Topic
Piecewise linear function
About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.
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TL;DR: This paper defines and analyze a posteriori error estimators for nonconforming approximations of the Stokes equations and proves that these estimators are equivalent to an appropriate norm of the error.
Abstract: In this paper we define and analyze a posteriori error estimators for nonconforming approximations of the Stokes equations. We prove that these estimators are equivalent to an appropriate norm of the error. For the case of piecewise linear elements we define two estimators. Both of them are easy to compute, but the second is simpler because it can be computed using only the right-hand side and the approximate velocity. We show how the first estimator can be generalized to higher-order elements. Finally, we present several numerical examples in which one of our estimators is used for adaptive refinement.
108 citations
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TL;DR: Two simple heuristic algorithms for piecewise-linear approximation of functions of one variable are described, which use a limit on the absolute value of error and strive to minimize the number of approximating segnents subject to the error limit.
Abstract: Two simple heuristic algorithms for piecewise-linear approximation of functions of one variable are described. Both use a limit on the absolute value of error and strive to minimize the number of approximating segnents subject to the error limit. The first algorithm is faster and gives satisfactory results for sufficiently smooth functions. The second algorithm is not as fast but gives better approximations for less well-behaved functions. The two algorithms are ilustrated by several examples.
108 citations
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TL;DR: Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening.
Abstract: Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of C 0 piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.
107 citations
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TL;DR: An extension of the well-known canonical representation for continuous piecewise-linear functions is introduced in this article, which is capable of describing all piecewise linear functions in two dimensions, being no longer subject to any restrictions.
Abstract: An extension of the well-known canonical representation for continuous piecewise-linear functions is introduced. This form is capable of describing all piecewise-linear functions in two dimensions, being no longer subject to any restrictions. Moreover, it is shown that just one nesting of absolute-value functions is sufficient to cover the whole class. >
107 citations
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03 Apr 1996TL;DR: In this paper, a structural description of an integrated circuit is converted into a constraint graph, and the constraint graph is then expanded by replacing edges having piecewise linear cost function with subgraphs constructed from the piecewiselinear cost function.
Abstract: A system, method, and software product in a computer aided design apparatus for system design, to simultaneously optimize multiple performance criteria models of the system, where the performance criteria models are characterized by convex cost functions based on linear dimensional characteristics of system being designed. One embodiment is provided in a computer aid design environment for integrated circuit design, and used to simultaneously optimize fabrication yield along with other performance criteria. Optimization is provided by converting a structural description of an integrated circuit into a constraint graph, compacting, and modifying the constraint graph to include convex cost functions for selected performance criteria to optimized, such as yield cost functions. The cost functions are then transformed to piecewise linear cost functions. The constraint graph is then expanded by replacing edges having piecewise linear cost function with subgraphs constructed from the piecewise linear cost function. The expanded constraint graph is then minimized using a network flow algorithm. Once minimized, the constraint graph describes the positions of circuit elements that maximize yield (and other selected performance criteria) given the cost functions.
107 citations