The use of negative penalty functions in linear systems of equations

2006 ◽  
Vol 462 (2074) ◽  
pp. 2965-2975 ◽  
Author(s):  
Harm Askes ◽  
Sinniah Ilanko
Keyword(s):  
Exact Solution ◽  
Linear Systems ◽  
Vibration Analysis ◽  
Linear Equations ◽  
Penalty Functions ◽  
Penalty Parameter ◽  
Systems Of Equations ◽  
Mathematical Proofs ◽  
Systems Of Linear Equations ◽  
Linear Systems Of Equations

Contrary to what is commonly thought, it is possible to obtain convergent results with negative (rather than positive) penalty functions. This has been shown and proven on various occasions for vibration analysis, but in this contribution it will also be shown and proven for systems of linear equations subjected to one or more constraints. As a key ingredient in the developed arguments, a pseudo-force is identified as the derivative of the constrained degree of freedom with respect to the inverse of the penalty parameter. Since this pseudo-force can be proven to be constant for large absolute values of the penalty parameter, it follows that the exact solution is bounded by the results obtained with negative and positive penalty parameters. The mathematical proofs are presented and two examples are shown to illustrate the principles.

scholarly journals Schwarz Methods for a Preconditioned WOPSIP Method for Elliptic Problems

2012 ◽  
Vol 12 (3) ◽  
pp. 241-272 ◽  
Author(s):  
Paola F. Antonietti ◽  
Blanca Ayuso de Dios ◽  
Susanne C. Brenner ◽  
Li-yeng Sung
Keyword(s):  
Boundary Value Problem ◽  
Linear Systems ◽  
Condition Number ◽  
Numerical Experiments ◽  
Boundary Value ◽  
Penalty Parameter ◽  
Systems Of Equations ◽  
Schwarz Methods ◽  
Interior Penalty ◽  
Linear Systems Of Equations

Abstract We propose and analyze several two-level non-overlapping Schwarz methods for a preconditioned weakly over-penalized symmetric interior penalty (WOPSIP) discretization of a second order boundary value problem. We show that the preconditioners are scalable and that the condition number of the resulting preconditioned linear systems of equations is independent of the penalty parameter and is of order H/h, where H and h represent the mesh sizes of the coarse and fine partitions, respectively. Numerical experiments that illustrate the performance of the proposed two-level Schwarz methods are also presented.

Assessing the Relevance of Processing Building Blocks in Evolutionary Computation: Experiments with Linear Systems of Equations

1999 ◽  
Vol 3 (5) ◽  
pp. 394-400
Author(s):  
David B. Fogel ◽  
◽  
Peter J. Angeline ◽  
Keyword(s):  
Evolutionary Computation ◽  
Linear Equations ◽  
Building Blocks ◽  
Small Population ◽  
Systems Of Equations ◽  
Uniform Crossover ◽  
Population Sizes ◽  
Systems Of Linear Equations ◽  
Linear Systems Of Equations ◽  
Better Than

Experiments are conducted to assess the utility of processing building blocks within a framework of evolutionary computation. Systems of linear equations are used for testing the efficiency of different recombination operators, including one- and two-point and uniform crossover. The consistent results indicate that uniform crossover, which disrupts building blocks maximally, generates statistically significantly better solutions than one- or two-point crossover. Moreover, for the cases of small population sizes, crossing over existing solutions with completely random solutions (i.e., macromutation) can perform as well or better than the traditional oneand two-point operators. The results do not support the building block hypothesis.

A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations

2013 ◽  
Vol 3 (2) ◽  
pp. 120-137 ◽  
Author(s):  
Jan Brandts ◽  
Ricardo R. da Silva
Keyword(s):  
Matrix Method ◽  
Eigenvalue Problems ◽  
Linear Equations ◽  
Computational Cost ◽  
Systems Of Linear Equations ◽  
Definition Of ◽  
Specific Sense ◽  
Low Computational Cost ◽  
Minimal Residual Methods ◽  
Linear Systems Of Equations

AbstractGiven two n × n matrices A and A0 and a sequence of subspaces with dim the k-th subspace-projected approximated matrix Ak is defined as Ak = A + Πk(A0 − A)Πk, where Πk is the orthogonal projection on . Consequently, Akν = Aν and ν*Ak = ν*A for all Thus is a sequence of matrices that gradually changes from A0 into An = A. In principle, the definition of may depend on properties of Ak, which can be exploited to try to force Ak+1 to be closer to A in some specific sense. By choosing A0 as a simple approximation of A, this turns the subspace-approximated matrices into interesting preconditioners for linear algebra problems involving A. In the context of eigenvalue problems, they appeared in this role in Shepard et al. (2001), resulting in their Subspace Projected Approximate Matrix method. In this article, we investigate their use in solving linear systems of equations Ax = b. In particular, we seek conditions under which the solutions xk of the approximate systems Akxk = b are computable at low computational cost, so the efficiency of the corresponding method is competitive with existing methods such as the Conjugate Gradient and the Minimal Residual methods. We also consider how well the sequence (xk)k≥0 approximates x, by performing some illustrative numerical tests.

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