Smoothers for ecient multigrid methods in IGA

Smoothers for ecient mutigrid methods in IGA Cemens Hofreither, Stefan Takacs, Water Zuehner DD23, Juy 2015 supported by The work was funded by the Austrian Science Fund (FWF): NFN S117 (rst and third author) and J3362-N25 (second author).

Outine Preiminaries Mode probem IGA discretization Mutigrid agorithm Mutigrid framework Basis-independent smoother Smoother for one dimension Smoother for two dimensions Numerica resuts Concusions and Outook

Poisson mode probem Domain Ω R d Given function f L 2 (Ω) Find u H 1 (Ω) such that u = f u n = 0 in Ω on Ω

Variationa formuation Domain Ω R d Given function f L 2 (Ω) Find u H 1 (Ω) such that for a ũ H 1 (Ω). (u, ũ) H 1 (Ω) = (f, ũ) L 2 (Ω)

Spine spaces in one dimension Let Ω = (0, 1)

Spine spaces in one dimension Let Ω = (0, 1) M is the uniform subdivision of Ω = (0, 1) in n = n 0 2 subintervas T i of ength h := h 0 2 for = 0, 1, 2,...

Spine spaces in one dimension Let Ω = (0, 1) M is the uniform subdivision of Ω = (0, 1) in n = n 0 2 subintervas T i of ength h := h 0 2 for = 0, 1, 2,... Spine space S p,k, (Ω) is the space of a spine functions in C k (Ω), which are piecewise poynomias of degree p on each subinterva in M.

Spine spaces in two and more dimensions Let Ω = (0, 1) d

Spine spaces in two and more dimensions Let Ω = (0, 1) d Tensor product B-spines: ϕ (i+m j) p, (x, y) = φ (i) p, (x)φ(j) p, (y)

Spine spaces in two and more dimensions Let Ω = (0, 1) d Tensor product B-spines: ϕ (i+m j) p, (x, y) = φ (i) p, (x)φ(j) p, (y) For Ω = (0, 1) d with d > 1: S p, (Ω) denotes tensor product spine space More genera domains: geometry mapping

Discretization Variationa formuation: Find u S p, (Ω) such that for a ũ S p, (Ω) (u, ũ ) H 1 (Ω) = (f, ũ ) L 2 (Ω)

Discretization Variationa formuation: Find u S p, (Ω) such that for a ũ S p, (Ω) (u, ũ ) H 1 (Ω) = (f, ũ ) L 2 (Ω) Matrix-vector notation: K u = f

Mutigrid method One step of the mutigrid method on grid eve k appied to iterate u (0,0) = u (0) and right-hand-side f to obtain x (1) is given by: Appy ν smoothing steps u (0,m) for m = 1,..., ν. = u (0,m 1) + τ Appy coarse-grid correction 1 K (f K u (0,m 1) ) Compute defect and restrict to coarser grid Sove probem on coarser grid Proongate and add resut If reaized exacty (two-grid method): u (1) = u (0,ν) + I 1 K 1 1 I 1 (f K k u (0,ν) ) Standard arguments: convergence of two-grid method convergence of the mutigrid method (W-cyce)

Intergrid transfer Nested spaces: S p, 1 (Ω) S p, (Ω)

Intergrid transfer Nested spaces: S p, 1 (Ω) S p, (Ω) The proongation I 1 is the canonica embedding Knot insertion agorithm

Intergrid transfer Nested spaces: S p, 1 (Ω) S p, (Ω) The proongation I 1 is the canonica embedding Knot insertion agorithm The restriction is its transpose: I 1 = (I 1 )T

Hackbusch-ike convergence anaysis Convergence of two-grid method with rate q, i.e., u u(1) L q u u(0) L,

Hackbusch-ike convergence anaysis Convergence of two-grid method with rate q, i.e., u u(1) L q u u(0) L, in matrix notation T S ν L q,

Hackbusch-ike convergence anaysis Convergence of two-grid method with rate q = C AC S ν, i.e., u u(1) L q u u(0) L, in matrix notation T S ν L q, is guaranteed by L 1/2 T K 1 L 1/2 C A (approximation property) L 1/2 K S ν L 1/2 C S ν 1 (smoothing property)

Basis-(in)dependent smoother Richardson smoother: i.e., K := I u (0,m) = u (0,m 1) + τ (f K u (0,m 1) ),

Basis-independent smoother Richardson smoother (in continuous understanding): u (0,m) = u (0,m 1) + τ where K := L is the Riesz isomorphism 1 K (f K u (0,m 1) ),

Basis-independent smoother Richardson smoother (in continuous understanding): u (0,m) = u (0,m 1) + τ where K := L is the Riesz isomorphism 1 K (f K u (0,m 1) ), Lemma Assume that K := L and that τ is chosen such that 0 < τ 1 L 1/2 K L 1/2. Then the smoothing property is satised for C S = τ 1. Proof: Standard eigenvaue anaysis.

