Stiffness matrices arising from IgA: a symbol approach A symbol approach in IgA matrix analysis (and in the design of efficient multigrid methods) Stefano Serra-Capizzano, Dept. of Science and high Technology, U. Insubria, Como Joint work with C. Garoni, C. Manni, F. Pelosi, H. Speleers C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der Mee () Cagliari, September 2–5 , 2013 1 / 23
Model problem Model problem � − ∆ u + β · ∇ u + γ u = f on (0 , 1) d (1) on ∂ ((0 , 1) d ) u = 0 f ∈ L 2 ((0 , 1) d ) , β ∈ R d , ( d ≥ 1 , γ ≥ 0) Weak form Find u ∈ H 1 0 ((0 , 1) d ) such that ∀ v ∈ H 1 0 ((0 , 1) d ) a ( u , v ) = F ( v ) (2) where � � a ( u , v ) := (0 , 1) d ( ∇ u · ∇ v + β · ∇ u v + γ u v ) F ( v ) := (0 , 1) d f v ∃ ! solution u ∈ H 1 0 ((0 , 1) d ) to (2), called the weak solution of (1). C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der Mee () Cagliari, September 2–5 , 2013 2 / 23
Galerkin method and IgA To approximate u we consider the Galerkin method. Galerkin method Choose a subspace W ⊂ H 1 0 ((0 , 1) d ) with dim W =: N < ∞ 1 Find u W ∈ W such that 2 a ( u W , v ) = F ( v ) ∀ v ∈ W (3) ⇓ ∃ ! solution u W ∈ W to (3) (whatever W ). Chosen a basis { ϕ 1 , ..., ϕ N } for W , u W has the representation u W = � N i =1 u i ϕ i , with [ u 1 u 2 · · · u N ] T =: u ∈ R N , and problem (3) is equivalent to the following: find u ∈ R N such that A u = f , where A = [ a ( ϕ j , ϕ i )] N i , j =1 is the stiffness matrix and f = [ F ( ϕ i )] N i =1 . In the IgA setting W is chosen as a space of splines. C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der M () Cagliari, September 2–5 , 2013 3 / 23
Spectral analysis (1D case)... the 2D case is similar To simplify both the notation and the presentation, we focus on the model problem (1) in the case d = 1. In this case, we made the following choices in the Galerkin method. W = W [ p ] n , where p ≥ 1, n ≥ 2 and � � W [ p ] s ∈ C p − 1 [0 , 1] : s | [ i := n ) ∈ P p ∀ i = 0 , ..., n − 1 , s (0) = s (1) = 0 n , i +1 n ⊂ H 1 0 (0 , 1) is the space of polynomial splines of degree p defined over the unifor grid n , i = 0 , ..., n , and vanishing at x = 0 , 1 (dim W [ p ] i = n + p − 2). n { ϕ 1 , ..., ϕ n + p − 2 } = basis formed by the polynomial B-splines vanishing at 0 , 1. A [ p ] n := stiffness matrix for the Galerkin problem resulting from these choices C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der Mee () Cagliari, September 2–5 , 2013 4 / 23
A guide to the talk... the importance of the symbol ① Construction of the matrix A [ p ] and computation of a proper symbol f p . n ② Taking the symbol in mind, analysis of the spectral properties of A [ p ] with particular n attention to the following: ✵ (Asymptotic) spectral distribution in the Weyl sense of the sequence of � � n A [ p ] 1 matrices n : n = 2 , 3 , 4 , ... , for fixed p ≥ 1 . ✵ Estimates for the extremal eigenvalues. ✵ Spectral conditioning κ 2 ( A [ p ] n ) . ③ Design of fast (iterative) solvers and constructive use of the symbol. C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der Mee () Cagliari, September 2–5 , 2013 5 / 23
Spectral distribution of a sequence of matrices { X n } We denote by µ m the Lebesgue measure in R m . Definition: (asymptotic) spectral distribution of a sequence of matrices Let { X n } be a sequence of matrices with increasing dimension ( X n ∈ C d n × d n with d n < d n +1 ∀ n ) and let f : D ⊂ R m → C be a measurable function defined on the measurable set D with 0 < µ m ( D ) < ∞ . We say that { X n } is distributed like f in the sense of the eigenvalues, and we write λ { X n } ∼ f , if � d n � 1 1 lim F ( λ j ( X n )) = F ( f ( x 1 , ..., x m )) dx 1 ... dx m ∀ F ∈ C c ( C , C ) µ m ( D ) d n n →∞ D j =1 C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der Mee () Cagliari, September 2–5 , 2013 6 / 23
