Inner and outer bounds for the solution set of parametric linear systems

Popova, Evgenija D.; Krämer, Walter

doi:10.1016/j.cam.2005.08.048

Cited by 28 publications

(22 citation statements)

References 5 publications

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“…2. A similar observation is also made in, e.g., Tichatschke et al (1989) and Popova and Krämer (2007).…”

Section: Lemmasupporting

confidence: 83%

“…For a comprehensive treatment and for references to the literature on interval linear systems one may refer to the books Neumaier (1990), , Fiedler et al (2006) and Moore et al (2009). Due to the NP-hardness of solving (3) exactly, many ingenious methods have been developed to obtain sufficiently close outer estimates of the solution set, e.g., Hansen (1992), Jansson (1997), Ning and Kearfott (1997), Rump (2010), Hladík (2014), Popova (2004Popova ( , 2014, Rohn (1981), Alefeld et al (1998), Calafiore and Ghaoui (2004), Popova and Krämer (2007) and Hladík (2012) consider interval linear systems with dependent data. We refrain here from listing papers dedicated to computing enclosures since they are simply too many.…”

Section: Introductionmentioning

confidence: 99%

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Centered solutions for uncertain linear equations

Zhen

Hertog

2017

Comput Manag Sci

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Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we derive convex representations for united and tolerable solution sets. Secondly, to obtain centered solutions for uncertain linear equations, we develop a new method based on adjustable robust optimization (ARO) techniques to compute the maximum size inscribed convex body (MCB) of the set of the solutions. In general, the obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We use recent results from ARO to characterize for which convex bodies the obtained MCB is optimal. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input-output model, Colley's Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method.

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“…2. A similar observation is also made in, e.g., Tichatschke et al (1989) and Popova and Krämer (2007).…”

Section: Lemmasupporting

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Centered solutions for uncertain linear equations

Zhen

Hertog

2017

Comput Manag Sci

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“…For both steps, double precision computations are sufficient for most practical cases (if the system matrix is very ill conditioned with a condition number of 10 16 or higher a different approach as discussed in [10,21] needs to be taken) and thus the highly optimized routines of BLAS and LAPACK can be used. This also has the added benefit that a modern BLAS and LAPACK implementation will already be multi-threaded an can thus directly make use of multi-core and multi-processor systems.…”

Section: Methodsmentioning

confidence: 99%

Solvers for the verified solution of parametric linear systems

2011

Self Cite

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We present a newly developed version of our solvers for the verified solution of dense parametric linear systems, i.e. linear systems whose system matrix and right-hand side depend affine-linearly on parameters that vary inside prescribed intervals. The solvers use our C++ class library for reliable computing, C-XSC. The C-XSC library provides many features, especially easy to handle data types for dense and sparse matrices and vectors and the ability to compute dot products and dot product expressions in arbitrary precision. The new solvers can use either sparse or dense matrices as the coefficient matrices for the parameters. The use of sparse coefficient matrices can result in huge improvements in both performance and memory consumption. BLAS and LAPACK routines are used where applicable, and OpenMP is used for the parallelization on multi-core and multi-processor systems. The solvers also provide the ability to compute not only an outer but also a componentwise inner enclosure of the solution set of the system and to choose between two versions of the algorithm, one being very fast and one giving sharp results and extending the range of solvable systems. We give some examples for parametric linear systems (also from real world examples such as worst-case tolerance analysis of linear electric circuits),

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“…A direct method was given by Skalna (2006), and a monotonicity approach by Popova (2006a), Rohn (2004), and Skalna (2008). Inner and outer approximations by a fixed-point method were developed by Rump (1994;, and implemented by Popova and Krämer (2007). A Mathematica package for solving parametric interval systems is introduced by Popova (2004a).…”

Section: Introductionmentioning

confidence: 99%

Enclosures for the solution set of parametric interval linear systems

Hladík

2012

International Journal of Applied Mathematics and Computer Science

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We investigate parametric interval linear systems of equations. The main result is a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for this case, comparing and refinement of both. We show that the latter bounds are not provable better, and that they are also sometimes too pessimistic. The presented form of both methods is suitable for combining them into one to get a more efficient algorithm. Some numerical experiments are carried out to illustrate performances of the methods.

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Inner and outer bounds for the solution set of parametric linear systems

Cited by 28 publications

References 5 publications

Centered solutions for uncertain linear equations

Centered solutions for uncertain linear equations

Solvers for the verified solution of parametric linear systems

Enclosures for the solution set of parametric interval linear systems

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