Sparse Tensor Galerkin Discretizations for First Order Transport Problems

Transcription

1 Sparse Tensor Galerkin Discretizations for First Order Transport Problems Ch. Schwab R. Hiptmair, E. Fonn, K. Grella, G. Widmer ETH Zürich, Seminar for Applied Mathematics IMA WS Novel Discretization Methods Oct 31-Nov Acknowledgement: Swiss Nat l Science Foundation and ERC W. Dahmen and G. Kutyniok Priority Research Programme 1324 German Research Foundation (DFG).

2 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 2 / 38 Outline 1 Problem 2 Variational Formulations 3 Galerkin discretization Without boundary conditions With boundary conditions 4 Numerical experiments 5 Tensor Product Wavelet Galerkin Approach (Widmer etal. JCP 2008) Inflow Boundary: Example Preconditioning Subspace Correction Preconditioner Experiment I 6 Adaptive Approximation Adaptivity: Rationale Numerical Experiment II Numerical Experiment III 7 Conclusion

3 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 3 / 38 The Radiative Transfer Problem Stationary monochromatic radiative transfer problem: (s x + κ(x)) u(x, s) = κ(x)i b (x), (x, s) D S d S u(x, s) = 0, x Γ (s) := D {s : s n(x) < 0}. x D R d D bounded, Lipschitz, s S d S 5-dimensional phase space (d D, d S ) = (3, 2) D R d D bounded: zero inflow BCs Model for general, parametric transport problems (UQ for Neutron Transport, v x ψ(x, v; t)...) Numerical approaches: MCM, discrete ordinates, P N approximation Complexity h d D D h d S S, slow convergence due to curse of dimensionality Sparse tensorization: Wavelets in S d S ( Widmer et al. (2008) ) Current: Sparse Adaptive Tensor Approximation with 1 spectral discretization in S d S ( this talk) 2 hierarchic, directional representation systems in D ( W. Dahmen s talk)

4 Variational formulation cg unstable = a) Stabilization or b) Petrov Galerkin (Talk of W. Dahmen) Stabilized LSQ formulation (Manteuffel, Ressel, Starke: SINUM(2000)): Find u V 0 such that with a(u, v) = l(v) v V 0 a(u, v) := (ɛ(s x u + κu), s x v + κv) L2 (D S), ɛ(x) = 1 κ(x) l(v) := (ɛκi b, s x v + κv) L2 (D S d S ) V := { u L 2 (D S d S ) : s x u L 2 (D S d S ) } V 0 := {u V : u = 0 on D, s n(x) < 0} a(, ) is continuous and coercive on V 0 V 0 Energy Norm u A := a(u, u) Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 4 / 38

5 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 5 / 38 Discretization Cartesian product domain D S d S span V with product basis functions from tensor product (TP) space H 1 (D) L 2 (S d S ) Inflow Boundary s-dependent: TP structure broken Hierarchical function spaces allow for incremental resolution refinement: On D: Piecewise polynomials (here p = 1) on dyadically refined mesh TD l V l D := S p,1 (D, T l D) On S d S : Spherical harmonics S (d S) n,m V l D = V l 1 D W l D up to order l H 1 (D) V l S := P d S l = span{s (d S) n,m : n = 0,..., l; m = 1,..., m n,ds } L 2 (S d S ) V l S = V l 1 S W l S Finite dimensional Full Tensor Product (FTP) space: V L = VD L VS N = 0 l D L 0 l S N W l D D W l S S H 1 (D) L 2 (S d S )

6 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 6 / 38 Structure of FTP space Illustration with VD L=3 = S (1,1) ([0, 1] [0, 1], TD 3 N=3 ) and VS = P 1 3 W l S S W 0 D W 0 S W 0 D W 1 S W 0 D W 2 S W 0 D W 3 S W 1 D W 0 S W 1 D W 1 S W 1 D W 2 S W 1 D W 3 S W 2 D W 0 S W 2 D W 1 S W 2 D W 2 S W 2 D W 3 S W 3 D W 0 S W 3 D W 1 S W 3 D W 2 S W 3 D W 3 S W l D D

