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Maria Ugryumova Direct Solution Techniques in Spectral Methods CASA Seminar, 13 December 2007

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Outline 1. Introduction 2. Ad-hoc Direct Methods 4. Direct methods 5. Conclusions 2/30 3. The matrix diagonalization techniques

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1. Introduction 3/30 Constant-coefficient Helmholtz equation Some generalizations Spectral descretization methods lead to the system Steady and unsteady problems;

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Outline 1. Introduction 2. Ad-hoc Direct Methods 3. The matrix diagonalization techniques 4. Direct methods 5. Conclusions 4/30

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2. Ad-hoc Direct Methods Fourier Chebyshev Legendre 1. To performe appropriate transform 2. To solve the system 3. To performe an inverse transform on to get. Solution process: 5/30 Approximations:

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2. Ad-hoc Direct Methods Fourier Chebyshev Legendre 1. To performe appropriate transform 2. To solve the system 3. To performe an inverse transform on to get. Solution process: 5/30 Approximations:

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2.1 Fourier Approximations Problem - the Fourier coefficients; Solution 1a - The Fourier Galerkin approximation The solution is - the trancated Fourier series; 6/30

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Solution 1b - a Fourier collocation approximation Given 7/30

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and - the discrete Fourier coefficients; Solution 1b - a Fourier collocation approximation Using the discrete Fourier transform (DFT is a mapping between ) Given 7/30

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and - the discrete Fourier coefficients; Solution 1b - a Fourier collocation approximation Using the discrete Fourier transform (DFT is a mapping between ) Given (3) is solved for Reversing the DFT 7/30

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Galerkin and collocation approximation to Helmholz problem are equally straightfoward and demand operations. 8/30

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2.3 Chebyshev Tau Approximation Solution 1 - Chebyshev Tau approximation: Problem 9/30

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2.3 Chebyshev Tau Approximation Solution 1 - Chebyshev Tau approximation: Rewriting the second derivative, where L is upper triangular. Solution process requires operations Problem 9/30

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Solution 2 - To rearrange the equations 2.3.1 More efficient solution procedure For q=2 For q=1 in combination with (5) will lead 10/30

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After simplification To minimize the round-off errors; quasi-tridiagonal system; not diagonally dominant; Nonhomogeneous BC. For even coefficients: 11/30

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1.Discrete Chebyshev transform; 2. To solve quasi-tridiagonal system; 3. Inverse Chebyshev transform on to get. 2.4 Mixed Collocation Tau Approximation 12/30 Solution process:

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2.5 Galerkin Approximation Solution: Legendre Galerkin approx. Problem 13/30

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2.5 Galerkin Approximation Solution: Legendre Galerkin approx. After integration by parts Problem (full matrices) 13/30

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The same system but An alternative set of basis functions produces tridiagonal system: Then expension is The right-hand side terms in (5) are related to Legendre coefficients : Two sets of tridiagonal equations; O(N) operations 14/30

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Transformation between spectral space and physical space: The standard Legendre coefficients of the solution can be found via 15/30

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2.6 Numerical example for Ad Hoc Methods in 1-D Galerkin method is more accurate than Tau methods Roundoff errors are more for Chebyshev methods, significantly for N>1024 Exact solution is 16/30

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Outline 1. Introduction 2. Ad-hoc Direct Methods 4. Direct methods 5. Conclusions 17/30 3. The matrix diagonalization techniques

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3.1 Schur Decomposition Collocation approx and Legendre G-NI approxim. lead Problem: Solving (6) by Schur decomposition [Bartels, Stewart, 1972] lower-triangular upper-triangular 18/30

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Computational cost: Solution process: Reduction and to Schur form Construction of F’ Solution of for U’ Transformation from U’ to U. 19/30

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3.2 Matrix Digitalization Approach Similar to Schur decomposition. The same solution steps. and are diagonalized Operation cost: 20/30

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3.3 Numerical example for Ad Hoc Methods in 2-D Problem: Matrix diagonalization was used for the solution procedure Haidvogel and Zang (1979), Shen (1994) Results are very similar to 1-D case 21/30

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Outline 1. Introduction 2. Ad-hoc Direct Methods 4. Direct methods 5. Conclusions 22/30 3. The matrix diagonalization techniques

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4. Direct Methods Matrix structure produced by Galerkin and G-NI methods ; How the tensor-product nature of the methods can be used efficiently to build matrices; How the sparseness of the matrices in 2D and in 3D can be accounted in direct techniques 23/30

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4.1 Multidimensional Stiffness and Mass Matrices + homogen. BC onProblem: Integral formulation: Galerkin solution: – stiffness matrix Let be a finite tensor-product basis in. The trial and test function will be chosen in 24/30

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Decomposition of K into its 1 st, 2 nd, 0 – order components 25/30

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then Decomposition of K into its 1 st, 2 nd, 0 – order components for a general, the use of G-NI approach with Lagrange nodal basis 26/30 will lead to diagonal marix

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Decomposition of K into its 1 st, 2 nd, 0 – order components - tensor-product function, - arbitrary, G-NI approach leads to sparse matrix (a matrix-vector multiply requires operations) 27/30 2D3D

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In 2D: matrix is in general full for arbitrary nonzero Decomposition of K into its 1 st, 2 nd, 0 – order components for arbitrary, G-NI approach (with Lagange nodal basis) leads (a matrix-vector multiply requires operations) In 3D: has sparse structure 28/30 to sparse matrix

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- decomposition 4. Gaussian Elimination Techniques 29/30 LU - decomposition 2D: special cases of Ad-hoс methods have lower cost Frontal and multifrontal [Davis and Duff 1999] To get benifit from sparsyty, reodering of matrix to factorization have to be done [Gilbert, 1992, Saad, 1996]

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Outline 1. Introduction 2. Ad-hoc Direct Methods 4. Direct methods 5. Conclusions 30/30 3. The matrix diagonalization techniques

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5. Conclusions 31/31 Approximation techniques; Galerkin approximations give more accurate results than other methods; Techniques, which can eliminate the cost of solution on prepocessing stage; Sparcity matrices

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Thank you for attention.

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Thank you for attention

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