Preconditioning Technique for Darcys Law in Porous Media
- Slides: 45
Preconditioning Technique for Darcys Law in Porous Media F. Fairag, H Tawfiq and M. Al-Shahrani Department of Math & Stat King Fahd University of Petroleum & Minerals Dhahran, Saudi Arabia Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Main Idea We begin with Sparse indefinite ill-conditioned Matrices A and B are generated from specific problem In this talk, we will present a preconditioner for this linear system. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Main Idea Powell-Silvester [2003] Bramble-Pasiak [1988] The eigenvalues of the preconditioned matrix the first interval is of small size The preconditioned matrix is nonsymmetric It is self-adjoint in the inner product H and + discrete represent P-MINRES can be applied Department of Mathematics and Statistics, KFUPM. PCG based on this inner product can be applied Nov 6, 2013
Outline 1) Definition ( all matrices A, B, D, N ) 2) self-adjointness (H-symm) 3) Clustering (eigenvalues) 4) Positive definiteness 5) Numerical Examples Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Matrices Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Main Idea We consider the solution of system of linear algebraic equations which is obtained from Raviart-Thomas mixed finite element formulation of Darcys equations. We begin with the mixed formulation of Darcy flow with some suitable boundary conditions u is the velocity p is the pressure K represents the permeability. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Saddle Point Problem We begin with the mixed formulation of Darcy flow We define the two subspaces Velocity and pressure space there exist positive constants Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Saddle Point Problem We begin with the mixed formulation of Darcy flow Multiplying by test functions, integrating the first equation by parts and imposing essential boundary conditions yield the variational problem, Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Saddle Point Problem Abstract Mixed Formulation Two Hilbert spaces we assume that Two binlinear forms Consider the following problem Thm: Babuska(1971) (Brezzi 1974) (*) has a unique solution.
Discrete Form We restrict our discussion to the lowest order Raviart-Thomas space On rectangles We start by dividing the domain into triangles or rectangles Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Saddle Point Problem discrete velocity space discrete pressure space give the problem Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Saddle Point Problem A+D is the matrix representation of the div-norm the velocity divergence matrix the pressure mass matrix N is the matrix representation of the 0 -norm Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Self-adjoint Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Self-Adjoint Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Extensions to the classical Bramble. Pasciak case: q Meyer and Steidten (2001). q Benzi and Simoncini (2006). q Liesen and Parlett (2006). q Stoll and Wathen (2008). (excellent survey) Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Self-Adjoint Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Positive Definiteness Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Positive Definite Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Positive Definite congruence transformation = Sylvester's law of inertia if positive Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Lemma 1 There exist two positive numbers and such that For some positive numbers and Lemma 2 Department of Mathematics and Statistics, KFUPM. for all Nov 6, 2013
Eigenvalues Clustering Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties We want to compute all eigenvalues We study the generalized eigenvalue problem We get Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties Assume that Hence, eigenvalues (multiplicity n). Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties If , then we have So that the remaining m eigenvalues satisfy This equation leads to four cases. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties Theorem 4. 3: If is a quasi-uniform mesh, the n+m eigenvalue of the generalized eigenvalue problem, where lie on the intervals, where is the minimum eigenvalue of the Schur complement and c is a constant independent of h. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties Example 4. 1 is an eigenvalue of multiplicity n the remaining m positive eigenvalues Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties Theorem 4. 4: If is a quasi-uniform mesh, the n+m eigenvalue of the generalized eigenvalue problem, where lie on the union of the intervals, where is the minimum eigenvalue of the Schur complement is the area of the smallest element in and c is a constant independent of h. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties Example 4. 2 is an eigenvalue of multiplicity n the remaining m positive eigenvalues Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties Theorem 4. 5: If is a quasi-uniform mesh, the n+m eigenvalue of the generalized eigenvalue problem, where lie on the union of the intervals, where is the minimum eigenvalue of the Schur complement is the area of the smallest element in independent of h. and c is a constant