Basis-independent smoother for one dimension Cassica anaysis: In L 2 (Ω)

Basis-independent smoother for one dimension Cassica anaysis: In L 2 (Ω) L = h 2 M is the propery scaed mass matrix

Basis-independent smoother for one dimension Cassica anaysis: In L 2 (Ω) L = h 2 M is the propery scaed mass matrix Proofs for standard smoothers (Richardson, Jacobi, Gauss-Seide) use that the mass matrix is spectray equivaent to its diagona. This is true for B-spines, but deteriorate for increased p.

Mass-Richardson smoother Use K := L := h 2 M

Mass-Richardson smoother Use K := L := h 2 M Condition for choice of τ reads as: sup u R m \{0} and C S = τ 1 u K h 1 u M = sup u S p, (Ω)\{0} u H 1 (Ω) h 1 u L 2 (Ω) τ 1/2.

Mass-Richardson smoother Use K := L := h 2 M Condition for choice of τ reads as: sup u R m \{0} and C S = τ 1 u K h 1 u M = sup u S p, (Ω)\{0} u H 1 (Ω) h 1 u L 2 (Ω) Need: robust inverse inequaity for spine space Loca Fourier anaysis suggests: Such a robust inverse inequaity hods Numerica experiments [Hofreither, Zuehner 2014]: This approach does not work we τ 1/2.

Inverse inequaity Theorem ([T., Takacs 2015]) For a N 0 and p N, u H 1 2 3h 1 u L 2 is satised for a u S p, (Ω), where S p, (0, 1) is the space of a u S p, (0, 1) whose odd derivatives vanish at the boundary: 2+1 x 2+1 u (0) = 2+1 x 2+1 u (1) = 0 for a N 0 with 2 + 1 < p.

Idea: boundary correction Theorem ([Hofreither, T., Zuehner 2015]) For a N 0 and p N, u H 1 2(1 + 4 6)h 1 ( u 2 L 2 + H Γ, u,γ 2 H 1 ) 1/2 is satised for a u = u Γ, + u I, S p, (Ω), where u I, S (I ) p, (Ω) S p, (Ω), u Γ, (S (I ) p, (Ω)) and H Γ, is the discrete harmonic extension. Proof: need inverse inequaity on S p, (Ω) and robust approximation error estimate aso on S p, (Ω) (cf. [T., Takacs 2015]).

Boundary correction Reorder variabes such that: ( KΓΓ, KI T K = Γ, K I Γ, K II, ), u = ( uγ, u I, )

Boundary correction Reorder variabes such that: ( KΓΓ, KI T K = Γ, K I Γ, K II, ), u = ( uγ, u I, Compute the discrete Harmonic extension: ( ) u H Γ, u Γ, H 1 (Ω) = Γ, K 1 II, K = u Γ, I Γ,u KΓΓ, K T Γ, I Γ, K 1 II, K. I Γ, K Generaized inverse inequaity: u K 2(1 + 4 6) u L ) with ( K := L := h 2 KΓΓ, KI T M + Γ, K 1 ) II, K I Γ, 0. 0 0 }{{} K :=

Boundary correction Reorder variabes such that: ( KΓΓ, KI T K = Γ, K I Γ, K II, ), u = ( uγ, u I, Compute the discrete Harmonic extension: ( ) u H Γ, u Γ, H 1 (Ω) = Γ, K 1 II, K = u Γ, I Γ,u KΓΓ, K T Γ, I Γ, K 1 II, K. I Γ, K Generaized inverse inequaity: with L 1/2 K L 1/2 2(1 + 4 6) 2 ( K := L := h 2 KΓΓ, KI T M + Γ, K 1 ) II, K I Γ, 0. 0 0 }{{} K := )

