spheres
– Green’s first passage
Laplace’s Equation with Dirichlet
Boundary Conditions
 u  0 on G
2
u  f ( x) on G
Finite Difference Approx.
• For the 2D problem via centered difference:
u j ,k  14 [u j 1,k  u j 1,k  u j ,k 1  u j ,k 1 ]
uj-1,k
uj,k-1
uj,k
Uj+1,k
Uj,k+1
Finite Difference Approx.
• How does it work?
– Start at a boundary an iterate over the grid
points until you reach the desired accuracy
• How can we build a Monte Carlo algorithm
from this?
– Notice that each grid point is the average of
its neighbors…..
Randomizing the algorithm
• Notice that the value at each grid point is the average its neighbors,
which are in turn the average of their neighbors, until a boundary
point is found
• Consider the value at one point in the grid.
– Randomly pick a neighbor with probability 0.25. We don’t know it’s
value, but we know it is also the average of its neighbors, so pick one of
those.
– Continue this until a boundary point is picked. The value at this
boundary point is now an estimate of the value of the previous point,
which is an estimate of the previous point, etc, etc, all the way back to
the original point.
– Repeat this N times, add the estimates, then divide by N.
– This is the random walk on the grid method
Markov Chain
•
The walk on the grid is a Markov Chain process and satisfies three
conditions:
– The probability of moving from one grid point to another depends
only on the current grid point, not on any previous moves (Markov
property)
– In order to converge it must satisfy ergodicity:
• The chain can reach any point in the grid from any other point
(irreducibility)
• The chain eventually terminates at a boundary point
(aperiodicity)
– P(x)T(x,y) = P(y)T(y,x) where P(x) is the stationary distribution of the
chain and T(x,y) is the transition probability of going from x to y.
(detailed balance)
Laplace’s Equation with Variable
Coefficients
• Consider the problem:
u xx  sin( x)u yy  0 on G
u  f ( x) on G
This is slightly more complicated, due to the variable coefficient
Finite Difference Approx.
• The central difference approximation for this is:
ui , j 
ui , j 1  ui , j 1  sin x j (ui 1, j  ui 1, j )
2(1  sin x j )
• Notice that the coefficients of the u’s are all positive and sum to 1
• The value at point is now the weighted average of its neighbors
• We can do the same thing we did before, except we now choose each
neighbor with probability equal to its weight
Can we do better?
• Consider what happens as we let the grid spacing go to
zero
– For Laplace’s equation this becomes simple Brownian
motion
– The value of u(x0) becomes the expected value of f at
the boundary point where the particle exits the
domain
• We can solve Laplace’s equation by directly simulating
Brownian motion but this is slow.
• What else can we do?
Walk on Spheres
• Consider a sphere centered around x0
– At some future time the particle has an equal chance
to land at any point on the surface of the sphere.
– If the process is ergodic, it must eventually end up on
the surface of the sphere
• So why simulate Brownian directly?
– Instead, directly simulate the jump from the center to
the surface.
– At each step, we now make much larger time steps
Walk on Spheres
• How large can our sphere be?
– As large as possible. Find the closest boundary point
and make that distance the radius of the sphere
• How do we reach the boundary?
– Introduce an epsilon layer just inside the boundary.
The particle reaches the boundary when it’s inside
this epsilon layer
– The epsilon layer introduces a bias into the estimate,
but we can control the size of the bias by the size of
epsilon
Walk on other shapes
• We are not constrained to walking on
spheres
– Spheres are nice in this case because of the
uniform distribution of exit points
• We can walk instead on any surface that
we can write p(x0,x) for in a convenient
way.
– Usually this is the intersection of simple
shapes
Green’s Function First Passage
• The transition probability for a surface is related to the Green’s
function for the operator and surface under consideration
p( x0 , x)  n G( x0 , x)
• If we know the Green’s function for the domain, we can directly
simulate the transition from the starting point to the boundary
• If the domain can be broken into sub domains where we know the
Green’s function for the sub domain, we can directly simulate the
transition from the starting point to either the boundary or another sub
domain
• For example – a union of spheres
Walk on the Boundary
• The idea:
– Rewrite the PDE as an integral equation
– Using the integral representation, write a
Monte Carlo estimator of the solution that
involves a Markov chain of random walks on
the boundary
An Example
• Consider the exterior Dirichlet problem:
• The solution to this can be written in integral equation
form as:
Example con’t
• Assume no charges outside of G
• The charge density satisfies:
Monte Carlo Estimate
• We can solve this with a walk on the boundary.
• Our transition probability is uniform in solid angle:
• The charge density can now be estimated as the
expected value of this walk on the boundary.
• An estimator for the potential can also be written
directly.
Why use Monte Carlo methods?
•
•
•
•
•
No “curse of dimensionality”
Very fast estimates of point values
Able to handle complex boundaries
No discretization errors
Simple to parallelize
My Current Research
• We’ve seen how we can derive a Monte Carlo method that
estimates a finite difference solution
• Can we do this with other deterministic techniques?
– Finite elements
– Boundary elements
• Can we combine Monte Carlo methods with deterministic
techniques to leverage the strengths of both?
– Split domain into sub domains
– Solve in each sub domain with the appropriate method
– Join the solutions using domain decomposition techniques
• What is the fundamental relationship between deterministic and
Monte Carlo techniques for PDE’s?