Some observations on weighted GMRES

We investigate the convergence of the weighted GMRES method for solving linear systems. Two different weighting variants are compared with unweighted GMRES for three model problems, giving a phenomenological explanation of cases where weighting improves convergence, and a case where weighting has no effect on the convergence. We also present new alternative implementations of the weighted Arnoldi algorithm which may be favorable in terms of computational complexity, and examine stability issues connected with these implementations. Two implementations of weighted GMRES are compared for a large number of examples. We find that weighted GMRES may outperform unweighted GMRES for some problems, but more often this method is not competitive with other Krylov subspace methods like GMRES with deflated restarting or BICGSTAB, in particular when a preconditioner is used

Natural preconditioners for saddle point systems

The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or discrete setting, so saddle point systems arising from discretization of partial differential equation problems such as those describing electromagnetic problems or incompressible flow lead to equations with this structure as does, for example, the widely used sequential quadratic programming approach to nonlinear optimization.\ud This article concerns iterative solution methods for these problems and in particular shows how the problem formulation leads to natural preconditioners which guarantee rapid convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness -- in terms of rapidity of convergence -- is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends

The antitriangular factorisation of saddle point matrices

Mastronardi and Van Dooren recently introduced the block antitriangular ("Batman") decomposition for symmetric indefinite matrices. Here we show the simplification of this factorisation for saddle point matrices and demonstrate how it represents the common nullspace method. We show the relation of this factorisation to constraint preconditioning and how it transforms but preserves the block diagonal structure of block diagonal preconditioning

On choice of preconditioner for minimum residual methods for nonsymmetric matrices

Existing convergence bounds for Krylov subspace methods such as GMRES for nonsymmetric linear systems give little mathematical guidance for the choice of preconditioner. Here, we establish a desirable mathematical property of a preconditioner which guarantees that convergence of a minimum residual method will essentially depend only on the eigenvalues of the preconditioned system, as is true in the symmetric case. Our theory covers only a subset of nonsymmetric coefficient matrices but computations indicate that it might be more generally applicable

On the eigenvalues and eigenvectors of block triangular preconditioned block matrices

Block lower triangular matrices and block upper triangular matrices are popular preconditioners for $2\times 2$ block matrices. In this note we show that a block lower triangular preconditioner gives the same spectrum as a block upper triangular preconditioner and that the eigenvectors of the two preconditioned matrices are related

Chic by Choice : a case study on pricing strategies and consumer perceptions

Millennials have been shaping markets and influencing the way companies are offering and communicating its products and services because, in contrast with the so called generation baby boomers, they are focused more on the experience, rather than on ownership of things. Chic by Choice is a Portuguese start-up, founded in May 2014, that decided to enter in a new online market category that has been witnessing significant growth and showing great potential when considering the reasoning millennials live up to: why buy if you can rent, use, have fun and then, return it? The Online Fashion Rental industry has one big player in America, who started operations in 2009, and in Europe, some smaller competitors in major markets such as the United Kingdom and Germany, where online shopping is something relatively embedded in their consumption behaviours. Being able to acquire one of the latter competitors, Chic by Choice was in a phase of rapid growth but, at the same time, it needed to keep its results positive and enticing to secure external investment. Therefore, in order to achieve these goals, the company had to make decisions regarding pricing, more specifically, the price and discounts given to the first customers so as to acquire them more easily, making them a more desirable and attractive value proposition and, by consequence, keeping revenue and average order values as high as possible.A geração “millennials” tem moldado mercados e influenciado a forma como as empresas comunicam e oferecem os seus produtos e serviços uma vez que, em contraste com o que acontece com a chamada geração “baby boomers”, os primeiros estão mais concentrados na experiência que obtêm dos bens e não na posse dos mesmos. Fundada em maio de 2014, a Chic by Choice é uma start-up Portuguesa que decidiu entrar numa nova categoria de mercado que tem assistido a um crescimento considerável e apresentado um grande potencial quando se toma em consideração o pensamento dos millennials: porquê comprar se é possível alugar, usar, aproveitar e depois simplesmente devolver? A indústria de Aluguer de Moda Online possui um grande concorrente na América, que iniciou operações em 2009, e na Europa, algumas empresas mais pequenas em mercados chave como o Reino Unido e a Alemanha, onde fazer compras online é algo que está relativamente incorporado nos seus comportamentos de consumo. Sendo capaz de adquirir um dos seus concorrentes mais pequenos, a Chic by Choice apresentava assim uma fase de crescimento acelerado, mas, ao mesmo tempo, precisava de manter os seus resultados positivos e atrativos, de modo a assegurar investimento externo. Assim sendo, para conseguir atingir estes objetivos, a empresa teve de tomar decisões relativas ao preço, mais especificamente, ao preço e descontos aplicados aos primeiros clientes para os captar mais facilmente, tornando a proposta de valor mais desejável e, consequentemente, manter os lucros e o valor médio de aluguer o mais altos possível

Null-space preconditioners for saddle point systems

The null-space method is a technique that has been used for many years to reduce a saddle point system to a smaller, easier to solve, symmetric positive-definite system. This method can be understood as a block factorization of the system. Here we explore the use of preconditioners based on incomplete versions of a particular null-space factorization, and compare their performance with the equivalent Schur-complement based preconditioners. We also describe how to apply the non-symmetric preconditioners proposed using the conjugate gradient method (CG) with a non-standard inner product. This requires an exact solve with the (1,1) block, and the resulting algorithm is applicable in other cases where Bramble-Pasciak CG is used. We verify the efficiency of the newly proposed preconditioners on a number of test cases from a range of applications

The antitriangular factorisation of saddle point matrices

Mastronardi and Van Dooren [SIAM J. Matrix Anal. Appl., 34 (2013), pp. 173--196] recently introduced the block antitriangular (``Batman'') decomposition for symmetric indefinite matrices. Here we show the simplification of this factorization for saddle point matrices and demonstrate how it represents the common nullspace method. We show that rank-1 updates to the saddle point matrix can be easily incorporated into the factorization and give bounds on the eigenvalues of matrices important in saddle point theory. We show the relation of this factorization to constraint preconditioning and how it transforms but preserves the structure of block diagonal and block triangular preconditioners

Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

Symmetric collocation methods with radial basis functions allow approximation of the solution of a partial differential equation, even if the right-hand side is only known at scattered data points, without needing to generate a grid. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. This cost can be alleviated somewhat by considering compactly supported radial basis functions and a multiscale technique. But the condition number and sparsity will still deteriorate with the number of data points. Therefore, we study certain block diagonal and triangular preconditioners. We investigate ideal preconditioners and determine the spectra of the preconditioned matrices before proposing more practical preconditioners based on a restricted additive Schwarz method with coarse grid correction (ARASM). Numerical results verify the effectiveness of the preconditioners