Boundary correction Reorder variabes such that: ( KΓΓ, KI T K = Γ, K I Γ, K II, ), u = ( uγ, u I, Compute the discrete Harmonic extension: ( ) u H Γ, u Γ, H 1 (Ω) = Γ, K 1 II, K = u Γ, I Γ,u KΓΓ, K T Γ, I Γ, K 1 II, K. I Γ, K Generaized inverse inequaity: with L 1/2 K L 1/2 2(1 + 4 6) 2 ( K := L := h 2 KΓΓ, KI T M + Γ, K 1 ) II, K I Γ, 0. 0 0 }{{} K := Smoothing property for 0 < τ 2 1 (1 + 4 6) 2 )

Robust approximation error estimate Theorem ([T., Takacs 2015]) For each u H 1 (Ω), each N 0 and p N: (I Π p, )u L 2 (Ω) 2 2 h u H 1 (Ω) is satised, where Π p, is the H 1 -orthogona projection to S p, (Ω). The approximation property foows using standard arguments (for the h 2 M part) u K = H Γ, u Γ, H 1 (Ω) u H 1 (Ω) = u K (for the K part)

Convergence theorem Theorem ([Hofreither, T., Zuehner 2015]) The two-grid method converges with rate C AC S ν if ν > C A C S smoothing steps are appied. The constants C A and C S do not depend on the grid size h and the spine degree p. The extension to the W-cyce mutigrid method is standard.

Matrices from tensor-product spines The mass matrix has a tensor-product structure: M = M M

Matrices from tensor-product spines The mass matrix has a tensor-product structure: M = M M The stiness matrix is the sum of two tensor-product matrices: K = K M + M K

Matrices from tensor-product spines The mass matrix has a tensor-product structure: M = M M The stiness matrix is the sum of two tensor-product matrices: K = K M + M K Tensor-product structure can be used for inverting M, but not for inverting K

Smoother for two dimensions K,orig = K M + M K = (h 2 M + K ) M + M (h 2 M + K ) = 2h 2 M M + K M + M K h 2 K }{{ K } h 2 K }{{ K } tensor-product ow-rank correction =: K, where K = h 1 M + K.

Shermann Morrisson Woodburry formua Let A R N N, B R n n, P R n N with fu rank. Then: (A PBP T ) 1 = A 1 + A 1 P(B 1 P T A 1 P) 1 P T A 1

Shermann Morrisson Woodburry formua Let A R N N, B R n n, P R n N with fu rank. Then: (A PBP T ) 1 = A 1 + A 1 P(B 1 P T A 1 P) 1 P T A 1 The inversion of A = K K can be done using tensor-product structure and sovers for one dimension The rest ives ony in the boundary ayer

One dimension p 1 2 3 4 5 6 7 8 = 10 22 20 20 21 21 20 20 20 = 11 23 20 20 21 20 20 20 20 = 12 23 20 20 20 20 20 20 20 p 9 10 12 14 16 18 20 = 10 20 20 20 18 18 17 17 = 11 20 19 19 18 19 18 17 = 12 20 20 19 18 18 18 18 ν pre + ν post = 1 + 1, τ = 0.14 Stopping criterion: Eucidean norm of the initia residua is reduced by a factor of ɛ = 10 8.

Two dimensions p 2 3 4 5 6 7 8 9 = 5 82 80 75 76 76 73 72 72 = 6 83 87 76 74 75 73 72 72 p 10 11 12 13 14 15 16 = 5 70 71 70 68 69 69 66 = 6 70 70 69 68 68 67 65 ν pre + ν post = 1 + 1, τ = 0.10 (for p 3) and τ = 0.11 (for p > 3) Stopping criterion: Eucidean norm of the initia residua is reduced by a factor of ɛ = 10 8.

Concusions (One and) two dimensions

Concusions (One and) two dimensions Robust convergence rates, robust number of smoothing steps

Concusions (One and) two dimensions Robust convergence rates, robust number of smoothing steps Optima compexity in the sense: as compex as the mutipication with K

Concusions (One and) two dimensions Robust convergence rates, robust number of smoothing steps Optima compexity in the sense: as compex as the mutipication with K Rigorous anaysis

Concusions (One and) two dimensions Robust convergence rates, robust number of smoothing steps Optima compexity in the sense: as compex as the mutipication with K Rigorous anaysis C. Hofreither, S. Takacs and W. Zuehner. A Robust Mutigrid Method for Isogeometric Anaysis using Boundary Correction. Submitted (preprint: G+S-Report 33/2105), 2015. S. Takacs and T. Takacs. Approximation error estimates and inverse inequaities for B-spines of maximum smoothness. Submitted (preprint: arxiv:1502.03733), 2015.