1. 1 Lesson 9: Solution of Integral Equations & Integral Boltzmann Transport Eqn Neumann self-linked equations Neumann self-linked equations Attacking the integral Boltzmann Transport Equation with our new methodology. Attacking the integral Boltzmann Transport Equation with our new methodology. Presentation of the integral equation Presentation of the integral equation Reformulation into Neumann Reformulation into Neumann Breaking it into convenient “event” pieces Breaking it into convenient “event” pieces MC attack of Neumann linked equations MC attack of Neumann linked equations Event-based tallies Event-based tallies

2. 2 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. 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: 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.

3. 3 Neumann series (2) If we have the general linear recurring integral equation: 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: We can “line up” the left hand and right hand terms in the following way:

4. 4 Neumann series (3) Obviously, the sum of the solutions of these coupled equations obeys the original equation. Obviously, the sum of the solutions of these coupled equations obeys the original equation. We solve them sequentially, eliminating the circular dependence 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. 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.

5. 5 Neumann series (4) Example: Develop an infinite sampling procedure for the recurring equation: 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: Answer: Integrating the differential equation over x from 0 to x (and applying the boundary condition) gives us the recurring integral equation:

6. 6 Neumann series (5) If we insert the infinite Neumann series for the function on both sides, we get the following coupled equations: If we insert the infinite Neumann series for the function on both sides, we get the following coupled equations:

7. 7 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:

8. 8 Neumann series (7) 3. Sample from using the above sample: …Sample from using the above sample:

9. 9 Neumann series (8) Observations: 1. 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? 2. We must remember that it is not a single sample of f 0 (x) or f 1 (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 i th sample

10. 10 Neumann series (9) Observations: 3. 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". 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".

11. 11 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 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: Convergence is guaranteed only if the eigenvalues of the recurrence operator K in: have magnitude less than one. have magnitude less than one.

12. 12 Tallies All of this discussion has been focused on sampling a function at a point. 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 functions— e.g., reaction rates in a cell. BUT, it is more common for us to be interested in INTEGRALS of the functions— e.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 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: 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:

13. 13 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. Although tallies are not mathematically necessary (one could keep the and for later use), almost all MC codes use them to save storage.

14. 14 Presentation of Boltzmann Equation The integral form of the Boltzmann Transport Equation: The integral form of the Boltzmann Transport Equation: NOTE: Throughout this lecture, to simplify the notation, I DO NOT use any vector symbols. You need to remember that and are position and direction, respectively. NOTE: Throughout this lecture, to simplify the notation, I DO NOT use any vector symbols. You need to remember that and are position and direction, respectively.

15. 15 Reformulation into Neumann Since flux appears on the right hand side as well as the left, we have the same “bootstrap” problem as the traditional recurrence operator equation—No place to start. Since flux appears on the right hand side as well as the left, we have the same “bootstrap” problem as the traditional recurrence operator equation—No place to start. We solve it the same way: By creating a Neumann sequence that begins with the equation that has a RHS source term that does NOT depend on either variable and then building from there. We solve it the same way: By creating a Neumann sequence that begins with the equation that has a RHS source term that does NOT depend on either variable and then building from there.

16. 16 Reformulation into Neumann (2) We get: We get: Note: It is handy to remember that the Neumann superscript corresponds to how many times the particle has collided Note: It is handy to remember that the Neumann superscript corresponds to how many times the particle has collided

17. 17 MC attack of Neumann linked equations So far, of course, we have done no Monte Carlo. We have only created a Neumann sequence for the Boltzmann Transport Equation. So far, of course, we have done no Monte Carlo. We have only created a Neumann sequence for the Boltzmann Transport Equation. We could, if we wanted to (and knew how), solve it deterministically. We could, if we wanted to (and knew how), solve it deterministically. Now we have to approximate this Neumann sequence one term at a time using the Monte Carlo tools that we have learned Now we have to approximate this Neumann sequence one term at a time using the Monte Carlo tools that we have learned

18. 18 Application of MC methodology Recall that the essence of our formal MC methodology is to replace functions of continuous variables with function that contain stochastic variables. Recall that the essence of our formal MC methodology is to replace functions of continuous variables with function that contain stochastic variables. For multidimensional Dirac approximation: For multidimensional Dirac approximation: Let us apply this to the Neumann coupled equation set Let us apply this to the Neumann coupled equation set

19. 19 Application of MC methodology The steps for this are straightforward, but fraught with tedious opportunities for error! The steps for this are straightforward, but fraught with tedious opportunities for error! For each coupled equation from Neumann: For each coupled equation from Neumann: 1. Create the formal MC approximation weight definition with the function (LHS of Neumann Eq evaluated at chosen values of independent variables) in the numerator and pdf’s for all of the variables in the denominator (with any variable NOT changing having a Dirac distribution at current value as its pdf) 2. Replace the numerator with the RHS equivalent from the Neumann Eq. 3. Replace any previously-MC-approximated term in the numerator with the MC approximation 4. Work out the resulting definition of the weight (with the pdf’s unspecified) 5. (Just for interest) Plug in some typical (e.g., physical) pdfs to see what the weights do.

