Multigrid preconditioning for nonlinear (degenerate) parabolic equations with application to monument degradation

Transcription

1 Multigrid preconditioning for nonlinear (degenerate) parabolic equations with application to monument degradation M. Donatelli 1 M. Semplice S. Serra-Capizzano 1 1 Department of Science and High Technology University of Insubria Department of Mathematics University of Torino ENUMATH013 Losanna, 6-30 August 013 M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 1 / 30

2 Outline 1 Marble sulfation t u = (D(u) u) Implicit discretization in time and finite differences in space 3 Uniform grids (spectral analysis of the Jacobian matrix, etc.) 4 Marble sulfation 5 Nonuniform grids 6 Future work M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 / 30

3 Marble sulfation model Ω 0 1 x c(t, x) concentration of CaCO 3, c(0, x) = c 0 s(t, x) concentration of SO, s(0, x) = 0 Porosity: ϕ(c) = αc + β, with α, β > 0 B.C.: Dirichlet for SO at x = 0, free-flow at x = 1 M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 3 / 30

4 Marble sulfation model Ω 0 1 x t ϕ(c)s t c = a m c ϕ(c)sc + d (ϕ(c) s) = a m s ϕ(c)sc c(t, x) concentration of CaCO 3, c(0, x) = c 0 s(t, x) concentration of SO, s(0, x) = 0 Porosity: ϕ(c) = αc + β, with α, β > 0 B.C.: Dirichlet for SO at x = 0, free-flow at x = 1 M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 3 / 30

5 Model problem We first consider a single equation of the form t u = (D(u) u), where D(u) is a nonnegative function. (D(u) differentiable and D (u) is Lipschitz continuous for convergence... ) Space discretization by finite differences 1 Uniform grid Nonuniform grid under the assumption that the grid points can be seen as the image of a uniform grid via an invertible function. M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 4 / 30

6 Uniform Grids 1D finite difference discretization u j 1 x j 1 x j 1 ( D(u(x)) u(x) ) x x u j u j+1 x j x j+1 x j+ 1 D(u(x)) u xj+1/ D(u(x)) u x h x xj 1/ D j+1/(u j+1 u j ) D j 1/ (u j u j 1 ) h where D j+1/ = D(u j+1) + D(u j ) = D(u(x j+1/ )) + O(h ). Thus the N N matrix of the spatial operator is L D(u) = tridiag N k [D k 1/(u), D k 1/ (u) D k+1/ (u), D k+1/ (u)] M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 5 / 30

7 θ-method Uniform Grids Discretizing in time by the θ-method u n u n 1 = θ t h L D(u n )u n + (1 θ) t h L D(u n 1 )u n 1 For θ > 0 (e.g., Eulero, Crank-Nicolson), it holds F (u n ) = 0 where F (u) = u θ t h L D(u) u (1 θ) t h L D(u n 1 )u n 1 u n 1 M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 6 / 30

8 Jacobian matrix Uniform Grids F (u) = u θ t h L D(u) u (1 θ) t h L D(u n 1 )u n 1 u n 1 where the Jacobian matrix is F (u) = I N θ t L h D(u) +Y N } {{ } X N X N (u) = s.p.d. Y N (u) = θ t h T N (u) diag N k (D k ) T N (u) = tridiag N k [u k 1 u k, u k 1 u k + u k+1, u k+1 u k ] = tridiag N k [O(h), O(h ), O(h)] M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 7 / 30

9 Uniform Grids Newton method Theorem (D., Semplice, Serra-Capizzano SIMAX(011)) If u is the sampling of a solution of t u = x (D(u) x u), with D and u Lipschitz. Then C 1 > 0 independent of h s.t. F (u) 1 C 1 (1) if t C h for a constant C > 0. Combining the previous result with Kantorovich theorem it follows Theorem (D., Semplice, Serra-Capizzano SIMAX(011)) The Newton method for computing u (n) converges if t = O(h) and u (0) = u n 1. M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 8 / 30

10 Uniform Grids Spectral analysis of the Jacobian matrix [SIMAX, 011] For the Jacobian matrix F (u) = I N θ t L h D(u) +Y N } {{ } X N If u is Lipschitz, Y N (u) C t h D, thus if t = O(h), Y N is negligible with respect to X N : Spectrum of F Y N (u) = O(1) while X N = O( 1 h ). κ (F ) = O(N) Σ(F ) [c, CN] i[ d, d] M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 9 / 30

11 Uniform Grids Spectral distribution {A N } λ (θ, G) where θ is a measurable function on G R d, if 1 F C 0 (C) : lim N N Eigenvalue cluster N 1 F (λ j (A N )) = 1 F (θ(t))dt µ(g) G If {A N } λ (θ = c = const, G), then it is weakly clustered at c, i.e. #{λ j (A N ) : λ j / D(c, ε)} = o(n), ε > 0. The cluster is strong if o(n) = O(1). M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

