Matrix assembly by low rank tensor approximation

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1 Matrix assembly by low rank tensor approximation Felix Scholz

2 References Angelos Mantzaflaris, Bert Juettler, Boris Khoromskij, and Ulrich Langer. Matrix generation in isogeometric analysis by low rank tensor approximation. In Curves and Surfaces, volume 9213 of LNCS, pages Springer, Angelos Mantzaflaris, Bert Juettler, Boris Khoromskij, and Ulrich Langer. Low rank tensor methods in galerkin-based isogeometric analysis. Technical report, Radon Institute for Computational and Applied Mathematics, W. Hackbusch. Tensor spaces and numerical tensor calculus. Springer, Berlin, M. Griebel and H. Harbrecht. Approximation of two-variate functions: Singular value decomposition versus sparse grids. IMA Journal of Numerical Analysis, Bert Jüttler and Dominik Mokriš. Low rank spline interpolation of boundary value curves. G+S Report No. 47, 2016.

3 Overview Motivation Singular Value Decomposition (SVD) of functions Discrete Singular Value Decomposition Numerical examples Generalisation to arbitrary dimensions

4 Motivation Given a parametrisation of the physical domain Ω by a regular tensor product B-Spline (or NURBS) function F : ˆΩ Ω we consider the weak formulation an elliptic equation as a model problem: Find u H0 1 (Ω), such that a(u, v) = (f, v) 0,Ω v H 1 0 (Ω), where a(u, v) = x u(x) (A(x) x v(x)) + cu(x)v(x)dx. Ω

5 Motivation We consider the 2D-case with ˆΩ = [0, 1] 2, A I and c 1. As the discrete space of functions on the parametric domain we choose a tensor spline space with the B-spline basis S p (Ξ) = S p1 (Ξ 1 ) S p2 (Ξ 2 ) ˆB i,p (ξ) = ˆB i1,p 1 (ξ 1 )ˆB i2,p 2 (ξ 2 ) and set the discrete space of functions on the physical domain to be (up to the boundary conditions) V h := span{ˆb i F 1 } = span{b i }.

6 Motivation Computing the L 2 -product of the basis elements we get the entries of the mass matrix: M ij = B i (x)b j (x)dx Ω = det ξ F (ξ) ˆB i (ξ)ˆb j (ξ)dξ ˆΩ = ω(ξ)ˆb i1 (ξ 1 )ˆB j1 (ξ 1 )ˆB i2 (ξ 2 )ˆB j2 (ξ 2 )dξ 1 dξ 2, where ω(ξ) = det ξ F (ξ). Analogously we get for the stiffness matrix S ij = x B i x B j dx = Ω 2 p,q= K pq (ξ) ˆB i ˆB j dξ, ξ p ξ q where K(ξ) = (det ξ F )( ξ F ) 1 ( ξ F ) T.

7 Motivation The complexity of computing the bivariate integrals in S ij and M ij by Gauss quadrature is in O(n 2 p 6 ) (if n = n 1 = n 2 and p 1 = p 2 ). We want to decompose the functions ω(ξ) and K pq (ξ) into products of univariate functions so we only need to compute univariate integrals where the complexity is O(np 3 ).

8 Singular value decomposition of a function Any bivariate continuous function f C([0, 1] [0, 1]) has a singular value decomposition f (ξ 1, ξ 2 ) = σ r u r (ξ 1 )v r (ξ 2 ), r=1 where σ 1 σ and {u r } r 1 and {v r } r 1 are L 2 ((0, 1))-orthonormal systems of continuous functions. The sum converges in L 2 ((0, 1) (0, 1)). The rank R-approximation f R (ξ 1, ξ 2 ) = R σ r u r (ξ 1 )v r (ξ 2 ) r=1 is the best approximation of f by a rank R function in the L 2 -norm.

9 Singular value decomposition of a function Lemma If f H s ((0, 1) (0, 1)), then (i) the singular values decay like (ii) the approximation error fulfils f f R L 2 = σ r r s. r=r+1 σ 2 r R 1 2 s Proof. Griebel, [4]

10 Discrete singular value decomposition A low rank approximation of the functions ω and K pq is computed as follows: 1. Project the function ω or K pq into a suitable spline space. 2. Decompose the coefficient tensor using matrix SVD. 3. Choose the rank R such that the overall approximation error is lower than a given constant ɛ, for example the discretisation error.

11 Projection into a spline space The function ω = det F is a tensor product spline function of higher polynomial degrees q d = 2p d 1 and lower smoothness than F. We can thus represent it exactly with respect to a B-Spline basis { B i } i (m1,m 2 ). ω(ξ) = Πω(ξ) = m 1 m 2 i 1 =1 i 2 =1 ω i1 i 2 B i1 (ξ 1 ) B i2 (ξ 2 ). For the functions K pq we choose a sufficiently refined spline space span{ B i } such that K pq ΠK pq L 2 (ˆΩ) = K pq m 1 m 2 i 1 =1 i 2 =1 (K pq ) i1 i 2 B i1 B i2 L 2 (ˆΩ) ɛ Π.

