The central theme of this thesis is the design of optimal balanced blackbox stopping criteria in iterative solvers of symmetric positivedefinite, symmetric indefinite, and nonsymmetric linear systems arising from finite element approximation of stochastic (parametric) partial differential equations. For a given stochastic and spatial approximation, it is known that iteratively solving the corresponding linear(ized) system(s) of equations to too tight algebraic error tolerance results in a wastage of computational resources without decreasing the usually unknown approximation error. In order to stop optimallyby avoiding unnecessary computations and premature stoppingalgebraic error and a posteriori approximation error estimate must be balanced at the optimal stopping iteration. Efficient and reliable a posteriori error estimators do exist for close estimation of the approximation error in a finite element setting. But the algebraic error is generally unknown since the exact algebraic solution is not usually available. Obtaining tractable upper and lower bounds on the algebraic error in terms of a readily computable and monotonically decreasing quantity (if any) of the chosen iterative solver is the distinctive feature of the designed optimal balanced stopping strategy. Moreover, this work states the exact constants, that is, there are no userdefined parameters in the optimal balanced stopping tests. Hence, an iterative solver incorporating the optimal balanced stopping methodology that is presented here will be a blackbox iterative solver. Typically, employing such a stopping methodology would lead to huge computational savings and in any case would definitely rule out premature stopping. The constants in the devised optimal balanced blackbox stopping tests in MINRES solver for solving symmetric positivedefinite and symmetric indefinite linear systems can be estimated cheaply onthe fly. The contribution of this thesis goes one step further for the nonsymmetric case in the sense that it not only provides an optimal balanced blackbox stopping test in a memoryexpensive Krylov solver like GMRES but it also presents an optimal balanced blackbox stopping test in memoryinexpensive Krylov solvers such as BICGSTAB(L), TFQMR etc. Currently, little convergence theory exists for the memoryinexpensive Krylov solvers and hence devising stopping criteria for them is an active field of research. Also, an optimal balanced blackbox stopping criterion is proposed for nonlinear (Picard or Newton) iterative method that is used for solving the finite dimensional NavierStokes equations. The optimal balanced blackbox stopping methodology presented in this thesis can be generalized for any iterative solver of a linear(ized) system arising from numerical approximation of a partial differential equation. The only prerequisites for this purpose are the existence of a cheap and tight a posteriori error estimator for the approximation error along with cheap and tractable bounds on the algebraic error.
Date of Award  31 Dec 2017 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  David Silvester (Supervisor) & Catherine Powell (Supervisor) 

 Inbuilt optimal stopping tests
 Finite Element Method
 Preconditioned iterative solversMINRES, CG, GMRES, BICGSTAB(L) etc.
 UQ
 (Stochastic) PDEsdiffusion, convectiondiffusion, Stokes, NavierStokes
Optimal iterative solvers for linear systems with stochastic PDE origins: Balanced blackbox stopping tests
Pranjal, P. (Author). 31 Dec 2017
Student thesis: Phd