An efficient algorithm for solving elliptic problems on percolation clusters

Transcription

1 An efficient algorithm for solving elliptic problems on percolation clusters Chenlin GU DMA/ENS, PSL Research University Les probabilités de demain May 13, 2020 Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

2 Motivation Outline for section 1 1 Motivation 2 Preliminary Monte-Carlo Markov chain Jacobi iterative method Homogenization 3 Algorithm 4 Numerical experience 5 Percolation setting Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

3 Motivation Random conductance model (Z d, E d ) standard d-dimension lattice, Q r := ( r 2, 2) r d Z d. a : E d [Λ 1, Λ] with Λ > 1. {a(e)} e Ed i.i.d. called random conductances. a is the discrete divergence operator defined by a u(x) := a(x, y)(u(x) u(y)). y x Object: Find an algorithm to solve the elliptic Dirichlet problem quickly for big r, { a u = f in int(qr ), u = g on Q r. (1.1) Motivation: Modelisation in disordered medium and heterogeneous material. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

4 Motivation Random stationary medium Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

5 Motivation Question: What is the challenge in this problem? Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

6 Preliminary Outline for section 2 1 Motivation 2 Preliminary Monte-Carlo Markov chain Jacobi iterative method Homogenization 3 Algorithm 4 Numerical experience 5 Percolation setting Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

7 Preliminary Recap of some classical methods Let us recall some classical method to solve this problem: Monte-Carlo Markov chain. Jacobi iterative method.. Homogenization. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

8 Preliminary Monte-Carlo Markov chain Outline 1 Motivation 2 Preliminary Monte-Carlo Markov chain Jacobi iterative method Homogenization 3 Algorithm 4 Numerical experience 5 Percolation setting Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

9 Preliminary Monte-Carlo Markov chain MCMC MCMC For the case f = 0, the solution of Dirichlet problem is u(x) = E x [g(x τ )] for (X n ) n 0 the Markov chain associated to the operator a and τ the hitting time of the boundary. Advantages: Dimension free, easy to program. Disadvantages: It takes time if we want u(x) for all x int(q r ). Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

10 Preliminary Monte-Carlo Markov chain MCMC MCMC For the case f = 0, the solution of Dirichlet problem is u(x) = E x [g(x τ )] for (X n ) n 0 the Markov chain associated to the operator a and τ the hitting time of the boundary. Advantages: Dimension free, easy to program. Disadvantages: It takes time if we want u(x) for all x int(q r ). Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

11 Preliminary Monte-Carlo Markov chain MCMC MCMC For the case f = 0, the solution of Dirichlet problem is u(x) = E x [g(x τ )] for (X n ) n 0 the Markov chain associated to the operator a and τ the hitting time of the boundary. Advantages: Dimension free, easy to program. Disadvantages: It takes time if we want u(x) for all x int(q r ). Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

12 Preliminary Jacobi iterative method Outline 1 Motivation 2 Preliminary Monte-Carlo Markov chain Jacobi iterative method Homogenization 3 Algorithm 4 Numerical experience 5 Percolation setting Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

13 Preliminary Jacobi iterative method Jacobi iterative method Jacobi iteration Jacobi iterative method = iteration of semigroup. a(x,y) P (x, y) := a(x,z), f(x) = f(x)/( z x a(x, z)). z x We do iteration u 0 = g, u n+1 = J(u n, f) J(u n, f) := P u n + f. (2.1) u is the unique solution of the equation u = P u + f, lim n u n = u. Advantages: Easy to program and converges exponentially. Disadvantages: It takes time when Q r is big: every iteration is a contraction (1 1 r 2 ), so at least O(r 2 ) iterations. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

14 Preliminary Jacobi iterative method Jacobi iterative method Jacobi iteration Jacobi iterative method = iteration of semigroup. a(x,y) P (x, y) := a(x,z), f(x) = f(x)/( z x a(x, z)). z x We do iteration u 0 = g, u n+1 = J(u n, f) J(u n, f) := P u n + f. (2.1) u is the unique solution of the equation u = P u + f, lim n u n = u. Advantages: Easy to program and converges exponentially. Disadvantages: It takes time when Q r is big: every iteration is a contraction (1 1 r 2 ), so at least O(r 2 ) iterations. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

15 Preliminary Jacobi iterative method Jacobi iterative method Jacobi iteration Jacobi iterative method = iteration of semigroup. a(x,y) P (x, y) := a(x,z), f(x) = f(x)/( z x a(x, z)). z x We do iteration u 0 = g, u n+1 = J(u n, f) J(u n, f) := P u n + f. (2.1) u is the unique solution of the equation u = P u + f, lim n u n = u. Advantages: Easy to program and converges exponentially. Disadvantages: It takes time when Q r is big: every iteration is a contraction (1 1 r 2 ), so at least O(r 2 ) iterations. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

