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LupoPasini_SIAMCSE15

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RX-Solvers Presentation at SIAM CSE15

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LupoPasini_SIAMCSE15

  1. 1. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Iterative Performance of Monte Carlo Linear Solver Methods Massimiliano Lupo Pasini (Emory University) In collaboration with: Michele Benzi (Emory University) Thomas Evans (Oak Ridge National Laboratory) Steven Hamilton(Oak Ridge National Laboratory) Stuart Slattery(Oak Ridge National Laboratory) SIAM Conference on Computational Science and Engineering 2015 March 17, 2015 Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 1
  2. 2. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Acknowledgments This material is based upon work supported by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research program. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 2
  3. 3. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Outline Motivations Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 3
  4. 4. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Outline Motivations Standard Monte Carlo vs. Monte Carlo Synthetic Acceleration (MCSA) Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 3
  5. 5. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Outline Motivations Standard Monte Carlo vs. Monte Carlo Synthetic Acceleration (MCSA) Choice of the preconditioner Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 3
  6. 6. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Outline Motivations Standard Monte Carlo vs. Monte Carlo Synthetic Acceleration (MCSA) Choice of the preconditioner Convergence conditions Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 3
  7. 7. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Outline Motivations Standard Monte Carlo vs. Monte Carlo Synthetic Acceleration (MCSA) Choice of the preconditioner Convergence conditions Numerical results Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 3
  8. 8. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Motivations Goal: development of reliable algorithms to solve highly dimensional (towards exascale) sparse linear systems in parallel Issues: occurrence of faults (abnormal operating conditions of the computing system which cause a wrong answer) hard faults soft faults [M. Hoemmen and M.A. Heroux, 2011; G. Bronevetsky and B.R. de Supinski, 2008; P. Prata and J.B. Silva, 1999 ] Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 4
  9. 9. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Motivations Resilience: ability to compute a correct output in presence of faults Different approaches to tackle the problem: adaptation of Krylov subspace methods (CG, GMRES, Bi-CGStab, ...) to fault tolerance by recover-restart strategies abandonment of deterministic paradigm in favor of stochastic approaches [E. Agullo, L. Giraud, A. Guernouche, J. Roman and M. Zounon, 2013] Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 5
  10. 10. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Mathematical setting Let us consider a sparse linear system Ax = b, (1) where A ∈ Rn×n, x ∈ Rn, b ∈ Rn. With left preconditioning (1) becomes P−1 Ax = P−1 b, P ∈ Rn×n . (2) (2) can be reinterpreted as a fixed point scheme x = Hx + f, H = I − P−1 A, f = P−1 b. (3) Assuming ρ(H) < 1, (3) generates a sequence of approximate solutions {xk}∞ k=0 which converges to the exact solution of (1). Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 6
  11. 11. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Mathematical setting The solution to (3) can be written in terms of a power series in H (Neumann series): x = ∞ =0 H f. By restricting the attention to a single component of x we have xi = ∞ =0 n k1=1 n k2=1 · · · n k =1 Hk0,k1 Hk1,k2 . . . Hk −1,k fk . (4) (4) can be reinterpreted as the sampling of an estimator defined on a random walk. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 7
  12. 12. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Mathematical setting: Forward (Direct) Method Goal: evaluate a functional such as J(x) = h, x = n i=1 hi xi , h ∈ Rn . Approach: define random walks to evaluate J. Consider a random walk whose state space S is the set of indices of the forcing term f: S = {1, 2, . . . , n} ⊂ N. [N. Metropolis and S. Ulam, 1949; G. E. Forsythe and A. Leibler, 1950; W. R. Wasow, 1952; J. H. Halton, 1962] Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 8
