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Spring 2003Prof. Tim Warburtontimwar@math.unm.edu

MA557/MA578/CS557Lecture 34

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Today’s Class

• A few ways to solve Ax=b for a sparse matrix A in Matlab.

• Followed by time for work on HW 10.

• On Friday 04/25/03 we will benefit from a special lecture on preconditioning for iterative methods presented by:– Dr D.M. Day of Sandia National Laboratories

• http://www.cs.sandia.gov/~dday/

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Recall: Summary of Temporal Implicit Schemes

• Backwards Euler is unconditionally stable for non-negative diffusion parameter D (i.e. any dt>=0) and first order in dt.

• Crank-Nicholson is unconditionally stable for non-negative diffusion parameter D (i.e. any dt>=0) and second order in dt.

• ESDIRK4 – generalizes to fourth order in dt.

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2 2n ndt dt

L LC M M C

11n ndt C M L MC

D N D Nx x y yD L M D D D D

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Backwards Euler Linear System

• Given Cn we wish to find a Cn+1 which satisfies:

• For simplicity we define:

• Note that A is a symmetric, positive matrix.

1n ndt M L C MC

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:

:

:then we wish to find such that:

n

n

dt

A M L

b MCx C

xAx b

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Lazy Way To Build The Matrix..

• Don’t tell anyone I told you this but here’s an easy way to program the construction of the DG operator.

• The first step is to understand that if I set up a vector which only has one non-zero entry in the n’th entry and then pass it to umDIFFUSIONop then the returned vector will be the n’th column of the A matrix.

( ) ( )

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j nj

j rank j rank

i ij j ij nj inj j

A A

x

v A x A A

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Laziness cont

• The next step is to use the sparsity of

• If I set one of the node values in the center white triangle to one and multiply by the Neumann DG derivative operator then the result vector will have non-zero entries in the red triangles and the original white triangle.

• If I take this result vector and premultiply by the Dirichlet DG derivative operator then there will be non-zero entries in the red, white and blue triangles..

Nx

Ny y

D DxD D DM DDA I D

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Finding The Neighbors and Their Neighbors of an Element

• In umMESH.m there is code which computes the sparse connectivity matrix umSparseEtoE.

• For: the matrix is:

• To find neighbors and neighbors of neighbors we consider the square of the connectivity matrix:

1 2 3 4 5

1 1 0 0 1 02 0 1 0 1 03 0 0 1 1 14 1 1 1 1 05 0 0 1 0 1

1 2

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1 0 0 1 0 1 0 0 1 0 2 1 1 2 00 1 0 1 0 0 1 0 1 0 1 2 1 2 00 0 1 1 1 0 0 1 1 1 1 1 3 2 21 1 1 1 0 1 1 1 1 0 2 2 2 4 10 0 1 0 1 0 0 1 0 1 0 0 2 1 2

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Example cont

• Double connectivity matrix:

• i.e. Element 1 is within two elements of 2, 3, 4• Element 2 is within two elements of 1, 3, 4• Element 3 is within two elements of 1, 2, 4, 5• Element 4 is within two elements of 1, 2, 3, 5• Element 5 is within two elements of 3, 4

1 2 3 4 5

1 2 1 1 2 02 1 2 1 2 03 1 1 3 2 24 2 2 2 4 15 0 0 2 1 2

1 2

34

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Matlab Implementation (Connectivity):

• Here we compute the square of the element connectivity matrix:

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Building DG MatrixumDIFFUSIONilu.m

16) Create sparse matrix

20-41) for each node, compute the row of the matrix

44) Compute Cholesky factorization

47) Compute incomplete Cholesky factorization.

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umDIFFUSIONpartop.m

• In this function we apply the operation of the matrix on a part of the vector.

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First Part of the Part Matrix MultiplyumDIFFUSIONpartop.m

• In this function we apply the Helmholtz operator to all the elements specified in the elmts argument.

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Applying the Helmholtz OperatorumDIFFUSIONpartop.m cont

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DriverumDIFFUSIONdemo.m• Driver for the implicit

DG diffusion solver.

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umDIFFUSIONrun.m

Note the changesto pcg and the call to: umDIFFUSIONilu

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In Action• It takes a while to build the matrix…

• We can look at the sparsity pattern of the matrix:

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Some Options For Solving Ax=b

• We will consider a couple of many options for accelerating the process of solving Ax=b

1) Cholesky factorization of A before time stepping and repeated backsolving.

2) Incomplete Cholesky factorization of A and using this as a preconditioner.

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Option 1: Cholesky factorization

• Use Cholesky factorization to decompose A into the product of a lower triangular matrix C and its transpose:

• Then every time we need to peform a backsolve twice:

• Each backsolve takes N^2 operations (N=total number of unknowns)

tA CC

tC y bCx y

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Option 1: Sparsity of Cholesky Factor

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Option 1: Sanity Check on Cholesky Factorization

• We can check to see how stable the computation of the Cholesky factorization was:

• Not bad – we lost about 4 decimal places…

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Option 1: Condition number

• We can use condest to estimate the condition number of the matrix. We see that the condition number is about 800 so we may well expect to lose 3 decimal places in computing the factorization..

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Option 1: Direct Solver Code

31-32) note two back solves..

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Option 2: Incomplete Cholesky Preconditioner• We can use an incomplete Cholesky preconditioner in

the PCG algorithm.

• The idea is to use cholinc to compute the Cholesky factorization with a drop tolerance used to determine if entries in the factor are created.

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Sparsity of Incomplete Cholesky Factor

cholinc(A, 1e-3)

Only 93047 non-zero entries Only 68213 non-zero entries

cholinc(A, ‘0’)

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Iteration Count Per Time Step

• Comparing unpreconditioned and preconditioned with incomplete Cholesky:

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Option 2: Incomplete Cholesky Preconditioner• We can use an incomplete Cholesky preconditioner in

the PCG algorithm.

30) Call to PCG usesincomplete Choleskyfactorization.

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Alternative Iterative Schemes in Matlab

• BICG, BICGSTAB, CGS, GMRES, LSQR, MINRES

• They all have the same interface.

• For details type:> help gmres