Lesson 6: Solution of Integral Equations

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Lesson 6 Solution of Integral Equations

• Application of our integration techniques to
  integral equations
• Introduction of Dirac notation
• Conversion of differential equations to integral
  equations
• Solution of integral equations
• Solution of linked equations
• Neumann self-linked equations

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Dirac notation

• In our integrations so far, I have simplified the
  mathematics a bit by always selecting a I was
  careful to always choose x between a and b.

• I was careful to always choose x between a and b.
  What if I had not done this?

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Dirac notation (2)

• A more rigorous way to approach this is to look
  at the Monte Carlo attack of the integral in TWO
  steps
• (1) an approximation of f(x) itself using
• (2) a substitution of this functional
  approximation into the integral

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Dirac notation (3)
This is the approach we will take from now on.
The notation has the advantage of giving
us not only the weight but also reminding us of
the selected point. This way we can think of a
sample as having these two piecesa weight
and a location
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Developing integral equations from differential
equations Simple

• We now know how to attack integrals with Monte
  Carlo
• We desire to be able to solve differential
  equations estimate functionals (usually
  integrals or point values) of the function that
  solves a given equation
• Traditional solution Convert them into integral
  equations and apply the MC integration rules to
  them
• Example Find the value of f(4), given the
  differential equation and boundary condition

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Simple integral equations (2)

• Answer We can integrate from 0 (the known value)
  to the desired value to get
• Now we apply one of the four integration methods
  to the integral in the equation

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Simple integral equations (2)

• NOTE From now on, I will skip the summation and
  division by N and just write

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Simple integral equations (3)

• The normal procedure for this method is to
• Choose a value of between a and b using a
  probability distribution p(x) (of YOUR choosing).
  
• Score
• So, lets do it.
• What PDF should we use?
• Lazy mans PDF uniform
• Optimum PDF ? (You tell me)

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Linked equations

• When you are faced with linked equation sets, the
  principles are the same, put you have to be more
  careful
• Putting in multiple boundary conditions
• Keeping up with multiple sampled variables (each
  equation will have one)
• Most tricky Realizing and adapting to CHANGING
  LIMITS on the integrals (after the first)
• MUCH more difficult to optimize the choice of the
  PDFs used

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Linked equation example

• Example Find f(2) for the second order
  differential equation
• In order to make it fit the category, we will
  start be re-writing as the linked set

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Linked equation example (2)

• Applying our tools to the second equation first,
  we begin by transforming it into an integral
  equation for the value at x2
• Using our MC integration approximation, we get
• How do we get the ? Answer We estimate
  it from the other equation.

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Linked equation example (3)

• Applying our tools to the first equation first,
  we begin by transforming it into an integral
  equation for the value at
• The resulting procedure is
• Choose a value of using
• Choose a value of using
• Score

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Linked equation example (4)

• Now lets do it.
• What PDFs to use?
• Flat
• Better than flat

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Sampling from recurring equations Neumann series

• Sampling from recurring equations introduces a
  complexity We cannot use the above procedure
  because, the procedure requires that we sample
  from f(x) on the right-hand side in order to
  sample from f(x) on the left-hand side.
• However, for linear occurrences of f(x) on the
  right-hand side, we can "bootstrap" a solution by
  representing f(x) as an infinite Neumann series
• on BOTH sides of the equation and properly
  lining up terms.

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Neumann series (2)

• If we have the general linear recurring integral
  equation
• with known source term q(x) and linear
  operator K(x,x), we can substitute to get
• We can line up the left hand and right hand
  terms in the following way

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Neumann series (3)

• Obviously, the sum of the solutions of these
  coupled equations obeys the original equation.
• We solve them sequentially, eliminating the
  circular dependence
• Of course, this procedure has an infinite number
  of steps for each sample of , so it will have to
  be truncated somehow, but -- before worrying
  about that -- let us first look at an example.

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Neumann series (4)

• Example Develop an infinite sampling procedure
  for the recurring equation
• Answer Integrating the differential equation
  over x from 0 to x (and applying the boundary
  condition) gives us the recurring integral
  equation

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Neumann series (5)

• If we insert the infinite Neumann series for the
  function on both sides, we get the following
  coupled equations

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Neumann series (6)

• Since the function f(x) is the infinite sum of
  these, the procedure to sample from is
• 1. Sample from using
• 2. Sample from using the above sample

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Neumann series (7)

• 3. Sample from using the above sample
• Sample from using the above sample

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Neumann series (8)

• Observations
• The procedure is infinite in theory, but not
  infinite in practice because as soon as we pick a
  value of x that is SMALLER than the one before
  it, then the weight will go to zero. Once this
  happens, of course, we can ignore the higher
  order f's because they will be zero as well.
  What else could we do to terminate the sequence?
• We must remember that it is not a single sample
  of f0(x) or f1(x) , etc., that constitutes our
  sample of the function, but ALL OF THEM together.
  Therefore, the i'th sample of f(x) is, formally
• Note We do NOT divide by the number of
  contributions to the ith sample

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Neumann series (9)

• Observations
• Therefore, if we improve our approximation by
  taking N samples, the combined best result would
  be
• As a practical matter, point 2 means that our
  coding must collect data in "sample bins" -- i.e,
  which collect data from individual Neumann terms
  within a single sample -- and, at the end of the
  sample, contribute from the "sample bins" to the
  overall "solution bins".

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Convergence of recurring equations

• You should be aware that there is a good
  possibility that a straight-forward application
  of the procedure for recurring equations will
  result in a divergent procedure
• Convergence is guaranteed only if the eigenvalues
  of the recurrence operator K in
• have magnitude less than one.

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Tallies

• All of this discussion has been focused on
  sampling a function at a point.
• BUT, it is more common for us to be interested in
  INTEGRALS of the functionse.g., reaction rates
  in a cell.
• These integrals are referred to as tallies and
  most of them represent a physical or mathematical
  value we want to know
• In our current sampling strategy, our samples
  include Dirac deltas, so our only choice for
  these tallies is to create integral tallies using
  the samples

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Tallies (2)

• Although tallies are not mathematically necessary
  (one could keep the and for later
  use), almost all MC codes use them to save
  storage.

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Homework P-6

• Analyze, develop a procedure, and implement to
  solve the following differential equations
• 1.
• 2. Solve as a linked set (2 equations)

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Homework P-6

• You have an initial inventory of one isotope of
  type A that decays to isotope B with a decay
  constant of lA1.0 sec-1.
• B decays to C with a decay constant of lB0.5
  sec-1.
• Write and run a MC program to determine much of C
  (per initial A atom) will be present after 3
  seconds.
• Assume there is initially no B or C.

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Homework P-6 (contd)

• I will help you get started. The equations are
• Your job is to find good (or bad, your choice)
  pdfs, work out the algebra, and code it to
  answer find how much C you have at 3 seconds.