Geometric Multigrid Methods for the Helmholtz equations

Transcription

1 Geometric Multigrid Methods for the Helmholtz equations Ira Livshits Ball State University RICAM, Linz, 4, 6 November 20 Ira Livshits (BSU) - November 20, Linz / 83

2 Multigrid Methods Aim: To understand geometric multigrid for the Helmholtz equations (difficulties and some solutions) Give a deeper understanding that led to the various choices made. Contents Basics of multigrid Basics of Local Fourier analysis Nonlinear multigrid (FAS) The Helmholtz equation Ira Livshits (BSU) - November 20, Linz 2 / 83

3 Helmholtz equation Lu = u(x) k 2 u(x), x Ω R d with large wave numbers k: 2π/k dim(ω) Ira Livshits (BSU) - November 20, Linz 3 / 83

4 A Comment on Terminology The indefinite Helmholtz equation is in contrast to Lu = ( k 2 )u = f, ( η)u = f, η > 0 often also called Helmholtz equation, or Helmholtz equation with the good sign, or positive definite Helmholtz equation. The subject today is the indefinite Helmholtz equation. Ira Livshits (BSU) - November 20, Linz 4 / 83

5 The 2D discrete Poisson equation in the unit square We consider a stationary diffusion equation with diffusion in two coordinate directions: 2 u(x, y) x 2 2 u(x, y) y 2 = f (x, y) for (x, y) Ω = (0, ) 2 with boundary conditions: u(x, y) = g(x, y) for (x, y) Ω The functions f C(Ω) and g C( Ω) are known. The Laplace Operator is commonly written as: := 2 x y 2 Ira Livshits (BSU) - November 20, Linz 5 / 83

6 The accuracy in x- and y-direction is O(h 2 x), O(h 2 y ) or, with h x = h y = h, O(h 2 ). Ira Livshits (BSU) - November 20, Linz 6 / 83 With mesh size h = /n a grid G is defined as Ω = {(x i, y i ) (x i, y j ) = (ih, jh) IR 2 ; i, j, = 0,,..., n} In analogy to the D case we approximate the partial derivatives in y-direction. For the second derivative in y-direction we obtain: 2 u u i,j 2u i,j + u i,j+ y 2 i,j hy 2

7 Gauss-Seidel and Jacobi for the Poisson equation The discrete Laplace operator reads: h u h = h 2 [u i,j + u i+,j + u i,j + u i,j+ 4u i,j ] or: h u h (x, y) = h 2 [u h(x h, y) + u h (x + h, y) + u h (x, y h) + u h (x, y + h) 4u h (x, y)] Jacobi iteration reads u m+ h (x i, y j ) = [ h 2 f h (x i, y j ) + uh m (x i h, y j ) + uh m (x i + h, y j ) 4 ] +uh m (x i, y j h) + uh m (x i, y j + h), (with (x i, y j ) Ω h ) Gauss-Seidel iteration reads u m+ h (x i, y j ) = 4 [ h 2 f h (x i, y j ) + u m+ h (x i h, y j ) + uh m (x i + h, y j ) ], +u m+ h (x i, y j h) + u m h (x i, y j + h) Ira Livshits (BSU) - November 20, Linz 7 / 83

8 Linear systems of equations in matrix form For the solution of a linear system of equations Au = b, Iterative method: u m+ = Qu m + s, (m = 0,, 2,...), so that the system u = Qu + s is equivalent to the original problem. Matrix Q is usually not computed explicitly! Ira Livshits (BSU) - November 20, Linz 8 / 83

9 From the matrix equation Au = b we construct a splitting A = B + (A B), so that B is easily invertible. With the help of a general nonsingular N N matrix B we obtain such iteration procedures from the equation Bu + (A B)u = b. If we put or, rewritten for u m+ ; Bu m+ + (A B)u m = b, u m+ = u m B (Au m b) = (I B A)u m + B b. We will see, that it is important, to have the eigenvalues of Q = I B A possibly small. This is the more likely, the better B resembles A. Ira Livshits (BSU) - November 20, Linz 9 / 83

10 Linear systems of equations in matrix form Definition: An iterative process u m+ = Qu m + s is called consistent, if matrix I Q is not singular and if (I Q) s = A b. If the iteration is consistent, the equation (I Q)u = s has exactly one solution u m u. Definition: An iteration is called convergent, if the sequence u 0, u, u 2,... converges to a limit, independent of u 0. Ira Livshits (BSU) - November 20, Linz 0 / 83