  13. 13. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Mathematical setting: Forward (Direct) Method Initial probability: pick k0 as initial step of the random walk: ˜p(k0 = i) = ˜pk0 = |hi | n i=1|hi | . Possible choices for the transition probability: p(:, i) : S → [0, 1] ∀i ∈ S p(ki = j|ki−1 = i) = Pi,j = |Hi,j | n i=1|Hi,j | weighted Pi,j = 0 if Hi,j = 0 1 #(non-zeros in the row) if Hi,j = 0 uniform Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 9
  14. 14. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Mathematical setting: Forward (Direct) Method A related sequence of weights is defined by wi,j = Hi,j Pi,j in order to build an auxiliary sequence W0 = hk0 ˜pk0 , Wi = Wi−1wki−1,ki ki−1, ki ∈ S. This allows us to define a random variable X(·) : Π → R. Π= set of realizations of a random walk γ defined on S: X(ν) = ∞ =0 W fk , ν = permutation of γ Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 10
  15. 15. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Mathematical setting: Forward (Direct) Method Define the expected value of X(·): E[X] = ν PνX(ν) (Pν = probability of the permutation ν). It can be proved that E[W fk ] = h, H f and consequently E ∞ =0 W fk = h, x . Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 11
  16. 16. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Mathematical setting: Forward (Direct) Method If h is a vector of the standard basis: h = ej ˜p(K0 = i) = δi,j , i, j = 1, . . . , n and the expected value assumes the following form: E ∞ =0 W fk = xi = ∞ =0 n k1=1 n k2=1 · · · n k =1 Pk0,k1 Pk1,k2 . . . Pk −1,k wk0,k1 wk1,k2 . . . wk −1,k fk . (5) Drawback: the estimator is defined entry-wise. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 12
  17. 17. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Mathematical setting: Adjoint Method Consider the adjoint linear system (P−1 A)T y = d. With a fixed point approach, we get y = HT y + d. By picking d = ej we have J∗ (y) := y, f = x, ej = xj , j = 1, . . . , n. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 13
  18. 18. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Mathematical setting: Adjoint Method The initial probability and transition matrix may be defined as ˜pk0 = p(k0 = i) = |fi | n i=1|fi | , Pi,j = |HT i,j | n k=1|HT i,k| = |Hj,i | n k=1|Hk,i | . Reintroduce weights as before: wi,j = Hj,i Pi,j ⇒ W0 = sign(fi ) f 1, Wj = Wj−1wki−1,ki , ki−1, ki ∈ S. Expected value of the estimator for the entire solution vector x: E ∞ =0 W dk = ∞ =0 n k1 n k2 · · · n k fk0 Pk0,k1 Pk1,k2 . . . Pk −1,K wk0,k1 . . . wk −1,k δk ,j . (6) Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 14
  19. 19. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Statistical constraints for convergence Expected value and variance of the estimator must be finite to apply the Central Limit Theorem. Forward Method: (H∗ )i,j = (Hi,j )2 Pi,j E ∞ =0 W fk < ∞ ⇔ ρ(H) < 1 Var ∞ =0 W fk < ∞ ⇔ ρ(H ∗ ) < 1 Adjoint Method: (H∗ )i,j = (Hj,i )2 Pi,j E ∞ =0 W dk < ∞ ⇔ ρ(H) < 1 Var ∞ =0 W dk < ∞ ⇔ ρ(H ∗ ) < 1 [J. H. Halton, 1994; H. Ji, M. Mascagni and Y. Li, 2013] Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 15
  20. 20. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers The choice of the transition probability Theorem (H. Ji, M. Mascagni and Y. Li) Let H ∈ Rn×n, where H ∞ < 1 for the Forward method and H 1 < 1 for the Adjoint method. Consider νk as the realization of a random walk γ truncated at the k-th step. Then, there always exists a transition matrix P such that Var X(νk) → 0 and Var ν X(νk) is bounded as k → ∞. Remark If the hypotheses hold, then Pi,j = |Hi,j | n k=1|Hi,k | (Forward) or Pi,j = |HT i,j | n k=1|HT i,k | (Adjoint) guarantee H∗ ∞ < 1 or H∗ 1 < 1 respectively. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 16
  21. 21. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers First approaches to a parallelization First attempts to apply Monte Carlo as a linear solver considered the fixed point formulation of the original linear system. + embarrassingly parallelizable (ideally we could employ as many processes as the amount of histories) − huge amount of time to compute a reasonably accurate solution E.g. r0 = b − Ax0 ≈ 1 then rk b < 10−m ⇒ Nhistories ≈ 102m because of the Central Limit Theorem. [S. Branford, C. Sahin, A. Thandavan, C. Weihrauch, V. N. Alexandrov and I. T. Dimov, 2008; P. Jakovits, I. Kromonov and S. R. Srirama, 2011] Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 17
