We consider the large-sparse symmetric linear systems of equations that arise in the solution of weak constraint four-dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3x3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2x2 block structure, or further to symmetric positive definite systems. In this paper, we analyse how sensitive the spectra of these matrices are to the number of observations of the underlying dynamical system. We also obtain bounds on the eigenvalues of the matrices. Numerical experiments are used to confirm the theoretical analysis and bounds.
We describe a method to evaluate integrals that arise in the asymptotic analysis when two saddle points may be close together. These integrals, which appear in problems from optics, acoustics or quantum mechanics as well as in a wide class of special functions, can be transformed into Airy-type integrals and we use the trapezoidal rule to compute these integrals numerically. The quadrature method, which remains valid when two saddle points coalesce, is illustrated with numerical examples.
This work is concerned with the iterative solution of systems of quasi-static multiple-network poroelasticity (MPET) equations describing flow in elastic porous media that is permeated by single or multiple fluid networks. Here, the focus is on a three-field formulation of the problem in which the displacement field of the elastic matrix and, additionally, one velocity field and one pressure field for each of the $n \ge 1$ fluid networks are the unknown physical quantities. Generalizing Biot's model of consolidation, which is obtained for $n=1$, the MPET equations for $n\ge1$ exhibit a double …
We develop robust solvers for a class of perturbed saddle-point problems arising in the study of a second-order elliptic equation in mixed form (in terms of flux and potential), and of the four-field formulation of Biot's consolidation problem for linear poroelasticity (using displacement, filtration flux, total pressure and fluid pressure). The stability of the continuous variational mixed problems, which hinges upon using adequately weighted spaces, is addressed in detail; and the efficacy of the proposed preconditioners, as well as their robustness with respect to relevant material properti…
The main focus of this paper is the characterization and exploitation of the asymptotic spectrum of the saddle--point matrix sequences arising from the discretization of optimization problems constrained by elliptic partial differential equations. We uncover the existence of a hidden structure in these matrix sequences, namely, we show that these are indeed an example of Generalized Locally Toeplitz (GLT) sequences. We show that this enables a sharper characterization of the spectral properties of such sequences than the one that is available by using only the fact that we deal with saddle--po…
For nonsymmetric block three-by-three singular saddle point problems arising from the Picard iteration method for a class of mixed finite element scheme, recently Salkuyeh et al. in (D.K. Salkuyeh, H. Aslani, Z.Z. Liang, An alternating positive semi-definite splitting preconditioner for the three-by-three block saddle point problems, Math. Commun. 26 (2021) 177-195) established an alternating positive semi-definite splitting (APSS) method. In this work, we analyse the semi-convergence of the APSS method for solving a class of nonsymmetric block three-by-three singular saddle point problems. Th…
The preconditioned iterative solution of large-scale saddle-point systems is of great importance in numerous application areas, many of them involving partial differential equations. Robustness with respect to certain problem parameters is often a concern, and it can be addressed by identifying proper scalings of preconditioner building blocks. In this paper, we consider a new perspective to finding effective and robust preconditioners. Our approach is based on the consideration of the natural physical units underlying the respective saddle-point problem. This point of view, which we refer to …
The saddle point matrices arising from many scientific computing fields have block structure $ W= \left(\begin{array}{cc} A & B\\ B^T & C \end{array} \right) $, where the sub-block $A$ is symmetric and positive definite, and $C$ is symmetric and semi-nonnegative definite. In this article we report a unobtrusive but potentially theoretically valuable conclusion that under some conditions, especially when $C$ is a zero matrix, the spectral radius of $W$ must be the maximum eigenvalue of $W$. This characterization approximates to the famous Perron-Frobenius property, and is called quasi-Perron-Fr…
We consider the iterative solution of regularized saddle-point systems. When the leading block is symmetric and positive semi-definite on an appropriate subspace, Dollar, Gould, Schilders, and Wathen (2006) describe how to apply the conjugate gradient (CG) method coupled with a constraint preconditioner, a choice that has proved to be effective in optimization applications. We investigate the design of constraint-preconditioned variants of other Krylov methods for regularized systems by focusing on the underlying basis-generation process. We build upon principles laid out by Gould, Orban, and …
Large linear systems of saddle-point type have arisen in a wide variety of applications throughout computational science and engineering. The discretizations of distributed control problems have a saddle-point structure. The numerical solution of saddle-point problems has attracted considerable interest in recent years. In this work, we propose a novel Braess-Sarazin multigrid relaxation scheme for finite element discretizations of the distributed control problems, where we use the stiffness matrix obtained from the five-point finite difference method for the Laplacian to approximate the inver…
We consider the problem of iteratively solving large and sparse double saddle-point systems arising from the stationary Stokes-Darcy equations in two dimensions, discretized by the Marker-and-Cell (MAC) finite difference method. We analyze the eigenvalue distribution of a few ideal block preconditioners. We then derive practical preconditioners that are based on approximations of Schur complements that arise in a block decomposition of the double saddle-point matrix. We show that including the interface conditions in the preconditioners is key in the pursuit of scalability. Numerical results s…
We propose an efficient solver for saddle point problems arising from finite element approximations of nonlocal multi-phase Allen--Cahn variational inequalities. The solver is seen to behave mesh independently and to have only a very mild dependence on the number of phase field variables. In addition we prove convergence, in three GMRES iterations, of the approximation of the two phase problem, regardless of mesh size or interfacial width. Numerical results are presented that illustrate the competitiveness of this approach.
We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved a…