11 The spectral radius Definition: The spectral radius ρ(q) of a (N, N) matrix Q is { } ρ(q) := max λ : λ eigenvalue of Q Lemma The spectral radius is the infimum over all (vector norm related) matrix norms: { } ρ(q) = inf Q Theorem: The iterative method u m+ = Qu m + s m = 0,, 2,... yields a convergent sequence {u m } for a general u 0, s ( the method converges ), if ρ(q) <. In the case ρ(q) <, matrix I Q is regular, there is for each s exactly one fix point of the iterative scheme. Ira Livshits (BSU) - November 20, Linz / 83

12 Error reduction General splitting of A (another way of writing): Bu m+ + (A B)u m = b = Au = Bu + (A B)u or: B(u m+ u ) = (B A)(u m u ), So, (u m+ u ) = Q(u m u ). By induction, we can obtain: (u m u ) = Q m (u 0 u ). Assume ρ(q) <. Ira Livshits (BSU) - November 20, Linz 2 / 83

13 Efficient solution methods for discrete equations Hierarchy One level methods, like Jacobi, Gauss-Seidel iterative methods are far from optimal: O(N α ), α >. O(N) complexity requires hierarchical approaches. Multigrid is the classical hierarchical approach Approximation on different scales Compute only those parts of a solution on fine scales that really require the fine resolution. Compute/correct the remaining parts on coarser scales Gives O(N) complexity Advantages increase with increasing problem size The idea behind multigrid is very general Ira Livshits (BSU) - November 20, Linz 3 / 83

14 Relaxation of u = (a) Random initial error, (b) Error after 0 GS iterations, (c) Error after 0 more GS iterations. Ira Livshits (BSU) - November 20, Linz 4 / 83

15 Convergence factor vs smoothing factor Convergence factor (GS, Jacobi, wjacobi, SOR) is ρ = O(h 2 ) Smoothing factor is a convergence factor for oscillatory components, and it seems to be good. Remaining error components are smooth. Ira Livshits (BSU) - November 20, Linz 5 / 83

16 Motivation to use coarse grids The remaining after relaxation, aka smoothing, error has a good coarse grid representation. Ira Livshits (BSU) - November 20, Linz 6 / 83

17 Detour to local Fourier analysis with a hint of trouble brewing for the Helmholtz The basic idea of any multigrid algorithm is to annihilate the high frequency error components efficiently on a fine grid by relaxation (smoothing) while the low frequencies are approximated from coarser grids. Smoothing analysis is used to analyze quantitatively this reduction of the high frequency error components on the fine grid. Assuming the ideal situation in which the relaxation does not affect the low frequencies all low frequencies are approximated more or less exactly on the coarse grid there is no interaction between high and low frequency error components it already leads to an estimate or prediction of the 2 level convergence factor. Ira Livshits (BSU) - November 20, Linz 7 / 83

18 Assumptions For a discretized PDE: L h U h = F h With the basic assumptions of any local Fourier analysis: linearize operator L h and/or freeze coefficients to local values neglect boundaries and boundary conditions consider the frozen, linearized operator on an infinite grid G h apply the linear constant coefficient local Fourier analysis Ira Livshits (BSU) - November 20, Linz 8 / 83

19 Components, the D case Infinite uniform mesh, meshsize h G h = {k h : k Z} D Fourier components: e iωx = cos ωx + i sin ωx On G h only Fourier components e iθx/h with θ ( π, π] are visible. Definition: A component e iθx/h is visible on G h if there is no frequency θ 0 with θ 0 < θ such that e iθ0x/h = e iθx/h for all x G h. A visible component e iθx/h does not coincide with any lower frequency on grid G h. θ frequency, 2πh θ period Ira Livshits (BSU) - November 20, Linz 9 / 83

20 Components, the 2D case Infinite uniform mesh, meshsize h = (h x, h y ) in x and y direction: G h = {(kh x, lh y ) : k, l Z} θ = (θ, θ 2 ) frequency On G h, only Fourier components e iθx/h := e iθx/hx e iθ2y/hy with θ := max ( θ, θ 2 ) π are visible. For higher frequencies θ > π, the Fourier components coincide with a lower frequency ( π, π]. For higher dimensions: analogously Ira Livshits (BSU) - November 20, Linz 20 / 83

21 Difference operators Let G h be an infinite uniform mesh, meshsize h ; u h : grid function on G h ; I = { ν,..., 0,..., ν}: set of indices. Difference operators L h : 2D x = (x, y) G h L h u h (x) = a µµ 2 u h (x + µ h x, y + µ 2 h y ) µ I µ 2 I Ira Livshits (BSU) - November 20, Linz 2 / 83

