Theory and Computation for Bilinear Quadratures
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1 Theory and Computation for Bilinear Quadratures Christopher A. Wong University of California, Berkeley 15 March 2015 Research supported by NSF, AFOSR, and SIAM C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
2 Classical and bilinear quadratures Classical quadrature (approximates linear functionals): Ω f (x) dµ = [ f (x 1) ] w 1... w n. f (x n) Image from Wikimedia Commons C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
3 Classical and bilinear quadratures Classical quadrature (approximates linear functionals): Ω f (x) dµ = [ f (x 1) ] w 1... w n. f (x n) Image from Wikimedia Commons Bilinear quadrature (approximates bilinear forms): f (x)g(x) dµ = [ f (y 1)... f (y ] w w 1n g(x 1) m).... Ω w m1... w mn g(x n) C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
4 Classical and bilinear quadratures Classical quadrature (approximates linear functionals): Ω f (x) dµ = [ f (x 1) ] w 1... w n. f (x n) Image from Wikimedia Commons Bilinear quadrature (approximates bilinear forms): f (x)g(x) dµ = [ f (y 1)... f (y ] w w 1n g(x 1) m).... Ω w m1... w mn g(x n) Useful for finite element/galerkin method. C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
5 Classical and bilinear quadratures Classical quadrature (approximates linear functionals): Ω f (x) dµ = [ f (x 1) ] w 1... w n. f (x n) Image from Wikimedia Commons Bilinear quadrature (approximates bilinear forms): f (x)g(x) dµ = [ f (y 1)... f (y ] w w 1n g(x 1) m).... Ω w m1... w mn g(x n) Useful for finite element/galerkin method. Previous work: Boland and Duris (1972), McGrath (1979), Gribble (1980), Chen (2012) C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
6 Advantages f s and g s can belong to different spaces. Can do any continuous bilinear form, e.g. f, g H 1 = fg + Du Dv, f, g = Very useful in Galerkin methods! Ω Ω fg + fg. Ω The right method for computing an orthogonal projection onto a fixed function space. Image courtesy of P.-O. Persson C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
7 Looking for a bilinear quadrature Choose L 2 inner product on some finite-dim. function space X 0 (e.g. polynomials) C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
8 Looking for a bilinear quadrature Choose L 2 inner product on some finite-dim. function space X 0 (e.g. polynomials) Pick orthonormal basis {f 1,..., f k } for X 0. Solve for W = (w ij ) and x = {x i } such that: F (x) T WF (x) = I k where F (x) := f 1(x 1)... f k (x 1).. f 1(x n)... f k (x n) C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
9 Looking for a bilinear quadrature Choose L 2 inner product on some finite-dim. function space X 0 (e.g. polynomials) Pick orthonormal basis {f 1,..., f k } for X 0. Solve for W = (w ij ) and x = {x i } such that: F (x) T WF (x) = I k where F (x) := f 1(x 1)... f k (x 1).. f 1(x n)... f k (x n) Approximate coefficients of Proj X0 (φ) using matvec F (x) T W φ(x) C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
10 Nonlinear optimization Find points x by minimizing an objective. Inspired by Chen, Rokhlin, and Yarvin (1999), Bremer, Gimbutas, Rokhlin (2010), Xiao and Gimbutas (2010) C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
11 Nonlinear optimization Find points x by minimizing an objective. Inspired by Chen, Rokhlin, and Yarvin (1999), Bremer, Gimbutas, Rokhlin (2010), Xiao and Gimbutas (2010) Choose orthonormal set {g 1,..., g p} orthogonal to X 0 and minimize the pairing with X 0. C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
12 Nonlinear optimization Find points x by minimizing an objective. Inspired by Chen, Rokhlin, and Yarvin (1999), Bremer, Gimbutas, Rokhlin (2010), Xiao and Gimbutas (2010) Choose orthonormal set {g 1,..., g p} orthogonal to X 0 and minimize the pairing with X 0. g 1(x 1)... g p(x 1) G(x) =.., g 1(x n)... g p(x n) then must solve Find x, W minimizing F (x) T WG(x) 2 subject to F (x) T WF (x) = I k C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
13 Nonlinear optimization Find points x by minimizing an objective. Inspired by Chen, Rokhlin, and Yarvin (1999), Bremer, Gimbutas, Rokhlin (2010), Xiao and Gimbutas (2010) Choose orthonormal set {g 1,..., g p} orthogonal to X 0 and minimize the pairing with X 0. g 1(x 1)... g p(x 1) G(x) =.., g 1(x n)... g p(x n) then must solve Find x, W minimizing F (x) T WG(x) 2 subject to F (x) T WF (x) = I k Don t have access to derivatives. Use a quasi-newton method (e.g. BFGS). Need many initial guesses to get close to the minimum. C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
14 Connection with familiar classical quadratures Theorem Given: n-point bilinear quadrature on an interval, exact for L 2 inner products on P n 1 P n 1, minimized over degree n polynomials, then W = diag(gauss weights) and x are the Gauss points from Gaussian quadrature! C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
15 Connection with familiar classical quadratures Theorem Given: n-point bilinear quadrature on an interval, exact for L 2 inner products on P n 1 P n 1, minimized over degree n polynomials, then W = diag(gauss weights) and x are the Gauss points from Gaussian quadrature! Theorem Given: n-point bilinear quadrature on [0, 2π], n odd, exact for L 2 inner products on T (n 1)/2 T (n 1)/2, minimized over span{sin n+1 n+1 x, cos x}, 2 2 then W = diag( π, 2π,..., 2π, π ) and x are uniformly spaced points on [0, 2π]! n n n n C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
16 Two numerical results The nodes of 28-point bilinear quadratures on P 6 P 6, minimized against P 7, for both the triangle and the square. Notice symmetry! C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
17 Accuracy Compare with two very good 28-point classical quadratures on triangles. Project onto P6. Calculate relative 2-norm error in projection coefficients for random functions from four different spaces. P5 9.38e e e-15 Dunavant (1985) Xiao et al. (2010) Bilinear Cauchy 4.96e e e-06 P6 3.97e e e-15 Trig 9.12e e e C. Wong (UC Berkeley) Bilinear Quadratures March /9
18 Conclusions Advantages: Ideal for computing orthogonal projections. Independent of domain or choice of functions. C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
19 Conclusions Advantages: Ideal for computing orthogonal projections. Independent of domain or choice of functions. Disadvantages A bilinear quadrature does less than a general-purpose classical quadrature. Expensive to construct; optimization is slow. C. Wong (UC Berkeley) Bilinear Quadratures 15 March / 9
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