12 Decomposition of the coefficient tensor We compute the SVD of the m 1 m 2 matrix W = (ω i1 i 2 ), i.e. W = UΣV T = min(m 1,m 2 ) r=1 σ r u r v T r, where U is an orthogonal m 1 m 1 -matrix with columns u r, V is an orthogonal m 2 m 2 -matrix with columns v r and Σ = diag(σ 1,..., σ min(m1,m 2 )). We assume σ 1... σ min(m1,m 2 ).

13 Decomposition of the coefficient tensor The rank-r approximation R W R = σ r u r vr T r=1 is the best approximation of W by a matrix of rank R in the Frobenius norm and fulfils W W R F = min(m 1,m 2 ) σr 2. r=r+1

14 Decomposition of ω We multiply the decomposed coefficient tensor with the basis of the projection space to get the decomposition min(m 1,m 2 ) m 1 m 2 Πω(ξ) = (u r ) i1 B i1 (ξ 1 ) (v r ) i2 B i2 (ξ 2 ) = r=1 min(m 1,m 2 ) r=1 σ r i 1 =1 U r (ξ 1 )V r (ξ 2 ) i 2 =1

15 Error estimate for the low rank approximation Lemma The rank R-approximation Λ R ω(ξ) = R U r (ξ 1 )V r (ξ 2 ) r=1 fulfils Πω Λ R ω L (ˆΩ) W W R F = min(m 1,m 2 ) r=r+1 σ 2 r. Thus for a given accuracy ɛ Λ we can choose the smallest rank R such that the approximation error is below ɛ Λ.

16 Assembly of the matrices The entries of approximated mass matrix are M ij M ij = R r=1 1 0 U r (ξ 1 )ˆB i1 (ξ 1 )ˆB j1 (ξ 1 )dξ 1 1 and thus M can be written in the Kronecker format M = R X r Y r r=1 0 V r (ξ 2 )ˆB i2 (ξ 2 )ˆB j2 (ξ 2 )dξ 2 where each X r is a n 1 n 1 and Y r a n 2 n 2 -matrix containing the univariate integrals. For the stiffness matrix we can proceed in the same way.

17 Computational complexity We assume n 1 = n 2 = n, m 1 = m 2 = m, p 1 = p 2 = p and q 1 = q 2 = q. The complexity is bounded from below by the number of non-zeros in the matrix, which is O(n 2 p 2 ). The complexity of computing the matrix SVD up to rank R is O(Rm 2 ). For assembling the matrices X r and Y r using univariate element-wise Gauss quadrature the complexity is O(Rnp 3 ) The complexity for computing the Kronecker sum R r=1 X r Y r is O(Rn 2 p 2 ). Since generally n p, the overall complexity is dominated by the last step and is thus O(Rn 2 p 2 ).

18 Numerical examples For the method to be efficient we need to be able to choose the rank low. Table: Rank values for accuracy ɛ Λ = ɛ Π = 10 8 n p (1, 2) (1, 2) (4, 4) (7, 7) rank(ω) ( 1 ) ( 1 ) ( 7 ) ( 8 ) rank(k)

19 Numerical examples Assembly time Decomposition method Gauss Slope x10 6 1x10 7 1x10 8 n 3 Assembly time Decomposition method Gauss Slope x10 6 1x10 7 n 3 (a) Quarter annulus, p = 2 (b) 2nd Coons surface (star), p = 7 Figure: Comparison of computation times for the stiffness matrix using the decomposition method and an element-wise Gauss method.

20 Numerical examples Assembly time Full Gauss Slope 2 Slope p Figure: Comparison of the p-dependence of the computation times for the stiffness matrix using the decomposition method and an element-wise Gauss method. Computed on the quarter annulus with = DOF.

21 Generalisation to arbitrary dimensions The tensor decomposition method can be generalised to arbitrary dimensions since any d tensor T W (n1,...,n d ) possesses a canonical representation where v r k Rn k. T = R v1 r v2 r... vd r, r=1 However, the truncation operator Λ R T = argmin T U 2 rank(u) R leads to a non-linear optimisation problem.

Low Rank Tensor Methods in Galerkin-based Isogeometric Analysis

Low Rank Tensor Methods in Galerkin-based Isogeometric Analysis Angelos Mantzaflaris a Bert Jüttler a Boris N. Khoromskij b Ulrich Langer a a Radon Institute for Computational and Applied Mathematics (RICAM),