22 The stencil notation, examples 2D finite differences for u, I = {, 0, } a a 0 a L h u h (x) = a 0 a 00 a 0 u h (x) a a 0 a = h 2 4 u h (x) 5 point stencil } {{ } h or for the biharmonic operator 2, I = { 2,, 0,, 2} h h u h (x) = h 4 3-point stencil Here: x = (x, y) G h, h x = h y = h u h(x) Ira Livshits (BSU) - November 20, Linz 22 / 83

23 Motivation to consider Biharmonic equation is Kaczmarz relaxation Kaczmarz relaxation (equivalent to GS applied to AA ) Implementation: Consider i th equation: A i x = b i and a current approximation x, here A i is the i th row of A. A new approximation is given by x x + r i A i, where the normalized residual is computed as r i = b i A i x A i A. i Kaczmarz relaxation always converges (indeed, AA t is always SPD) but it is often slow (much slower than the GS on original matrix) and thus should be used only as the last reserve (µ = 0.8 for Laplace vs GS s µ = 0.25). Ira Livshits (BSU) - November 20, Linz 23 / 83

24 Difference operators applied to Fourier components On an infinite grid G h, the Fourier components e iθx/h are eigenfunctions of any scalar difference operator L h with constant coefficients. D: L h e iθx/h = a µ e iθµ µ I }{{} Lh (θ) e iθx/h 2D: L h e iθx/h = a µµ 2 e iθµ e iθ2µ2 µ I µ 2 I }{{} Lh (θ) e iθx/h Lh (θ) is the eigenvalue, also called the Fourier symbol of L h. Ira Livshits (BSU) - November 20, Linz 24 / 83

25 Examples D: L h = h 2 [ 2 ] = L h e iθx/h = ( e iθ h e iθ) e iθx/h = 2 (cos θ ) h2 }{{} Lh (θ) e iθx/h 2D: θ = (θ, θ 2 ) L h = h = h 2 4 = L h (θ) = 2 h 2 (cos θ + cos θ 2 2) or for the biharmonic operator L h = h h (3-point stencil): Lh (θ) = 4 h 4 (cos θ + cos θ 2 2) 2 Ira Livshits (BSU) - November 20, Linz 25 / 83

26 Smoothing analysis; a general class of relaxations: Consider a linear problem L h U h (x) = F h on grid G h Then: Many (not all!) relaxation schemes for this problem are based on an operator splitting L h = A h + B h where A h and B h are again difference operators. Then: A relaxation sweep starting from the initial approximation u h ( old values ) produces a new approximation u h ( new values ) by solving A h u h (x) + B h u h (x) = F h (x) at each gridpoint x G h. Ira Livshits (BSU) - November 20, Linz 26 / 83

27 Smoothing analysis; a general class of relaxations: What does that mean for the errors before and after the relaxation sweep? Let e h = U h u h be the error before and e h = U h u h after the sweep. Then: A h e h (x) + B h e h (x) = 0 at each gridpoint x G h. How do these relaxations act on Fourier components? Let e h = Ae iθx/h : error before relaxation; then e h (x) = Ae iθx/h error after relaxation. A, A R are the error amplitudes before and after relaxation. Ira Livshits (BSU) - November 20, Linz 27 / 83

28 Smoothing analysis; a general class of relaxations: From A h e h (x) + B h e h (x) = 0 follows à h (θ)a + B h (θ)a = 0 and therefore ) (Ãh (θ) A = A B h (θ) where Ãh(θ), B h (θ) C are the Fourier symbols of A h, B h. µ(θ) := Ãh(θ) B h (θ) is the amplification factor of the component θ. Ira Livshits (BSU) - November 20, Linz 28 / 83

29 D Example:Gauss-Seidel relaxation Corresponding splitting of L h : [ 2 ] = [0 0 ] + [ 2 0] h2 }{{} h2 }{{} h2 }{{} L h A h B h Then: at gridpoint x G h for the errors Ae iθ(x h)/h 2Ae iθx/h + Ae iθ(x+h)/h = 0 ( e iθ 2 ) } {{ } B h (θ) A + ( e iθ) }{{} Ã h (θ) A = 0. This yields: A = eiθ e iθ 2 A Ira Livshits (BSU) - November 20, Linz 29 / 83

30 Questions What is smoothing? How can the smoothing property of a given relaxation scheme be measured quantitatively? How does the smoothing property (or its quantitative measure) influence the performance of multigrid cycling? Can the performance of a multigrid cycle be predicted in terms of the smoothing measures? Note: In a multigrid cycle, relaxations are used to smooth the highly oscillating error components which cannot be approximated on coarser grids. Therefore, smoothing roughly means convergence of high frequency error components. Ira Livshits (BSU) - November 20, Linz 30 / 83