16 Preliminary Outline 1 Motivation 2 Preliminary Monte-Carlo Markov chain Jacobi iterative method Homogenization 3 Algorithm 4 Numerical experience 5 Percolation setting Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

17 Preliminary Efficient method for a const. i.e. for the problem u = f. 1 Try to solve u = f, we do the Jacobi iteration and u 1 = J M (u 0, f). 2 f 1 = f ( u 1 ), coarsen the grid by 2, and u 2 = J M/2 (0, f 1 ). 3 f 2 = f 1 ( u 2 ), coarsen the grid by 2, and u 3 = J M/4 (0, f 2 ). 4 û = u 1 + u 2 + u 3. Iterate this procedure. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

18 Preliminary Efficient method for a const. i.e. for the problem u = f. 1 Try to solve u = f, we do the Jacobi iteration and u 1 = J M (u 0, f). 2 f 1 = f ( u 1 ), coarsen the grid by 2, and u 2 = J M/2 (0, f 1 ). 3 f 2 = f 1 ( u 2 ), coarsen the grid by 2, and u 3 = J M/4 (0, f 2 ). 4 û = u 1 + u 2 + u 3. Iterate this procedure. Advantages: Only O(log(r)) iterations are required in Q r. Disadvantages: a has to be constant. Probabilistic interpretation: coarsened grid random walk with big step size. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

19 Preliminary Efficient method for a const. i.e. for the problem u = f. 1 Try to solve u = f, we do the Jacobi iteration and u 1 = J M (u 0, f). 2 f 1 = f ( u 1 ), coarsen the grid by 2, and u 2 = J M/2 (0, f 1 ). 3 f 2 = f 1 ( u 2 ), coarsen the grid by 2, and u 3 = J M/4 (0, f 2 ). 4 û = u 1 + u 2 + u 3. Iterate this procedure. Advantages: Only O(log(r)) iterations are required in Q r. Disadvantages: a has to be constant. Probabilistic interpretation: coarsened grid random walk with big step size. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

20 Preliminary Efficient method for a const. i.e. for the problem u = f. 1 Try to solve u = f, we do the Jacobi iteration and u 1 = J M (u 0, f). 2 f 1 = f ( u 1 ), coarsen the grid by 2, and u 2 = J M/2 (0, f 1 ). 3 f 2 = f 1 ( u 2 ), coarsen the grid by 2, and u 3 = J M/4 (0, f 2 ). 4 û = u 1 + u 2 + u 3. Iterate this procedure. Advantages: Only O(log(r)) iterations are required in Q r. Disadvantages: a has to be constant. Probabilistic interpretation: coarsened grid random walk with big step size. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

21 Preliminary Homogenization Outline 1 Motivation 2 Preliminary Monte-Carlo Markov chain Jacobi iterative method Homogenization 3 Algorithm 4 Numerical experience 5 Percolation setting Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

22 Preliminary Homogenization Homogenized solution Homogenized solution For r very big, one can use an effective conductance ā which is a constant matrix and the homogenized solution as an approximation { ā ū = f in int(qr ), (2.2) u = g on Q r, then we have u ū L 2 (Q r) := 1 Q r u(x) ū(x) 2 x Q r 1 2 o(r). ā E[a] and can be solved much more quicker. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

23 Preliminary Homogenization Homogenized solution Early work of Kozlov, Papanicolaou, Varadhan, Yurinski etc. Advantages: It is as fast as multigrid method. Disadvantages: Loss of microscale information: Locally, ū and u is never similar. Limit of precision: For r fixed, 1 r u ū L 2 (Q r) r 1 2 cannot be better. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

24 Preliminary Homogenization Homogenized solution Early work of Kozlov, Papanicolaou, Varadhan, Yurinski etc. Advantages: It is as fast as multigrid method. Disadvantages: Loss of microscale information: Locally, ū and u is never similar. Limit of precision: For r fixed, 1 r u ū L 2 (Q r) r 1 2 cannot be better. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

25 Preliminary Homogenization Homogenized solution Early work of Kozlov, Papanicolaou, Varadhan, Yurinski etc. Advantages: It is as fast as multigrid method. Disadvantages: Loss of microscale information: Locally, ū and u is never similar. Limit of precision: For r fixed, 1 r u ū L 2 (Q r) r 1 2 cannot be better. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

26 Algorithm Outline for section 3 1 Motivation 2 Preliminary Monte-Carlo Markov chain Jacobi iterative method Homogenization 3 Algorithm 4 Numerical experience 5 Percolation setting Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