Spectral Properties Theorem 4. 6: If is a quasi-uniform mesh, the n+m eigenvalue of the generalized eigenvalue problem, where lie on the union of the intervals, Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties Example 4. 4 Then the eigenvalue of the preconditioned matrix lie on the union of the following intervals Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties We study the problem on with uniform mesh The observed eigenvalues of the preconditioned matrix system are listed in the following table Our goal is to verify that the bounds (obtained) agree with the computed bounds. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties left bound 0. 3124 0. 0124 0. 3124 0. 2125 100 Right bound 1. 0095 1. 0101 1. 1619 1. 1763 1. 1619 1. 765 The last four columns confirm that the bounds in Theorem (4. 3) and Theorem (4. 5) are tight. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
left bound Right bound 0. 3124 2 2 1. 9047 1. 9987 1. 9047 2. 0000 0. 0774 2 2 1. 9039 1. 9997 1. 9039 2. 0000 0. 3124 9 9 1. 0714 1. 1234 1. 0714 1. 1250 0. 0774 9 9 1. 0709 1. 1248 1. 0709 1. 1280 The last four columns confirm that the bounds in Theorem (4. 3) and Theorem (4. 5) are tight. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Spectral Properties In this example, we consider the same problem in Example (1) and we compute the eigenvalues of the original matrix and the preconditioned matrix. The Figure shows the eigenvalues clustering for the original matrix in (TOP) and the preconditioned matrix using preconditioner from case 2 with (BOTTOM) Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Numerical Experiments Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
GMRES We solve the resulting linear system in three different ways. 1) without a preconditioner. 2 -3) using GMRES method with two different set of parameters The Table shows GMRES iterations for these different ways. 46 83 136 Preconditioner with 3 3 4 Preconditioner with 3 3 3 Without preconditioner Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
PCG based on H inner product v In this example, we apply H-CG with two different sets of parameters. These two sets ensures that the matrices H and are symmetric positive definite. These choices of parameters make the preconditioned matrix self-adjoint and positive definite with respect to the inner product H·, The first set of parameters is (γ 1, γ 2, α) = (0. 001, 0. 5, 0. 002). v Theorem (4. 5) implies that the eigenvalues of the preconditioned matrix lie on the union of the intervals, [1. 0009, 1. 0010) ∪ [1000]. The second set of parameters is (γ 1, γ 2, α) = (0. 0001, 0. 9999). v Theorem (4. 3) implies that the eigenvalues of the preconditioned matrix lie on the union of the intervals, [0. 9525, 1. 0001) ∪ [10000]. v We carried out computations with various mesh sizes. v For our numerical computations we have chosen TOL = 10− 3 and the initial guess x 0 to be the zero vector. v Iteration counts are reported in Table (6. 4). This table shows that the number of iterations is insensitive to the linear system size. Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
PCG based on H inner product Mesh Sys. Size set 1 set 2 8 x 8 208 4 5 10 x 10 320 4 5 12 x 12 456 4 5 16 x 16 800 4 5 20 x 20 1240 4 5 32 x 32 3136 4 5 Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
P-MINRES based on H inner prod The following experiment report on P-H_MINRES convergence with different parameters. We apply P-H-MINRES (Algorithm 2) to the system with a stopping tolerance of 10^-6 on the residual norm with respect to H. If the number of iterations is more than 1, 000, we use the symbol *. Example 6. 5 We consider the same problem in Example 6. 1 and we solve it using both the classical MINRES on the unpreconditioned system and the HMINRES on the preconditioned system with a stopping tolerance of 10^-6. We choose the parameters These values are chosen such that H is symmetric positive definite. The number of iterations are reported in the following Table.
P-MINRES based on H inner prod h Unpreconditioned system Preconditioned system 63 3 89 3 111 3 145 3 181 3 275 3 * 3 Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Example 6. 6: We test, in this example, the performance of the preconditioner in the presence of discontinuous coefficients. Let with Iteration counts are in the Table. * 3 5 5 3 3 3 3 3 4 4 4 * 3 5 5 3 3 3 4 4 4 5 6 * 3 7 5 3 3 4 4 4 5 5 9 * 3 * 5 3 3 4 4 4 6 Department of Mathematics and Statistics, KFUPM. Nov 6, 2013
Conclusions and Remarks q We present a preconditioner so that PCG based on H-inner product can be applied. q Several numerical tests confirm that the computational bounds agree with theoretical bounds. q Numerical experiments illustrate good convergence properties. q The k-th error can be written as q We study the eigenvalues of the matrix Lemma 1 [Pestana and Wathen 2013]. instead of q We found the parameters experimentally (with no relations).
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