7 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 7 / 38 Structure of Sparse Tensor Product (STP) space Illustration with VD L=3 = S (1,1) ([0, 1] [0, 1], TD 3 N=3 ) and VS = P 1 3 W l S S W 0 D W 0 S W 0 D W 1 S W 0 D W 2 S W 0 D W 3 S W 1 D W 0 S W 1 D W 1 S W 1 D W 2 S W 1 D W 3 S W 2 D W 0 S W 2 D W 1 S W 2 D W 2 S W 2 D W 3 S W 3 D W 0 S W 3 D W 1 S W 3 D W 2 S W 3 D W 3 S W l D D

8 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 8 / 38 Sparse Tensor Product (STP) Space FTP Finite Element FTP space: Finite Element STP space: V L = ˆV L := 0 l D L 0 l S N 0 f (l D,l S ) L with cutoff function f : [0, L] [0, N] R. Dimensionality: W l D D W l D D W l S S W l S S, dim(v L ) N D N S dim( ˆV L ) N D log N S + N S log N D for f (l D, l S ) = l D + l S L N

9 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 9 / 38 Approximation properties Error estimates for approximation u L of u H s+1,t (D S), s [0, p], t N 0 : Full tensor case u L V L : u u L H 1,0 (D S) max{n t/d S S, N s/d D } u (H 1,t H 1+s,0 )(D S) Sparse tensor case u L ˆV L with f (l D, l S ) = l D + L log 2 (l S +1) log 2 (N+1) : u u L H 1,0 (D S) log 2 ( N 1/d D ) max{n t/d S S, N s/d D } u H 1+s,t (D S) Convergence rate (almost) retained while number of dofs reduced. To equilibrate contributions: N h s/t

10 Product basis functions satisfying boundary conditions Zero inflow conditions: u(x, s) = 0 for x D, s n(x) < 0 Physical basis functions and corresponding outflow regions: x 0 0 x x x x π π π 2 s s s π 3π 2 π 3π 2 π x 1 0 3π 2 Partition of sphere into angular regions S q, mesh T S = {S q } n S q=1 : S 2 π 2 S 1 π 0 S 3 S 4 3π 2 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 10 / 38

11 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 11 / 38 Tensor product space with boundary conditions To maintain TP structure and strongly satisfy b.c.s add restricted angular functions to VS L: Introduce space P d S,T S N of angular functions restricted to elements of mesh T S, e. g. (d S = 1) Compressed spherical harmonics Legendre polynomials on S q. Split physical spaces into boundary and non-boundary subspaces: V l D = V l D V l D,0 Tensorize V 0 -conforming combinations only: V0 L := VD,0 L P d S N V D L P d S,T S N Sparse version: ˆV L 0 := V L D,0 ˆ P d S N V L D ˆ P d S,T S N

12 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 12 / 38 Structure of TP spaces with boundary conditions Active dofs for L = 3, N = 8 for FTP and STP space (here d D + d S = 2 + 1): Physical dofs FTP l D S S 1 S 2 S 3 S Angular dofs Physical dofs STP Angular dofs 1853 dofs 275 dofs

13 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 13 / 38 Solution of discretized variational problem Benefits of TP structure: Separable quadrature for stiffness matrices over D and S System of equations is matrix equation: B n UC n = L Matrix-vector multiplication faster CG to solve system (currently no preconditioning). To evaluate solution: Look at Relative error of u in X = H 1,0, A, L 2 -norm on D S: err X = u u L X / u X n