� n A [ p ] � 1 Spectral distribution of : n = 2 , 3 , 4 , ... in the sense of Weyl: n Szeg¨ o type results (Garoni, Manni, Pelosi, S., Speleers, 2012, under revision for NUMER.MATH.) ∀ p ≥ 0 denote by φ [ p ] : R → R the cardinal B-spline of degree p over the uniform knot sequence { 0 , 1 , ..., p + 1 } : χ [0 , 1) ( x ) se p = 0 φ [ p ] ( x ) = p φ [ p − 1] ( x ) + p + 1 − x x φ [ p − 1] ( x − 1) se p ≥ 1 p ∀ p ≥ 1 let f p : [ − π, π ] → R , � � p − 1 � f p ( θ ) = (2 − 2 cos θ ) φ [2 p − 1] ( p ) + 2 φ [2 p − 1] ( p − k ) cos( k θ ) k =1 Theorem � 1 � n A [ p ] λ ∀ p ≥ 1 , n : n = 2 , 3 , 4 , ... ∼ f p i.e. � � 1 �� � π n + p − 2 � 1 = 1 n A [ p ] lim λ j F ( f p ( θ )) d θ ∀ F ∈ C c ( C , C ) F n n + p − 2 2 π n →∞ − π j =1 C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der Mee () Cagliari, September 2–5 , 2013 7 / 23
Properties of the symbol f p � � n A [ p ] 1 f p is called the symbol of the sequence of matrices n : n = 2 , 3 , 4 , ... ; ∀ p ≥ 1 , ❊ f p (0) = 0 and θ = 0 is the only zero of f p over [ − π, π ]; f p ( θ ) ❊ lim = 1 ⇒ θ = 0 is a zero of order 2; θ 2 θ → 0 ❊ f p ( θ ) > 0 ∀ θ ∈ [ − π, π ] \{ 0 } . Figure: graph of the normalized symbol f p / M f p for p = 1 , 2 , 3 , 4 , 5 , where M f p = θ ∈ [ − π,π ] f p ( θ ) max C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der Mee () Cagliari, September 2–5 , 2013 8 / 23
Estimates for the extremal eigenvalues and for the spectral conditioning (Garoni, Manni, Pelosi, S., Speleers, 2012, under revision for NUMER. MATH.) Theorem ∀ p ≥ 1 there exists a constant C p > 0 such that � � n ) ≥ C p ( π 2 + γ ) � � � λ min ( A [ p ] � ≥ λ min (Re A [ p ] n ) ∀ n ≥ 2 n where λ min ( A [ p ] n ) is an eigenvalue of A [ p ] with minimum modulus; n T n = A [ p ] n + A [ p ] Re A [ p ] n ; 2 γ is the parameter appearing in (1). Theorem ∀ p ≥ 1 there exists a constant α p such that κ 2 ( A [ p ] n ) ≤ α p n 2 ∀ n ≥ 2 C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der Mee () Cagliari, September 2–5 , 2013 9 / 23
Properties of the symbol f p When p increases, the value f p ( π ) / M f p decreases and, apparently, converges exponentially to 0 as p → ∞ . ε p p 1 1.0000 2 0.8889 3 0.4941 4 0.2494 5 0.1289 6 0.0570 7 0.0264 8 0.0120 9 0.0054 10 0.0024 Table: computation of ε p := f p ( π ) / M f p for increasing values of p . Notice that, for p = 3 , ..., 10 , we have (roughly) ε p ≈ 1 2 · ε p − 1 . C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der Mee () Cagliari, September 2–5 , 2013 10 / 23
Properties of the symbol f p When p increases, the symbol shows some non-canonical behavior: Non-canonical behavior at θ = π and for large p of f p , when compared with the symbols occurring in the FD/FE approximating matrices. Ill-conditioning at the low frequencies ( θ = 0: canonical) and, for large p , at the high frequencies ( θ = π : non-canonical). ⇓ Difficulty in the design of efficient multigrid methods. C. Garoni, C. Manni, F. Pelosi, S. Serra, H. Speleers: Nonlinear Evolution Equations and Linear Algebra, in the occasion of the 60th birthday of Cornelis van der Mee () Cagliari, September 2–5 , 2013 11 / 23
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