27 Algorithm Object of the algorithm Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

28 Algorithm Iterative algorithm An iterative algorithm Initial guess u 0 := g. Solve the following equations with the null Dirichlet boundary condition: (λ 2 a )u 1 = f + a u 0 in int(q r ), ā ū = λ 2 u 1 in int(q r ), (λ 2 a )u 2 = (λ 2 ā )ū in int(q r ), û := u 0 + u 1 + u 2 and we put û in the place of u 0 to restart the iteration. (3.1) All the three equations are easy to solve. The second one can be handled by the multigrid method, while the first and third one take less time (O( 1 λ 2 ) iterations) thanks to the regularization. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

29 Algorithm Main theorem Z = sup u0,f,g (û u) L 2 (Qr) (u 0 u) L 2 (Qr). l(λ) = 1 for d 3 and l(λ) = log 1 2 (λ) for d = 2. Theorem ((Armstrong, Hannukainen, Kuusi, Mourrat 18)(Gu 19)) For any s (0, 2), there exists a constant C(Λ, s, d) such that for any y > 0 ( ( ) s ) y P[Z y] exp. Cλ 1 2 l(λ) log 1 s (r) It suggests a practical choice of λ that 1 r λ 1 log 1 2 (r). Complexity := O(log(r)) iterations, very close to the one of multigrid. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

30 Algorithm Idea of proof Combing the first and second step of the iteration and we obtain ā ū = a (u u 0 u 1 ). The third equation gives (λ 2 a )u 2 = (λ 2 ā )ū. The first order corrector: {φ ek } 1 k d : a (l ek + φ ek ) = 0 in Z d. Two-scale expansion w := ū + d k=1 (D ek ū)φ ek. Key point: (w u) L 2 (Q r) o(1). We have û u = u (u 0 + u 1 + u 2 ) (u u 0 u 1 ) w + w u 2, so it suffices to know how close the two-scale expansion can be. First rigorous result in periodic homogenization : Allaire. Quantitative analysis in stochastic homogenization setting: Armstrong, Kuusi, Mourrat, Smart, Gloria, Neukamm and Otto etc. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

31 Numerical experience Outline for section 4 1 Motivation 2 Preliminary Monte-Carlo Markov chain Jacobi iterative method Homogenization 3 Algorithm 4 Numerical experience 5 Percolation setting Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

32 Numerical experience Numerical experience d = 2, size = , a { 1 2, 2} with law Bernoulli( 1 2 ). f = 1 and g = 0. λ = 0.1. The first 22 rounds of iteration give a convergence of errors ε n := f ( a u n ) L 2 (Q r). {ε n } 1 n 22 = {34.43, 18.56, 9.99, 5.38, 2.89, 1.56, 0.84, 0.45, 0.24, 0.13, , , , , , , , , , , , } Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

33 Numerical experience Figure: A numerical experience of the algorithm gives a very high precision of the solution. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

34 Percolation setting Outline for section 5 1 Motivation 2 Preliminary Monte-Carlo Markov chain Jacobi iterative method Homogenization 3 Algorithm 4 Numerical experience 5 Percolation setting Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

35 Percolation setting Dirichlet problem on percolation cluster Apply the same algorithm on the same problem on percolation setting. (Gu 19+) a : E d {0} [Λ 1, 1]. a(e) > 0 represents an open bond and a(e) = 0 represents a closed bond. Supercritical percolation P[a 0] = p > p c (d). Dirichlet problem on the maximal cluster in the cube Q r. More technical: the random conductance also influences the domain of the solution, and the random graph structure is challenging for PDE analysis. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

36 Percolation setting Dirichlet problem on percolation cluster Figure: Can you tell all the connected components in the graph? Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

37 Percolation setting Dirichlet problem on percolation cluster Figure: The cluster in blue is the maximal cluster in the cube Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

38 Percolation setting Calderón-Zygmund decomposition on cluster A technique to decompose the cluster into small cubes so in every cube the behavior is good. See the work Armstrong and Dario (18) for regularity of harmonic function on the cluster, Dario (18+) for the corrector on the cluster, Dario and Gu (19+) for estimate of Green function on the cluster. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

39 Percolation setting Calculate the corrector in a cube d = 2, size = , p = 0.6, a {0} [0.5, 1], λ = 0.1. a φ e1,l = ae 1 with null boundary condition. This example cannot be captured by homogenized solution. Initial error ε 0 = Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

40 Percolation setting Calculate the corrector in a cube round errors Figure: A table of errors Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

41 Percolation setting Calculate the corrector in a cube Figure: A simulation of the corrector on the maximal cluster of a cube Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35

42 Percolation setting Figure: A simulation of the corrector on the maximal cluster of a cube Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35 Calculate the corrector in a cube

43 Percolation setting Thank you for your attention. Chenlin GU (DMA/ENS) Algorithm on percolation cluster May 13, / 35