id	summary	reporter	owner	description	type	status	priority	milestone	component	version	resolution	keywords	cc
481	Minimal Residual Method (MINRES) for R4D-Var and 4D-PSAS	arango	arango	"A new C-preprocessing option was added '''MINRES''' to accelerate the convergence of the dual formulation data assimilation algorithms ('''R4D-Var''' and '''4D-PSAS''') using the Minimal Residual Method ('''!MinRes''') proposed by El Akkraoui and Gauthier (2010, QJRMS, 136, 107-115).  This method is described by Paige and Saunders (''Sparse Indefinite Systems of Linear Equations'', 1975, SIAM Journal on Numerical Analysis, 617-619). Specifically we use equations 6.10 and 6.11 of this paper.

This is a remarkable algorithm and makes '''R4D-Var''' and '''4D-PSAS''' practical in strong- and weak-constraint applications.  The convergence rate is similar to that of the primal form '''I4D-Var'''.  This is illustrated in the following plot:

[[Image(https://www.myroms.org/trac/minres.png, 500)]]

This algorithm also works for the associated observation sensitivity and observation impact.  We still need to work on the posterior analysis routines '''posterior.F''' and '''posterior_var.F'''.  These routines will be updated to '''MINRES''' later.


Many thanks to Andy Moore for his help in coding and testing this new algorithm.  Also many thanks to Amal El Akkraoui for bringing this minimization algorithm to our attention at the last International Adjoint Workshop."	upgrade	closed	major	Adjoint Based Algorithms	Adjoint	3.4	Done