12 Uniform Grids Spectral distribution of F A N = hf = L D(u) + R N (u) and Ω is the domain of the PDE. {A N } λ (θ, Ω [0, π]) θ(x, s) =D(u(x))( cos(s)) The negligible term R N = hi N + hy N {R N } λ (0, G) { RN h } λ ( 1 D (u(x))i sin(s), Ω [0, π] ) M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

13 Uniform Grids Multigrid Preconditioner Preconditioner P N = L D(u) + hi N, Σ(P 1 N A N) [ 1 c 1 h, 1 + c h ] i [ d, d ]. with c 1, c, and d independents of N. Multigrid Classical geometric multigrid (V-cycle with bilinear interpolation) with a standard smoother (Gauss-Seidel) is an optimal solver for P N. A robust and efficient solver for a linear system with the Jacobian matrix is GMRES with a V-cycle iteration as preconditioner. M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 1 / 30

14 Uniform Grids D case {A N } λ (θ, Ω [0, π] [0, π]) θ =D(u(x, y))( cos(s))( cos(t)) The Jacobian matrix has not a tensor product structure but they share the same sparsity pattern. All results about convergence of the Newton method, spectral estimation and preconditioning hold similarly to the 1D case... M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

15 Uniform Grids Numerical example Example Results Test problem: D(u) = 4u 3 Ω R 3 choices of t Newton converges always within 3 7 iterations (4 5 iterations for t = h). Number of GMRES iterations without preconditioning O( 1/h) M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

16 Numerical example Uniform Grids (a) (b) Average, min, and max number of iterations of (a) GMRES without preconditioning (b) GMRES preconditioned with one V-cycle iteration. M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

17 Marble sulfation Marble sulfation s j 1 c j 1 s j c j+ 1 s j+1 xj 1 x j 1 xj x j+ 1 xj+1 x (ϕ(c) x s) xj = ϕ(c j+1/)(s j+1 s j ) ϕ(c j 1/ )(s j s j 1 ) ϕ(c)sc xj = ϕ ( cj 1/ +c j+1/ ϕ(c)sc xj+1/ = ϕ(c j+1/ )c j+1/ s j +s j+1 h ) cj 1/ +c j+1/ s j M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

18 Marble sulfation Marble sulfation s j 1 c j 1 s j c j+ 1 s j+1 xj 1 x j 1 xj x j+ 1 xj+1 x (ϕ(c) x s) xj = ϕ(c j+1/)(s j+1 s j ) ϕ(c j 1/ )(s j s j 1 ) ϕ(c)sc xj = ϕ ( cj 1/ +c j+1/ ϕ(c)sc xj+1/ = ϕ(c j+1/ )c j+1/ s j +s j+1 h ) cj 1/ +c j+1/ s j Newton method for 0 = F (s) (s n, c n ) = Φ n s n + θ t a m c C n s n + θ tdl ϕ ns n 0 = F (c) (s n, c n ) = Φ n 1 s n 1 + (1 θ) t a m c C n 1 s n 1 + (1 θ) tdl ϕ n c n + θ t a m s S n c n c n 1 + (1 θ) t a m s S n 1 c n 1 where Φ n, C n, S n are diagonal matrices. M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

19 Jacobian matrix Marble sulfation u = ( ) [ s, J = F J s = s Jc s c Js c Jc c ], [Js s ] j,k = sk F (s) j, [Jc s ] j,k = ck+1/ F (s) j { } { } Js s ϕj+1/ +ϕ = diag j 1/ + θ t a ϕj+1/ c m c diag j+1/ +ϕ j 1/ c j 1/ + θ d t { } tridiag h j ϕj 1/, ϕ j 1/ + ϕ j+1/, ϕ j+1/ M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

20 Jacobian matrix Marble sulfation u = ( ) [ s, J = F J s = s Jc s c Js c Jc c ], [Js s ] j,k = sk F (s) j, [Jc s ] j,k = ck+1/ F (s) j { } { } Js s ϕj+1/ +ϕ = diag j 1/ + θ t a ϕj+1/ c m c diag j+1/ +ϕ j 1/ c j 1/ + θ d t { } tridiag h j ϕj 1/, ϕ j 1/ + ϕ j+1/, ϕ j+1/ { } Jc s = tridiag 0, ϕ j+1/ s j, +ϕ j 1/ s j { } + θ t a m c tridiag 0, (ϕ j+1/ c j+1/+ϕ j+1/ ), (ϕ j 1/ c j 1/+ϕ j 1/ )δ j,k+1 { } + θ d t tridiag 0, ϕ h j+1/ (s j+1 s j ), ϕ j 1/ (s j s j 1 ) Js c =θ t a m s tridiag { ϕ j+1/ c j+1/, ϕ j+1/ c j+1/, 0 } { } Jc c = diag 1 + θ t a m s (ϕ j+1/ c j+1/ + ϕ j+1/ ) s j+1+s j M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