14 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 14 / 38 Numerical experiments D = [0, 1] [0, 1], S = S 1, κ(x) = 1. Experiment 1 Solution prescribed: u(x, ϕ) = U(x)Y (ϕ) Experiment 2 U(x) = ( 1(x 1 1) 2 + 1)( 4(x 2 1/2) 2 + 1)x 1 x 2 ( 4(x 2 1)) 2 π ϕ + 1 if 0 ϕ < π 2 2 Y (ϕ) = π (ϕ 3π 2 ) if 3π 2 < ϕ < 2π 0 else U(x) as in Exp. 1 ( ) 2 sin(8ϕ) Y (ϕ) = 8ϕ if 0 ϕ < π 2 3π 2 < ϕ < 2π 0 else

15 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 15 / 38 Convergence rates Experiment Error u (testside2), FT/ST, N=32, L refined 10 0 Error u (testside2), FT/ST, L=5, N refined u u N / u H10 error FT A error FT L2 error FT H10 error ST A error ST L2 error ST # Dofs p=2 p=1/2 p=1 u u N / u In this setup: error limited by physical contribution p= # Dofs In physical space convergence rate 1/2 for H 1,0 -error as expected In angular space resolution refinement not effective ST method achieves better error per employed dofs ratio p=1/2 p=1

16 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 16 / 38 Convergence rates Experiment Error u (testside1), FT/ST, N=32, L refined 10 0 Error u (testside1), FT/ST, L=5, N refined u u N / u H10 error FT A error FT L2 error FT H10 error ST A error ST L2 error ST # Dofs p=2 p=1/2 p=1 u u N / u Here error limited by angular contribution Physical refinement not effective H10 error FT A error FT L2 error FT H10 error ST A error ST L2 error ST p= Angular convergence rate between 1/2 and 1 for H 1,0 -error # Dofs p=1/2 ST LSQ FEM achieves better error per dofs ratio for angular refinement p=1

17 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 17 / 38 Tensor Product Wavelet Galerkin D R 2 disc, dimd = 2, d S = 3 Meshes T 3 D, T 3 S 2 :

18 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 18 / 38 Multiscale Decompositions Nested meshes T l D nested spaces V l D := S0 1 (T l D ) TS l nested spaces VS l := S 1 0 (TS l) Complement ( detail, hierarchical surplus ) Spaces V l D = W l D V l 1 D V l S = W l S V l 1 S

19 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 19 / 38 Multiscale Decompositions In D R d : + In S d 1 :

20 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 20 / 38 Multiscale Decompositions Full Tensor Galerkin trial/test space V L = VD L VS L = WD i W j S 0 i,j L Sparse tensor product space: V L := WD i W j S dim V L = O(ND L log NS L + NS L log ND) L 0 i+j L

21 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 21 / 38 Sparse Tensor Product Space V L := WD i W j S 0 i+j L

22 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 22 / 38 Graded ST Space Problem: Homogeneous Essential BC s in Γ (s) destroy TP structure Solution: 1 Enforce Dirichlet BC s weakly via Multipliers 2 Enforce Dirichlet BC s strongly: local isotropic mesh refinement along Γ (s), s S 1. Graded ST space

23 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 23 / 38 φ Inflow Boundary: Example y 4 3 x 2 γ(x) φ D Tangent in x to D y x

24 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 24 / 38 Graded Sparse Tensor Product Space D S1 gamma Augment V L 0 := V 0 V L by tensor product nodal basis functions (in neighborhood of γ) multiscale tensor product is dictionary, frame (no Riesz basis anymore) dd Moderate increase of #d.o.f. by a logarithmic factor L. dim V 0 L = O(L 2 ( M L ) O(LML ) ) Approximation rate (proved for n = 2, d = 1, 2): C > 0 : L : û L V L 0 : u û L S C log h L h L u H 2,1 (D S 2 ).