  22. 22. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Monte Carlo Synthetic Acceleration (MCSA) Data: A, b, H, f, x0 Result: xnum xl = x0; while not reached convergence do xl+1 2 = Hxl + f; rl+1 2 = b − Axl+1 2 ; δxl+1 2 = (I − H)−1rl+1 2 ; "Solved" with Standard MC xl+1 = xl+1 2 + δxl+1 2 ; end xnum = xl+1 [S. Slattery, 2013 (PhD Thesis); T. M. Evans, S. R. Slattery and P. P. H. Wilson, 2013] Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 18
  23. 23. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Adaptivity for the number of random walks Forward Method: θi ∈ R θi = E ∞ l=0 Wl bkl σi = Var[θi ] Find ˜Ni s.t. σ ˜Ni i |ˆxi | < ε i = 1, . . . , n Adjoint Method: θ ∈ Rn θi = E ∞ l=0 Wl dkl δkl ,i σi = Var[θ]i Find ˜N s.t. σ ˜N 1 ˆx 1 < ε i = 1, . . . , n Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 19
  24. 24. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Applicable preconditioning techniques Remark: explicit knowledge of Hij is needed In fact the entry Pij of the transition matrix is defined in terms of Hij (Forward method) or Hji (Adjoint method). This limits the viable choices of preconditioners: diagonal preconditioners block diagonal preconditioners sparse approximate inverse preconditioners (AINV) Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 20
  25. 25. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Examples Strictly diagonally dominant matrices (s.d.d.) with diagonal preconditioning. A ∈ Rn×n s.d.d. by rows: P = diag(A) H = I − P−1A, H ∞ < 1 (⇔ H∗ ∞ < 1) A ∈ Rn×n s.d.d. by columns: P = diag(A) H = I − AP−1, H 1 < 1 (⇔ H∗ 1 < 1) Strict diag. dominance Forward Method Adjoint method by rows converges not guaranteed by columns not guaranteed converges Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 21
  26. 26. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers Examples Other examples: A M-matrix ⇒ ∃D diagonal s.t. AD is s.d.d. with preconditioner P = block_diag(A):    p j=1 j=i A−1 ii Aij ∞ < 1. ∀i = 1, . . . , p ⇒ Forward method with block Jacobi converges    p i=1 i=j A−1 ii Aij 1 < 1. ∀j = 1, . . . , p ⇒ Adjoint method with block Jacobi converges Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 22
  27. 27. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results 2D parabolic problems: test case 1 Give an open and bounded set Ω ⊂ R2 s.t. Ω = (0, 2)2 (1, 2)2    ∂u ∂t − µ∆u + β(x) · u = 0, x ∈ Ω, t ∈ (0, T] u(x, 0) = u0, x ∈ Ω u(x, t) = 0, x ∈ ΓD, t ∈ (0, T] ∂u ∂n + χu = 0, x ∈ ΓR, t ∈ (0, T] (7) where µ = 3 200 , χ = 3, β(x) = [x, −y]T . ΓR = {{x = 1} × (1, 2)} ∪ {(1, 2) × {y = 1}}, ΓD = ∂Ω ΓR . Discretization (FreeFem++ employed): Triangular FEM spatial discretization step hmax = 0.018, 37,177 d.o.f.’s time discretization step ∆t = h2 max Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 23
  28. 28. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results 2D parabolic problems: test case 1 Preconditioning: sparse factorized AINV with drop tolerance τ = 0.05. Code provided by courtesy by Miroslav Tuma. [M. Benzi and M. Tuma, SISC, 1998] Numerical setting: Adjoint MCSA employed as linear solver relative residual tolerance: ε1 = 10−7 weighted transition probability maximal # steps per history: 10 statistical error-based adaptive parameter: ε2 = 0.01 granularity of the adaptive approach: nhistories = 100 simulations run on laptop Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 24
  29. 29. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results 2D parabolic problems: test case 1 Stiffness matrix: A (n = 37, 177), s.d.d. by rows and columns. AINV preconditioner: M. Drop tolerance τ = 0.05 for both factors. Iteration matrix: H = I − MA. Solution computed just for a generic time step. nnz(H) nnz(A) = 2.984 H 1 = 0.597 ( ρ(H) ≈ 0.286, ρ(H∗) ≈ 0.104 ) # MCSA iterations: 8 relative error= 2.71 · 10−8 # total random walks employed: 90,000 Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 25
  30. 30. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results 2D parabolic problems: test case 2 Give an open and bounded set Ω = (0, 1) × (0, 1)    ∂u ∂t − µ∆u + β(x) · u = 0, x ∈ Ω, t ∈ (0, T] u(x, 0) = u0, x ∈ Ω u(x, t) = 0, x ∈ ∂Ω, t ∈ (0, T] (8) where µ = 3 200 , β(x) = [2, sin(x)]T . Discretization (FreeFem++ employed): Triangular FEM spatial discretization step hmax = 0.014, 40,501 d.o.f.’s time discretization step ∆t = h2 max Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 26
  31. 31. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results 2D parabolic problems: test case 2 Stiffness matrix: A (n = 40, 501), s.d.d. by rows and columns. AINV preconditioner: M. Drop tolerance τ = 0.05 for both factors. Iteration matrix: H = I − MA. Solution computed just for a generic time step. nnz(H) nnz(A) = 3.106 H 1 = 0.185 ( ρ(H) ≈ 0.083, ρ(H∗) ≈ 0.017 ) # MCSA iterations: 6 relative error= 5.148 · 10−8 # total random walks employed: 66,800 Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 27