31 What are high and low frequencies? Let G h be a fine grid and G H a coarse grid. Then: A Fourier component e iθx/h on grid G h is called a high frequency (with respect to G H ) if its restriction (injection) to the coarse grid G H is not visible there. Otherwise it is called a low frequency. Take one dimensional standard coarsening: G h = {kh : k Z}, G H := G 2h = {2kh : k Z} Let θ π; then by injection from G h to G 2h : iθx/h inject e = e i2θx/2h This component e i2θx/2h is visible on G 2h only if 2θ π, i.e. θ π 2 Ira Livshits (BSU) - November 20, Linz 3 / 83

32 = The high frequencies on G h (with respect to G 2h ) are those with π θ π 2. Fourier components in 2D: e iθx/h = e iθx/h e iθ2y/h, θ = (θ, θ 2 ) with θ := max( θ, θ 2 ) π Standard coarsening G h = {(kh, lh) : k, l Z} G H = G 2h = {(2kh, 2lh) : k, l Z} = High frequencies: π 2 θ π Low frequencies: θ < π 2 Note: The definition of high and low frequencies depends on the coarsening. Ira Livshits (BSU) - November 20, Linz 32 / 83

33 What is smoothing? Smoothing stands for convergence of high frequency error components which cannot be approximated from the coarse grids in a multigrid cycle. For a relaxation scheme with amplification factors µ(θ) of Fourier components e iθx/h the smoothing rate µ is defined by: µ := max{ µ(θ) : θ = high frequencies} Note: The above definition of µ depends on the coarsening. The amplification or reduction factors µ(θ) are also called smoothing factors. Ira Livshits (BSU) - November 20, Linz 33 / 83

34 Example: Gauss Seidel relaxation for Poisson s equation: h U h = h 2 4 Relaxation in error terms: h 2 e h (x) + h 2 }{{} B h Symbols: θ = (θ, θ 2 ) U h = F h } {{ } A h B h e iθx/h = ( e iθ h 2 + e iθ2 4 ) }{{} B h (θ) e h (x) = 0 e iθx/h A h e iθx/h = ( e iθ h 2 + e iθ2) }{{} e iθx/h à h (θ) Ira Livshits (BSU) - November 20, Linz 34 / 83

35 = Amplification factors: µ(θ) = Ãh(θ) B h (θ) = e iθ + e iθ2 e iθ + e iθ2 4 = Smoothing rate: µ = max{ µ(θ) : π 2 θ π} 0.5 = The high frequency error components are reduced by a factor of (at least) µ per relaxation sweep. Note: For low frequencies, the reduction per relaxation sweep is much worse: µ(θ) if θ 0 Ira Livshits (BSU) - November 20, Linz 35 / 83

36 Now to Helmholtz : the stencils and the relaxations D finite differences for Lu = u + k 2 u, L h = h 2 [ 2 + (kh)2 ] 2D finite differences for Lu = u + k 2 u, L h = 4 + (kh) 2 h 2 Here: x = (x, y) G h, h x = h y = h. 5 point stencil Ira Livshits (BSU) - November 20, Linz 36 / 83

37 Fourier symbols D: L h = h 2 [ 2 + k 2 h 2 ] = L h e iθx/h = h 2 (2 cos θ 2 + k2 h 2 ) }{{} Lh (θ) e iθx/h 2D: θ = (θ, θ 2 ) = L h e iθx/h = h 2 (2 cos θ + 2 cos(θ 2 ) 4 + k 2 h 2 ) }{{} Lh (θ) e iθx/h or for the biharmonic operator (think Kaczmarz!) Lh (θ) = h 4 (2 cos θ + 2 cos θ k 2 h 2 ) 2 Ira Livshits (BSU) - November 20, Linz 37 / 83

38 D Example: Gauss-Seidel relaxation of Relaxation: Corresponding splitting of L h : h 2 [ 2 + k2 h 2 ] U h (x) = F h (x) }{{} L h u h = u h u h (x h) (2 k 2 h 2 )u h (x) + u h (x + h) h 2 [ 2 + k2 h 2 ] }{{} L h h 2 = [0 0 ] h2 }{{} A h Then: at grid point x G h for the errors = F h (x) + h 2 [ 2 + k2 h 2 0] }{{} B h Ae iθ(x h)/h (2 k 2 h 2 )Ae iθx/h + Ae iθ(x+h)/h = 0 Ira Livshits (BSU) - November 20, Linz 38 / 83

