1. PETE 603 Lecture Session #29 Thursday, 7/29/10

2. 29.1 Iterative Solution Methods Older methods, such as PSOR, and LSOR require user supplied iteration parameters (such as ) for optimal performance. Minimization methods, such as conjugate gradient (not discussed here) and ORTHOMIN, overcome this disadvantage by automatic calculation of the iteration parameters, and almost always guarantee convergence. Minimization methods are iterative and are used in conjunction with “preconditioning” (such as approximate factorization).

3. 29.2 Minimization Methods Suppose we are solving the system, Ap = q, and that k iterations have been completed. The error in p, (p - p k ), leads to an error, r k, in q equal to: A(p - p k ) = (q - Ap k ) = r k We could calculate the error in p exactly using a direct solution method but wish to avoid that, so we estimate the error in p using an approximation (preconditioner) (A’) -1 rather than A -1, i.e. t k = (A’) -1 r k  (p - p k )

4. 29.3 Minimization Methods Of course, we don’t actually invert A’ either, calculate (A’) -1 r k. We could directly use this estimate of (p-p k ) to obtain an updated estimate of p, but convergence of the process can be accelerated by including iteration parameters,  and , such that the updated estimate of p is calculated as p k+1 = p k +  k s k, where: s k = t k +  k s k-1 and  and  are selected to minimize the error in some fashion. In addition,  may be used to influence the direction of the change in p.

5. 29.4 ORTHOMIN The coefficient matrices in reservoir simulation problems are normally not symmetric, therefore it is more appropriate to minimize the innner product (dot product) of the estimated residuals (least squares analysis), by minimizing: Then,

6. 29.5 ORTHOMIN In addition, let s k = t k +  k s k-1 where the  i are orthogonality coefficients.

7. 29.6 ORTHOMIN Consider this system and the 2-D finite difference equation Ap = q. OTHOMIN uses an LDU preconditioning to calculate the approximation t k. where

8. 29.7 ORTHOMIN Example: A 2 x 2 grid has the following flow coefficients and no-flow boundaries: E = -1 S = -2 W = -3 N = -4 C = 1 - ( E +S + W + N ) q = 10

9. 29.8 ORTHOMIN The approximate factors of A are:

10. 29.9 ORTHOMIN The matrix equation, Ap = q, is and

11. 29.10 ORTHOMIN For the solution, let then,

12. 29.11 ORTHOMIN Notice that solution of (LDU)t 0 = r 0 requires the solution of three systems

13. 29.12 ORTHOMIN And Since k = 0, it is not necessary to calculate  Also, the sum is zero, so s 0 = t 0

14. 29.13 Now calculate  and update pressure ORTHOMIN

15. 29.14 Preconditioning Calculating an approximate solution for the system of linear equations is an important step (the t k vector), this is called preconditioning. A good preconditioning method will be: –accurate (minimizes number of iterations required) –fast (minimizes cost of each iteration)

16. 29.15 Application of Minimization Methods Minimization methods (like ORTHOMIN) are well suited to modern computers due to the use of matrix/vector multiplies and inner products between vectors –CUDA (many parallel pipelines) –Cell Processor (vector processing) –Multi-core processors (OpenCL)