21 Block preconditioner Marble sulfation J c s Theorem has entries that decay as O( t), J c c is the identity matrix plus a diagonal matrix with O( t) entries. If t = O(h), the block upper triangular part of J, namely [ J s P = s Jc s 0 Jc c ], () is an optimal preconditioner for J if ϕ(c) is strictly positive (i.e., the porosity of the marble-gypsum mixture does not vanish). M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

22 Marble sulfation Numerical results for a D corner (a) (b) Number of iterations (average, min, and max) for (a) GMRES without preconditioning (b) Preconditioner P and one step of AMG for J s s M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30

23 Marble sulfation Numerical results for a D corner M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 0 / 30

24 Nonuniform grids A nonuniform grid can help... M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 1 / 30

25 Nonuniform grids Discretizzation on a nonuniform grid ( ) x D(u) u xj x D u j+1 u j j+1/ h j+1 D j 1/ u j u j 1 h j (h j + h j+1 )/ L (1) D(u) = diagn k [ ] [ h k +h k+1 tridiag N k D k 1/ h k, D k 1/ h k ] + D k+1/ h k+1, D k+1/ h k+1 Properties of L D(u) it is not symmetric it can be symmetrized, by a similarity transformation, only in the 1D case. M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 / 30

26 Nonuniform grid Nonuniform grids Consider the grid points as a map of a uniform grid x j = g(j/n), g : [0, 1] Ω If g is piecewise C 1, the discretizzation of x (D(x) x u) on the nonuniform grid is spectrally equivalent to the discretization of ) x (w D (x) u x on a uniform grid, with w D (x) = D(g(x)) [g (x)] [Serra Capizzano, Tablino Possio Lin. Alg. Appl. (003)] M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 3 / 30

27 Jacobian matrix Nonuniform grids X N is an M-matrix, F = I N θ tl wd (u,g) +Y N } {{ } X N Y N is negligible with respect to X N. Choosing t = 1 N Ω R {X 1 N F N } λ (1, G) Ω R {X 1 N F N } σ (1, G) (singular value distribution) It follows that the Algebraic Multigrid (AMG) is an affective preconditioner for the GMRES. M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 4 / 30

28 Conditioning of F Nonuniform grids 1 N F = 1 N I N L wd + negligible nonsymmetric perturbation ( ) { 1 N F } λ θ D(g(x)) ( cos(s)), Ω [0, π] [g (x)] λ min 1 N λ max grows like the maximum of D(g(x)) and hence like N q if g has a [g (x)] zero of order q where the numerator does not vanish. κ (F ) N q+1 Note: g (x) vanishes where the grid points accumulate. M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 5 / 30

29 Nonuniform grids 1D GMRES iterations Newton GMRES CPU time cond(j) mesh no P Jac AMG iterations no P Jac AMG unif N 0.97 N 0.4 N Shish N 1.14 N 0.46 N Bakh N 0.85 N 0.9 N g 1/ N 0.94 N 0.38 N g N 3.00 N 0.76 N g 3 N 5.00 N 0.83 N n/a g α (t) = sign(t) t α t = x (u x u) M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 6 / 30

30 Marble sulfation Nonuniform grids Number of iterations of Newton and PGMRES on uniform grids (red) and nonuniform grids (black). M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 7 / 30

31 Nonuniform grids Profile of the solution along the diagonal x = y M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 8 / 30

32 Nonuniform grids Future work 3D case Finite element discretization Application to 3D laser scanning of monuments (e.g., Donatello s David) Free boundary models of marble sulfation (the computational domain enlarges with the time) M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/013 9 / 30

33 References Nonuniform grids M. Donatelli, M. Semplice, S. Serra-Capizzano Analysis of multigrid preconditioning for implicit PDE solvers for degenerate parabolic equazions SIAM J. Matrix Anal. Appl. 3 (011) M. Semplice Preconditioned implicit solvers for nonlinear PDEs in monument conservation SIAM J. Sci. Comp. 3 (010) M. Donatelli, M. Semplice, S. Serra-Capizzano AMG preconditioning for nonlinear degenerate parabolic equations on nonuniform grids with application to monument degradation Appl. Numer. Math. 68 (013) 1 18 M. Donatelli et al. (Univ. Insubria) Grid adaptivity 6-30/8/ / 30