39 ( e iθ 2 + k 2 h 2) }{{} B h (θ) A + ( e iθ) }{{} Ã h (θ) A = 0. This yields: A = e iθ e iθ (2 k 2 h 2 ) A What is the difference compared to Laplace? What happens when θ = 0? What happens when θ is between π/2 and π? What happens when θ kh? Happens when kh grows, i.e., k is fixed and h is coarsened? Ira Livshits (BSU) - November 20, Linz 39 / 83

40 Answers If kh i.e., exp(±ikx) are smooth: other smooth components θ 0 remain almost unchanged by relaxation; For θ = ±kh there is a slight divergence; Oscillatory components π/2 < θ π converge almost as fast as in Laplace (convergence does depend on kh); If kh = O() divergence worsens! Results for kh = 0. and kh = Ira Livshits (BSU) - November 20, Linz 40 / 83

41 More examples: 2D Helmholtz equation: Gauss-Seidel Relaxation in error terms: k 2 h 2 0 h 2 e h (x) + h 2 }{{} B h } {{ } A h e h (x) = 0 Symbols: θ = (θ, θ 2 ) B h e iθx/h = ( e iθ h 2 + e iθ2 4 + k 2 h 2) }{{} B h (θ) e iθx/h A h e iθx/h = ( e iθ h 2 + e iθ2) }{{} Ã h (θ) e iθx/h Ira Livshits (BSU) - November 20, Linz 4 / 83

42 = Amplification factors: µ(θ) = Ãh(θ) B h (θ) = e iθ + e iθ2 e iθ + e iθ2 4 + k 2 h 2 = Smoothing rate (what is smooth here?): µ µl for kh ; µ deteriorates (grows) as kh grows; Ira Livshits (BSU) - November 20, Linz 42 / 83

43 Questions What is smooth? Is θ = 0 smooth? What is the smoothing factor? What do we want from relaxation? Ira Livshits (BSU) - November 20, Linz 43 / 83

44 Kaczmarz relaxation c = 4 + k 2 h 2 h 4 h h u h (x) = h c 4 + c c 2 } {{ } B h 2 2c 2 2c 4 + c 2 2c 2 2c 2 e h (x) + h 4 u h(x) 2 2c c } {{ } A h e h (x) = 0 Ira Livshits (BSU) - November 20, Linz 44 / 83

45 Symbols and amplification Symbol B h e iθx/h = ) ( (e 2iθ + e 2 iθ 2 ) + 2(e iθ iθ 2 + e iθ +iθ 2 ) + 2c(e iθ + e iθ 2 ) + (4 + c 2 ) e iθx/h h 4 } {{ } B h (θ) Symbol A h e iθx/h = ( h 4 (e 2iθ + e 2iθ 2 ) + 2(e iθ iθ 2 + e iθ +iθ 2 ) + 2c(e iθ + e iθ ) 2 ) e iθx/h }{{} Ã h (θ) Ira Livshits (BSU) - November 20, Linz 45 / 83

46 Good news: µ(θ) for any θ Q: But for which is works the best? Results for kh = (high frequencies and overall convergence). Ira Livshits (BSU) - November 20, Linz 46 / 83

47 More on convergence Consider ordered eigen pairs of A: (λ j, v j ): Av j = λ j v j such that λ λ 2 λ N. For errors representation e m+ = αn m+ v n and e m = αn m v n, Due to linearity (of everything), α m+ n v n = αn m Qv n and thus overall convergence is defined by max α m+ n n αn m The smoothing factor is a convergence factor for α n/2,..., α n, i.e., for the high energy eigenfunctions v n/2,... v n Remaining non reduced low energy eigenfunctions v,..., v n/2 are smooth. Ira Livshits (BSU) - November 20, Linz 47 / 83

48 Reason for using coarse grids Coarse grids can be used to compute an improved initial guess for the fine-grid relaxation. This is advantageous because: Relaxation on the coarse-grid is much cheaper (/2 as many points in D, /4in 2D, /8 in 3D); Relaxation on the coarse grid has a marginally better convergence rate, for example O(4h 2 ) instead of O(h 2 ); Smooth eigenfunctions on the finest grid become more oscillatory on the coarse grid. Ira Livshits (BSU) - November 20, Linz 48 / 83

49 Multigrid components Relaxation (smoother): efficiently dumps high-energy (eigenfunctions with large eigenvalues) error components; leaves unchanged low eigenmodes; A coarse-grid system operator: accurately resolve the low eigenmodes; Restriction operator (averaging): transfers fine-grid information (residual) to coarse grid; Prolongation operator (interpolation): transfers coarse-grid correction back to the fine grid; has to be accurate for low eigenmodes. Ira Livshits (BSU) - November 20, Linz 49 / 83

