A Residual-Based HP-Mesh Optimization Technique for Petrov-Galerkin Schemes with Optimal Test Functions

Test case selection helps in improving quality of test suites by removing ambiguous, redundant test cases, thereby reducing the cost of software testing. Various works carried out have chosen test cases based on single parameter and optimized the test cases using single objective employing single strategies. In this article, a parameter selection technique is combined with an optimization technique for optimizing the selection of test cases. A two-step approach has been employed. In first step, the fuzzy entropy-based filtration is used for test case fitness evaluation and selection. In second step, the improvised ant colony optimization is employed to select test cases from the previously reduced test suite. The experimental evaluation using coverage parameters namely, average percentage statement coverage and average percentage decision coverage along with suite size reduction, demonstrate that by using this proposed approach, test suite size can be reduced, reducing further the computational effort incurred.

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Analysis of the discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2013.07.012 ◽

2014 ◽

Vol 66 (12) ◽

pp. 2570-2586 ◽

Cited By ~ 10

Author(s):

Victor M. Calo ◽

Nathaniel O. Collier ◽

Antti H. Niemi

Keyword(s):

Galerkin Method ◽

Mindlin Plate ◽

Plate Bending ◽

Test Functions ◽

Optimal Test

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Alternative Enriched Test Spaces in the DPG Method for Singular Perturbation Problems

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2018-0207 ◽

2019 ◽

Vol 19 (3) ◽

pp. 603-630 ◽

Cited By ~ 1

Author(s):

Jacob Salazar ◽

Jaime Mora ◽

Leszek Demkowicz

Keyword(s):

Singular Perturbation ◽

Perturbation Parameter ◽

Standard Test ◽

Test Functions ◽

Optimal Test ◽

Test Norms ◽

Local Test ◽

Linear Problems ◽

Perturbation Problems ◽

Convection Dominated

AbstractWe propose and investigate the application of alternative enriched test spaces in the discontinuous Petrov–Galerkin (DPG) finite element framework for singular perturbation linear problems, with an emphasis on 2D convection-dominated diffusion. Providing robust {L^{2}} error estimates for the field variables is considered a convenient feature for this class of problems, since this norm would not account for the large gradients present in boundary layers. With this requirement in mind, Demkowicz and others have previously formulated special test norms, which through DPG deliver the desired {L^{2}} convergence. However, robustness has only been verified through numerical experiments for tailored test norms which are problem-specific, whereas the quasi-optimal test norm (not problem specific) has failed such tests due to the difficulty to resolve the optimal test functions sought in the DPG technology. To address this issue (i.e. improve optimal test functions resolution for the quasi-optimal test norm), we propose to discretize the local test spaces with functions that depend on the perturbation parameter ϵ. Explicitly, we work with B-spline spaces defined on an ϵ-dependent Shishkin submesh. Two examples are run using adaptive h-refinement to compare the performance of proposed test spaces with that of standard test spaces. We also include a modified norm and a continuation strategy aiming to improve time performance and briefly experiment with these ideas.

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A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions

Numerical Methods for Partial Differential Equations ◽

10.1002/num.20640 ◽

2010 ◽

Vol 27 (1) ◽

pp. 70-105 ◽

Cited By ~ 117

Author(s):

L. Demkowicz ◽

J. Gopalakrishnan

Keyword(s):

Galerkin Methods ◽

Test Functions ◽

Optimal Test

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Trace operators of the bi-Laplacian and applications

IMA Journal of Numerical Analysis ◽

10.1093/imanum/draa012 ◽

2020 ◽

Cited By ~ 1

Author(s):

Thomas Führer ◽

Alexander Haberl ◽

Norbert Heuer

Keyword(s):

Galerkin Method ◽

Numerical Experiments ◽

Space Dimension ◽

Dirichlet Condition ◽

Plate Bending ◽

Test Functions ◽

Optimal Test ◽

Trace Space ◽

Three Space ◽

Well Posed

Abstract We study several trace operators and spaces that are related to the bi-Laplacian. They are motivated by the development of ultraweak formulations for the bi-Laplace equation with homogeneous Dirichlet condition, but are also relevant to describe conformity of mixed approximations. Our aim is to have well-posed (ultraweak) formulations that assume low regularity under the condition of an $L_2$ right-hand side function. We pursue two ways of defining traces and corresponding integration-by-parts formulas. In one case one obtains a nonclosed space. This can be fixed by switching to the Kirchhoff–Love traces from Führer et al. (2019, An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation. Math. Comp., 88, 1587–1619). Using different combinations of trace operators we obtain two well-posed formulations. For both of them we report on numerical experiments with the discontinuous Petrov–Galerkin method and optimal test functions. In this paper we consider two and three space dimensions. However, with the exception of a given counterexample in an appendix (related to the nonclosedness of a trace space) our analysis applies to any space dimension larger than or equal to two.

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