  32. 32. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results 2D parabolic problems: test case 3 Give an open and bounded set Ω = (0, 1) × (0, 1)    ∂u ∂t − µ∆u + β(x) · u = 0, x ∈ Ω, t ∈ (0, T] u(x, 0) = u0, x ∈ Ω u(x, t) = uD(x), x ∈ ∂Ω, t ∈ (0, T] (9) where µ = 3 200 , β(x) = [2y(1 − x2 ), −2x(1 − y2 )]T , uD = 0 on {{x = 0} × (0, 1)}, {(0, 1) × {y = 0}}, {(0, 1) × {y = 1}}. Discretization (IFISS toolbox employed): Quadrilateral FEM spatial discretization step h = 2−8 ⇒ 66,049 d.o.f.’s time discretization step ∆t = h2 Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 28
  33. 33. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results 2D parabolic problems: test case 3 Stiffness matrix: A (n = 66, 049). Not s.d.d. AINV preconditioner: M. Drop tolerance τ = 0.05 for both factors. Iteration matrix: H = I − MA. Solution computed just for a generic time step. nnz(H) nnz(A) = 4.736 H 1 = 0.212 ( ρ(H) ≈ 0.171, ρ(H∗) ≈ 0.032 ) # MCSA iterations: 8 relative error= 1.7153 · 10−8 # total random walks employed: 88, 400 Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 29
  34. 34. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Conclusions and future developments Conclusions MC solvers motivated by resilience hypotheses for convergence difficult to satisfy in general currently best results obtained using AINV Future developments extension of the set of matrices for which MC solvers is guaranteed to converge a priori analysis of how ρ(H) and ρ(H∗) affect convergence refinement of the adaptive selection of histories Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 30
  35. 35. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Bibliography I I. Dimov, V. Alexandrov, and A. Karaivanova. Parallel resolvent Monte Carlo algorithms for linear algebra problems. Mathematics and Computers in Simulation, 55(55):25–35, 2001. T.M Evans, S.W. Mosher, S.R. Slattery, and S.P. Hamilton. A Monte Carlo synthetic-acceleration method for solving the thermal radiation diffusion equation. Journal of Computational Physics, 258(November 2013):338–358, 2014. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 31
  36. 36. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Bibliography II T.M. Evans, S.R. Slattery, and P.P.H Wilson. Mixed Monte Carlo parallel algorithms for matrix computation. International Conference on Computational Science 2002, 2002. T.M. Evans, S.R. Slattery, and P.P.H Wilson. A spectral analysis of the domain decomposed Monte Carlo method for linear systems. International Conference on Mathematics and Computational Methods Applied to Nuclear Science & Engineering, 2013. J.H. Halton. Sequential Monte Carlo. Mathematical Proceeding of the Cambridge Philosophical Society, 58(1):57–58, 1962. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 32
  37. 37. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Bibliography III J.H. Halton. Sequential Monte Carlo techniques for the solution of linear systems. Journal of Scientific Computing, 9(2):213–257, 1994. J. Hao, M. Mascagni, and L. Yaohang. Convergence analysis of Markov chain Monte Carlo linear solvers using Ulam-von Neumann algorithm. SIAM Journal on Numerical Analysis, 51(4):2107–2122, 2013. N. Metropolis and S. Ulam. The MOnte Carlo ethod. Journal of th American Statistical Association, 44(247):335–341, 1949. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 33
  38. 38. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Bibliography IV S. Slattery. Parallel Monte Carlo Synthetic Acceleration Methods For Discrete Transport Problems. PhD thesis, University of Wisconsin-Madison, 2013. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 34
  39. 39. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Monte Carlo Linear Solvers The choice of the transition probability Theorem (H. Ji, M. Mascagni and Y. Li) Let H ∈ Rn×n with spectral radius ρ(H) < 1. Let H+ ∈ Rn×n where H+ i,j = |Hi,j |. Consider νk as the realization of a random walk γ truncated at the k-th step. If ρ(H+) > 1, there does not exist a transition matrix P such that Var X(νk) converges to zero as k → ∞. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 35