50 Nice example: u = f Relaxation: Gauss-Seidel. A coarse-grid operator: discretization of the differential Laplace operator on coarse scale (2h, 4h,... ) Restriction operator: full weighting of the residual (annihilates remaining oscillatory components), transferring only smooth ones to coarser grids; Interpolation: polynomial (linear) which is accurate for smooth components. Works great because the lowest eigenmodes are physically smooth. Ira Livshits (BSU) - November 20, Linz 50 / 83

51 One picture is worth a thousand words Ira Livshits (BSU) - November 20, Linz 5 / 83

52 Back to 2d Helmholtz : discretization error Discrete wave number vs exact wave number: k h k ( + k2 h 2 ), 24 γ 48. γ The total (accumulated) phase error is E(kd, kh) = kd k2 h 2 2πγ. For the discrete solutions to be accurate approximation to the differential solutions need E(kd, kh). For fixed k, d this means small h and large problem size, N. Ira Livshits (BSU) - November 20, Linz 52 / 83

53 Bottlenecks for multigrid Dominant phase discretization error; Quality of standard coarse-grid and prolongation operators deteriorates on coarse grids; Poor multigrid relaxation - standard fast relaxation scheme diverge; Ira Livshits (BSU) - November 20, Linz 53 / 83

54 Lowest eigenmodes Lowest eigenmodes for k = 6π and k = 8π. Ira Livshits (BSU) - November 20, Linz 54 / 83

55 More bottlenecks for multigrid Oscillatory low eigenmodes; There are many of them; They are different from each other; On sufficiently coarse grids, there is no single prolongation or coarse grid operator of a reasonable sparsity that works for all such eigenmodes. Ira Livshits (BSU) - November 20, Linz 55 / 83

56 One-dimensional Helmholtz equation: slow to converge components, aka lowest eigenmodes (a) a random initial error; (b)-(d) examples of remaining errors, obtained by applying three multigrid V-cycles to the homogenous Helmholtz equation (periodic boundary conditions) Ira Livshits (BSU) - November 20, Linz 56 / 83

57 Assumption on the error The error unreduced by standard multigrid e(x) = exp(iwx) = α w exp(iwx) + β w exp( iwx) w k w k w k where the value of w is determined by the relaxation history. One can drive it to be ( γ )k w ( + γ 2 )k, where typical are values γ j 0.3 Ira Livshits (BSU) - November 20, Linz 57 / 83

58 Dual representation If we agreed on the previous page, we can also agree that e(x) = e (x)exp(ikx) + e 2 (x)exp( ikx), where e (x) = w k α w exp(i(w k)x) and e 2 (x) = w k β w exp( i(w k)x). Obviously, envelope functions e and e 2 are smooth (compared to exp(±ikx)). Indeed the highest frequencies are w k γ j k Remark : Representation is not unique - one has to work to guarantee the statement above. The idea here is simple Lets try approximate smooth e (x) and e 2 (x) instead of oscillatory e(x) (which we cannot do well anyway) and hope for the best. Ira Livshits (BSU) - November 20, Linz 58 / 83

59 Defect correction If we assume the error in the form e(x) = e (x)exp(ikx) + e 2 (x)exp( ikx), then naturally r(x) = Le(x) = L(e (x)exp(ikx) + e 2 (x)exp( ikx)) = L(e (x)exp(ikx)) + L(e 2 (x)exp( ikx)) = exp(ikx)l e (x) + exp( ikx)l 2 e 2 (x) Ira Livshits (BSU) - November 20, Linz 59 / 83

60 Differential operators and residual representation Two conclusions from the previous page: Note that we use the differential Helmholtz operators and its kernel components, resulting in differential equations for the envelope functions e and e 2, L and L 2 : L u = u xx + 2iku x and L 2 u = u xx 2iku x. Trick! Operator L has two kernel components (exp(±ikx)) and so do L j ( and exp( 2ikx)) But we consider grids where only one (a constant) is visible! Also, given the form of e j and L j, we can claim that r(x) = r (x)exp(ikx) + r 2 (x)exp(ikx) where r j are smooth (as smooth as e j in fact). Ira Livshits (BSU) - November 20, Linz 60 / 83

61 From Waves to Rays To reduce the irreducible wave error e(x): Le = r we approximate two ray errors L e = r and L 2 e 2 = r 2 and then reconstruct back to the wave e(x) = e (x)exp(ikx) + e 2 (x)exp( ikx) Where is the gain: To compute the ray functions one should start on the grid with kh π (very coarse) To accelerate, one may use standard multigrid on each L j e j = r j if wishes to go coarser. Ira Livshits (BSU) - November 20, Linz 6 / 83

