Monte Carlo Methods in Partial Differential Equations

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Monte Carlo MethodsinPartial Differential
Equations
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Random Walk Methods

• Based on diffusion algorithms
• Applicable to elliptic and parabolic eqns
• Variations
• Random walks on grids
• Brownian motion simulation
• Walk on spheres
• Greens first passage

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Laplaces Equation with Dirichlet Boundary
Conditions
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Finite Difference Approx.

• For the 2D problem via centered difference

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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..

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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 dont know its 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

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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)

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Laplaces Equation with Variable Coefficients

• Consider the problem

This is slightly more complicated, due to the
variable coefficient
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Finite Difference Approx.

• The central difference approximation for this is

• Notice that the coefficients of the us 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

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Can we do better?

• Consider what happens as we let the grid spacing
  go to zero
• For Laplaces 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 Laplaces equation by directly
  simulating Brownian motion but this is slow.
• What else can we do?

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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

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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
  its 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

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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

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Greens Function First Passage

• The transition probability for a surface is
  related to the Greens function for the operator
  and surface under consideration

• If we know the Greens 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 Greens 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

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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

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An Example

• Consider the exterior Dirichlet problem

• The solution to this can be written in integral
  equation form as

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Example cont

• Assume no charges outside of G

• The charge density satisfies

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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.

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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

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My Current Research

• Weve 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
  PDEs?