25 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 25 / 38 Preconditioning Multiplicative subspace correction preconditioner

26 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 25 / 38 Preconditioning Multiplicative subspace correction preconditioner transformation in standard basis

27 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 25 / 38 Preconditioning Multiplicative subspace correction preconditioner transformation in standard basis solution of decoupled, low dimensional transport problems (AMG)

28 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 25 / 38 Preconditioning Multiplicative subspace correction preconditioner transformation in standard basis solution of decoupled, low dimensional transport problems (AMG) Effort d.o.f. (Overhead of AMG ignored)

29 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 26 / 38 Preconditioning: Numerical Experiment Regular ST space Absorption coefficient: κ(x) = 1 Blackbody radiation: I b (x) = e 4 x 2

30 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 27 / 38 Numerical Experiment I Throughout: 2.5 D setting, D = {x R 2 x < 1}, n = d = 2 smooth solution u(x, s) due to large (constant) absorption 10 Emission f(r) distance to origin r Blackbody intensity f ( x ) Heat flux q(x)

31 Numerical Experiment I Efficacy of ST a priori DOF selection criterion: 1.18e+00 1 Level in Solid Angle e e 03 Size of multiscale contributions Ch. Schwab (SAM, ETH Zurich) 1.13e 01 l1l 2 l 3 Level in Physical Space l 1.00e largest multiscale contributions of solution in full tensor product space Sparse Tensor Approximation Novel Discretization Methods 28 / 38

32 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 29 / 38 Numerical Experiment I Numerical solutions: Incident Radiation G(x 1,0) Level 0 Full Level 1 Full Level 1 Sparse Level 2 Full Level 2 Sparse Level 3 Full Level 3 Sparse x 1 Incident radiation Net emission (div q)

33 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 30 / 38 Numerical Experiment I Convergence (T l D, T l S by uniform regular refinement): Relative error of the incident radiation in the L 2 (D)-norm Relative error of the heat flux in the L 2 (D)-norm

34 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 31 / 38 Adaptivity: Rationale Propagation of singularities No elliptic lifting theorems for radiative transfer BVP!

35 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 31 / 38 Adaptivity: Rationale Propagation of singularities No elliptic lifting theorems for radiative transfer BVP! Example (dimd = 2, d S emitting region = 1): small compact

36 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 31 / 38 Adaptivity: Rationale Propagation of singularities No elliptic lifting theorems for radiative transfer BVP! Example (dimd = 2, d S emitting region = 1): small compact Note: more smoothness with (isotropic) scattering

37 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 32 / 38 Numerical Experiment II highly non-uniform absorption/source u non-smooth flux Hea

38 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 33 / 38 Numerical Experiment II (Limited) efficacy of a priori selection criterion: Size of multiscale contributions largest multiscale contributions of solution in full tensor product space

39 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 34 / 38 Numerical Experiment II Numerical solutions: Incident radiation Net emission

40 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 35 / 38 Numerical Experiment II Convergence (T l D, T l S by uniform regular refinement): Relative error of the incident radiation in the L 2 (D)-norm Relative error of the heat flux in the L 2 (D)-norm

41 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 36 / 38 Numerical Experiment III { 5, x 0.08 Absorption: κ(x, y) = 3, otherwise. { e 20 x x 0 2 e Source: I b (x) = 5, x x , otherwise. x 0 = ( ) Threshold ν = 0.04

42 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 37 / 38 Conclusion STP combined with spectral and MR discretization of S d S essential b.c.s on Γ (s) strongly satisfied & TP structure maintained STP method results in discretization with far fewer dofs than FTP Theoretical error estimates without b.c.s shown to be (almost) identical for FTP and STP Error estimates with b.c.s numerically confirm theoretical estimates Adaptive mesh refinements in D and in S d S Essential BC s and Directional Adaptivity: stable frame discretizations of transport problems in bounded domains D

43 Ch. Schwab (SAM, ETH Zurich) Sparse Tensor Approximation Novel Discretization Methods 38 / 38 References Konstantin Grella, Christoph Schwab. Sparse tensor spherical harmonics approximation in radiative transfer. Technical report, SAM, ETH Zürich, reports G. Widmer, R. Hiptmair, and Ch. Schwab. Sparse adaptive finite elements for radiative transfer. Journal of Computational Physics, 227: , Gisela Widmer. Sparse Finite Elements for Radiative Transfer. PhD thesis, ETH Zürich, No