62 Residual separation and why on kh π Consider the wave (Helmholtz) residual with r and r 2 are smooth r(x) = r (x)exp(ikx) + r 2 (x)exp( ikx). Clearly, such representation is not unique and some others allow r(x) = r (x)exp(ikx) + r 2 (x)exp( ikx) with oscillatory r j. We want to approximate smooth ray residuals (on coarse grids). Ira Livshits (BSU) - November 20, Linz 62 / 83

63 Separation (implementation considered on grids) Compute fine-grid wave residual r h = f h L h u h (h is fine enough for small phase error kd(kh) 2 ) Transfer it to the grid with kh s π/2 using full weighting (annihilates physically oscillatory error components), resulting in r Hs = r Hs Hs exp(ikx) + r2 exp( ikx) To approximate r : Multiply by exp( ikx) exp( ikx)r Hs = r Hs + r Hs 2 exp( 2ikx) Note that first term is smooth and the second one is extremely oscillatory 2kH s π. Applying one more full weighting results in the desired ray residual I 2Hs H s (exp( ikx)r Hs ) = I 2Hs H s (r Hs ) + I 2Hs H s (r Hs 2Hs 2 exp( 2ikx)) r Ira Livshits (BSU) - November 20, Linz 63 / 83

64 Ray operator The discrete operators for the ray functions are derived from the differential operators: L u = u + 2iku = f considered on the grid with kh π. The choice of discretization is defined by the principality of terms: The second-order term : H 2 The first-order term: 2k H If kh π the first-order term is principal and should be accurately discretized; Thus using staggered grids. [ L H j = H 2 ( 2ik H ) ( 2ik H 2 H ) H 2 H 2 ] Ira Livshits (BSU) - November 20, Linz 64 / 83

65 Wave-ray cycle Wave cycle: Reduces all but problematic characteristic components; Produces residual for the ray cycles (on one of the fine grids with small phase error); Finest grid: (kd)(k 2 h 2 ) Coarsest grid: kh > π (Used to effectively wipe out constants and alikes, stencil [ (k 2 H 2 2) ] Correction Scheme; GS one pre- and one post relaxation on fine grid and on the coarsest grid; Kaczmarz four pre- and one post relaxation sweeps everywhere else; Linear interpolation and full weighting; Two ray cycles to reduce the characteristic components. Ira Livshits (BSU) - November 20, Linz 65 / 83

66 Ray cycles Staggered grid, short central differences for u x ; Marching scheme in the propagation direction; CS or FAS multigrid scheme depending on interpretation; Natural introduction of boundary conditions in terms of rays. Ira Livshits (BSU) - November 20, Linz 66 / 83

67 Before we proceed, detour to FAS Full Approximation Scheme Allows coarse grid approximation of the solution u H rather than correction e H ; Considers full equation L H u H = f H residual equation L H e H = r h ; Developed for solving non linear equations. Ira Livshits (BSU) - November 20, Linz 67 / 83

68 FAS details Consider ũ h a current fine-grid solution and r h = f h L h ũ h is its corresponding residual; CS: Coarse grid variable correction: v H Coarse grid system: L H v H = R H = Ih H r h Coarse grid correction: ũ h NEW = ũ h + IHv h H FAS: Coarse grid variable solution: ũ H = Ĩ h H ũ h + v H Coarse grid system: L H u H = f H = L H (Ĩ h H u h ) + R H Coarse grid correction: ũ h NEW = u h + IH(ũ h NEW H Ĩ h H ũ h ) If L H resolves all solution components then ũnew h = IH(ũ h NEW H ) Ira Livshits (BSU) - November 20, Linz 68 / 83

69 τ-correction Consider the c.g. equation as L H u H = f H + τ H h where Term τ H h f H = I H h f h, τ H h = L H (Ĩ H h u h ) I H h (L h u h ) is a fine-to-coarse defect correction. Nonlinear equations: The correction equation is L h (ũ h + v h ) L h ũ h = r h. On coarse grid its reads: L H (Ĩ H h ũh + v H ) L H (Ĩ H h ũh ) = I H h r h which is exactly the top equation. Ira Livshits (BSU) - November 20, Linz 69 / 83

70 Rays, directions and BC Ira Livshits (BSU) - November 20, Linz 70 / 83

71 Back to Boundary conditions Assumptions: Away from the source (near the boundary) the solution has a pure ray character so can be described using ray operators; The typical boundary conditions are the radiation BC, sometimes combined with the Dirichlet boundary conditions. The rays can enter the computational domain from the infinity and leave the computational domain freely unless there is an obstacle, media change, etc. The entering part is to be defined by the Direchlet boundary conditions at the entrance (for each ray); the exiting part is controlled by the Sommerfield boundary conditions at the exit (the latter can be replaced by an interior equation appropriately discretized). If no rays (waves) enter from infinity, the Dirichlet boundary conditions are homogenous. Ira Livshits (BSU) - November 20, Linz 7 / 83