20. 20 0 th order term We start with: We start with: Since there is an integral we have to sample something inside. The obvious choice is the S(..) term, which we sample using: Since there is an integral we have to sample something inside. The obvious choice is the S(..) term, which we sample using: where where

21. 21 0 th order term Substituting this gives us: Substituting this gives us: We still have an integral in R, so we sample the exponential: We still have an integral in R, so we sample the exponential: And substitute to give us: And substitute to give us:

22. 22 0 th order term where: where: You may wonder why the total cross section isn’t just incorporated into the weight, You may wonder why the total cross section isn’t just incorporated into the weight, The answer is just that it is traditional to not do this, so that analog procedures will keep the weight at 1.00 The answer is just that it is traditional to not do this, so that analog procedures will keep the weight at 1.00 (You may have noticed how well the numerators fit the “traditional” or “natural” PDFs we have studied (You may have noticed how well the numerators fit the “traditional” or “natural” PDFs we have studied

23. 23 i+1 st term For the other Neumann term(s), we use an inductive procedure where we assume that we have a MC approximation for the ith flux: For the other Neumann term(s), we use an inductive procedure where we assume that we have a MC approximation for the ith flux: and substitute it into the i+1 st equation to get: and substitute it into the i+1 st equation to get:

24. 24 i+1 st term (cont’d) Which reduces to: Which reduces to: This is still continuous in direction and energy, so we sample these variables (through the scattering cross section) to get: This is still continuous in direction and energy, so we sample these variables (through the scattering cross section) to get:

25. 25 i+1 st term (cont’d) Again, the one remaining integral is dealt with by sampling R: Again, the one remaining integral is dealt with by sampling R: which we substitute to get: which we substitute to get: where where

26. 26 Putting it together Putting the pieces together, we get: Putting the pieces together, we get: which is a sample of a SINGLE history (through its N scatters) which is a sample of a SINGLE history (through its N scatters)

27. 27 Putting it together (2) As we discussed before, we can deal with the infinite summation BY: As we discussed before, we can deal with the infinite summation BY: 1. Recognizing (from the fact that each weight is a multiple of the previous one) that once a weight goes to 0, all remaining weights need not be computed; and 2. Forcing a weight to zero (with non-zero probability) using one of two techniques: A. Define the PDFs so that there is a non-zero probability that the NUMERATOR at the chosen point will be zero; or B. Introduce an artificial “choice” with non-zero probability of going to zero. Russian Roulette works well (and can be introduced anywhere you want it):

28. 28 From general to specific MC algorithm Since the previous equations for the weights include unspecified PDFs, we can convert them into a programmable SPECIFIC algorithm by choosing the PDFs Since the previous equations for the weights include unspecified PDFs, we can convert them into a programmable SPECIFIC algorithm by choosing the PDFs Each pdf used must obey three rules: Each pdf used must obey three rules: 1. The pdf must be non-negative. 2. The integral of the pdf over its selection domain must be 1. (Integration of a function over the complete problem domain will be denoted as.) 3. The pdf must be non-zero for all values of its selection domain for which a non-zero contribution to any tally is possible.

29. 29 Analog specific algorithm To tie us into the previous event-based development, we next develop the “analog” specific algorithm: To tie us into the previous event-based development, we next develop the “analog” specific algorithm: One can either interpret the word “analog” in it physical sense: Following nature’s rules or in the Monte Carlo sense of “keeping the weights equal to 1” One can either interpret the word “analog” in it physical sense: Following nature’s rules or in the Monte Carlo sense of “keeping the weights equal to 1” To get started we bring together the three weight equations from previous slides: To get started we bring together the three weight equations from previous slides:

30. 30 Analog specific algorithm The analog approach starts off by making the denominators equal the numerators (with proper normalization), that is: The analog approach starts off by making the denominators equal the numerators (with proper normalization), that is:

31. 31 Analog specific algorithm Substituting these gives us the resulting weights of: Substituting these gives us the resulting weights of: Since it is our desire in “analog” simulations to always be following particles with weight of 1, we “fix” these by the setting the initial particle weight to 1 (leaving it to the user or coder to multiply times the total source strength) and use: Since it is our desire in “analog” simulations to always be following particles with weight of 1, we “fix” these by the setting the initial particle weight to 1 (leaving it to the user or coder to multiply times the total source strength) and use:

32. 32 Cell-averaged tallies Once we have a MC estimate of the flux, then the MC estimate integral tallies are just a substitution Once we have a MC estimate of the flux, then the MC estimate integral tallies are just a substitution This is most easily shown for a cell average tally where: This is most easily shown for a cell average tally where: Substituting Substituting

33. 33 Homework

34. 34 Homework