  40. 40. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Simplified PN equations Steady-state, multigroup, one-dimensional, eigenvalue-form of Boltzmann transport equation: µ ∂ψg (x, µ) ∂x + σg (x)ψg (x, µ) = Ng g =1 4π σgg s (x, ˆΩ · ˆΩ )ψg (x, Ω )dΩ + + 1 k Ng g =1 χg 4π 4π νσg f (x)ψg (x, Ω )dΩ . (10) ψg (x, µ) = angular flux for group g σg = total interaction cross section σgg s (x, ˆΩ · ˆΩ ) = scattering cross section from group g → g Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 36
  41. 41. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Simplified PN equations Legendre polynomial spectral discretization (PN equations) consider just odd sets of PN equations removing lower order gradient terms from each equation (SPN equations) − · Dn Un + N+1 2 m=0 AnmUm = 1 k N+1 2 m=1 FnmUm, n = 1, . . . , N + 1 2 (11) Um is a linear combination of moments. Discretized with Finite Volumes Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 37
  42. 42. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Simplified PN equations Sparsity pattern and zoom on the structure of a SP1 matrix Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 38
  43. 43. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Simplified PN equations Properties of matrices: discretization of SP1 equations all nonzero diagonal entries Numerical treatment: block Jacobi preconditioning Kind of matrix size nnz Prec. block size nnz after prec SP1 (a) 18,207 486,438 63 2,644,523 SP1 (b) 19,941 998,631 69 1,774,786 [Matrices provided by courtesy by Thomas Evans, Steven Hamilton, Stuart Slattery, ORNL] Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 39
  44. 44. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Simplified PN equations MCSA parameter setting: Adjoint method residual relative tolerance: ε1 = 10−7 almost optimal transition probability maximal # steps per history: 10 statistical error-based adaptive parameter: ε2 = 0.5 granularity of the adaptive approach: nhistories = 1, 000 simulations run in serial mode Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 40
  45. 45. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Simplified PN equations matrix ρ(H) ρ(H∗) relative err. # iterations SP1 (a) 0.9779 0.9758 9.97 · 10−6 340 SP1 (b) 0.9798 0.9439 3.89 · 10−5 209 Figure 1 : # histories for (a) at each iteration. Figure 2 : # histories for (b) at each iteration. Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 41
  46. 46. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Simplified PN equations Issues raised for a general SPN matrix: hard to find a block Jacobi preconditioner s.t. ρ(H∗) < 1 for every SPN problem attempt to use Approximate Inverse preconditioners was not effective for sparse preconditioners ρ(H∗ ) < 1 is not respected attempt to use ILU preconditioners was not effective for ILU(0) we got ρ(H∗ ) > 1 massive fill-in with ILUT reordering and scaling did not facilitate the convergence requirements Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 42
  47. 47. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Diagonally dominant matrices Set of matrices obtained by a diagonal shift of matrices associated with SP1, SP3 and SP5 equations. (A + sD), s ∈ R+ , D = diag(A). All the matrices have been turned into s.d.d. by columns. Initial matrix s ρ(H) Forward − ρ(H∗) Adjoint − ρ(H∗) SP1 (a) 0.3 0.7597 0.7441 0.6983 SP1 (b) 0.4 0.7046 1.1448 0.5680 SP3 0.9 0.5869 0.4426 0.3727 SP5 1.6 0.5477 0.3790 0.3431 Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 43
  48. 48. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Diagonally dominant matrices (A + sD), s ∈ R+ , D = diag(A). matrix nnz s relative err. # iterations SP1 (a) 486,438 0.3 6.277 · 10−7 36 SP1 (b) 998,631 0.4 9.885 · 10−7 21 SP3 846,549 0.9 5.919 · 10−7 18 SP5 1,399,134 1.6 4.031 · 10−7 19 Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 44
  49. 49. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Diagonal shift Strict diagonally dominance is a sufficient condition for convergence but not necessary. (A + sD), s ∈ R+ , D = diag(A) such that ρ(H) < 1 and ρ(H∗) < 1. Initial matrix s ρ(H) Forward − ρ(H∗) Adjoint − ρ(H∗) SP1 (a) 0.2 0.8230 0.8733 0.8195 SP1 (b) 0.2 0.8220 1.5582 0.7731 SP3 0.3 0.8126 0.9459 0.7961 SP5 0.7 0.8376 0.8865 0.8026 Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 45
  50. 50. Motivations Standard MC vs. MCSA Choice of preconditioners Convergence conditions Numerical results Numerical experiments Diagonal shift (A + sD), s ∈ R+ , D = diag(A). matrix s relative err. # iterations SP1 (a) 0.2 6.394 · 10−7 48 SP1 (b) 0.2 2.59 · 10−6 36 SP3 0.3 5.35 · 10−7 45 SP5 0.7 4.21 · 10−7 70 Massimiliano Lupo Pasini SIAM Conference on Computational Science and Engineering 2015 MCSA 46

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