72 Full ray formulation (for two-sided Sommerfield BC) Exiting Sommerfield BC: At x = 0: u + 2iku = 0 and at x = d: u 2iku = 0 Positively propagating ray (corresponding to exp(ikx)) L a(x) = a (x) + 2ika (x) = f (x), a (0) = a,0, a (d) = 0 Negatively propagating ray (corresponding to exp( ikx)) L 2 a(x) = a 2 (x) + 2ika 2(x) = f 2 (x), a 2 (d) = a 2,0, a (0) = 0 Two systems are solved independently; If there is a Dirichlet BC on one side, the two has two be combined (incident wave defines the amplitude of the reflected wave); Ira Livshits (BSU) - November 20, Linz 72 / 83

73 Wave-ray interaction Because of BC the ray cycles are implemented in FAS (solution representation); At the interior, regular FAS correction: ũ h NEW = ũh + exp(ikx)(i h H (ãh,new ãh )) + exp( ikx)(i h H (ãh 2,NEW ãh 2 )); At the boundary, direct replacement ũ h NEW = exp(ikx)(i h H ãh,new ) + exp( ikx)(i h H ãh 2,NEW ); Ira Livshits (BSU) - November 20, Linz 73 / 83

74 Conclusions on d The solver is a combination of the wave and the ray approaches; The complexity is similar to one for Laplace, it is O(N). Worked for jumping and slightly varying wave numbers; Ira Livshits (BSU) - November 20, Linz 74 / 83

75 Wave-Ray algorithm Geometric solver for the 2D Helmholtz equation with constant wave numbers Brandt, Livshits: Wave-ray multigrid method solving standing wave equations, 997 Livshits, Brandt: Accuracy Properties of the Wave-Ray Multigrid Algorithm for Helmholtz Equations, 2006 Special feature: Separate treatment of oscillatory kernel e ikx, k = k on coarse grids. Ira Livshits (BSU) - November 20, Linz 75 / 83

76 Directions of propagation e ikx Ira Livshits (BSU) - November 20, Linz 76 / 83

77 Discretization of the propagation directions: few chosen direction k l Ira Livshits (BSU) - November 20, Linz 77 / 83

78 Wave-ray approach Components not changed (at best) by the standard multigrid: exp(iwx), w k, w, x R 2 Ray representation for such components u(x) and residual r(x) u(x) = l u l (x)exp(ik l x) and r(x) = l r l (x)exp(ik l x) Instead of oscillatory u(x) compute smooth u l (x); Differential operators for u l (x) are obtained directly from L. Ira Livshits (BSU) - November 20, Linz 78 / 83

79 Phase error: Wave vs Rays Waves (as discussed): E(kd, kh) kd k2 h 2, 24 γ 48 2πγ Rays: E(kd, L) kd β 2πL 2, β 0.2 This is assuming very aggressive coarsening with L n = 2 n+2 : Propagation direction ξ: h n ξ = (L n ) 2 /(6k); Orthogonal direction η: h n η = CL n /4k For L = 8, kh ξ 4 and kh η 2 Ira Livshits (BSU) - November 20, Linz 79 / 83

80 Main features Smaller wave computational domain; Larger ray domains; grids aligned with propagation directions; Agressive ray coarsening; Ray equations a ξξ + a ηη + 2ika ξ, ξ = k l Ray residual separation using consecutive full weighting; FAS for both waves and rays - boundary values for waves come directly from rays; Ira Livshits (BSU) - November 20, Linz 80 / 83

81 More about rays Ira Livshits (BSU) - November 20, Linz 8 / 83

82 Algorithm Repeat until happy CS standard wave V cycle FAS ray cycle: computation of wave residual kh ; separation into ray residuals starting kh ; forming FAS ray rhs, line relaxation kh ξ 4, kh η 2; reconstructing to the wave form kh ; interpolating (with relaxation) to the finest wave grid kh. Ira Livshits (BSU) - November 20, Linz 82 / 83

83 Requirements and advantages Requirements Helmholtz PDE Structured grids Analytical knowledge of Lu 0 Advantages Local processes - wave representation on fine grids Geometrical optics processes - ray representation on coarse grids and larger domains Natural intro of radiation boundary conditions Ira Livshits (BSU) - November 20, Linz 83 / 83