Iterative Methods Mostly for SPD systems Iterative Linear - PDF document

Iterative Methods Mostly for SPD systems Iterative Linear  conjugate gradient and its variants System Solvers for Large, Sparse Matrices  minimize through successive line searches CS 176 Spring 2011 CS 176 Spring 2011 1 2 Steepest Descent Steepest Descent Decrease function How much?  negative of gradient  how much? h h CS 176 Spring 2011 CS 176 Spring 2011 3 4 Steepest Descent Let’s do Better Algorithm Conjugate directions  don’t undo earlier gains CS 176 Spring 2011 CS 176 Spring 2011 5 6

Conjugate Gradient Conjugate Gradient Directions A-orthogonal to error Next direction?  orthogonal to all others  easy update: CS 176 Spring 2011 CS 176 Spring 2011 7 8 Algorithm Iterative Methods What you’ll need What happened to d? (see section 7.3 of Shewchuk)  you supply Sweeps  given topological datastructure matrix multiply dot product acc.  implement matrix multiplies i l i l i li update dot product acc. update  stopping criterion: relative residual CS 176 Spring 2011 CS 176 Spring 2011 9 10 Typical Implementation Iterative Methods What lives where? Other issues  degrees of freedom at vertices  number of iterations proportional to  (steepest descent),  (CG)  need additional variables for solver  preconditioning  preconditioning  matrix entries on edges t i t i d  use approximate inverse Matrix multiplies  diagonal (Jacobi)  iterate over mesh!  incomplete Cholesky,  mesh is sparse matrix data structure hierarchical,… CS 176 Spring 2011 CS 176 Spring 2011 11 12

Other System Types SPD  conjugate gradients  Shewchuk, “CG w/o the Agony” non-S PD & non-S non-PD  bi-conjugate gradients  see Numerical Recipes  http://www.nr.com/ (chapter 2.7) CS 176 Spring 2011 13