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.
Papers published on a yearly basis
Papers
More filters
TL;DR: In this paper, a homogenization technique based on Transformation Field Analysis (TFA) for the study of heterogeneous composite media characterized by nonlinear response is presented. But the assumption of uniformity of the inelastic strain distribution is removed, so that a non-uniform inelastically strain field, better representing the nonlinearity distribution in the composite, is considered.
43 citations
06 Feb 1998
TL;DR: This AASERT program supported research in robust simulation, hierarchical uncertainty representation, and novel methods for robustness analysis of uncertain systems, and describes new bounds on a spherical mu problem that allows for correlations between uncertainties in an LFT framework.
Abstract: : This AASERT program supported research in robust simulation, hierarchical uncertainty representation, and novel methods for robustness analysis of uncertain systems. In the context of this program, robust simulation means simulating simultaneously sets of initial conditions and disturbance or noise signals. Thus sets of state space must be propagated by the dynamics of the model. Initial investigations have focused on piecewise linear discrete time systems, which map polyhedra to polyhedra at each time step. Linear programming can be used to refine the resulting bounds. This is important if the potentially exponential growth in set descriptions is to be overcome. Hierarchical uncertainty modeling is a new framework to include explicit representation of uncertainty in component modeling. The focus has been on LFTs and implicit (DAE) representations. A variety of examples including parasitics and non linearities illustrate the key ideas. Finally, this report describes new bounds on a spherical mu problem that allows for correlations between uncertainties in an LFT framework. Interestingly, this setting provides quite elegant bounds and simplified computation.
42 citations
TL;DR: This paper establishes almost pointwise stability of the Ritz projection and the discrete resolvent estimate in $L^\infty$ norm and extends to convex polyhedral domains.
Abstract: The main goal of the paper is to establish that the $L^1$ norm of jumps of the normal derivative across element boundaries and the $L^1$ norm of the Laplacian of a piecewise polynomial finite element function can be controlled by corresponding weighted discrete $H^2$ norm on convex polyhedral domains. In the finite element literature such results are only available for piecewise linear elements in two dimensions and the extension to convex polyhedral domains is rather technical. As a consequence of this result, we establish almost pointwise stability of the Ritz projection and the discrete resolvent estimate in $L^\infty$ norm.
42 citations
TL;DR: This work develops three non-convex optimization models: two yield the minimal number of breakpoints, and another in which the breakpoints are placed such that the maximal deviation is minimized.
Abstract: For univariate functions, we compute optimal breakpoint systems subject to the condition that the piecewise linear approximator, under-, and over-estimator never deviate more than a given $$\delta $$?-tolerance from the original function over a given finite interval. The linear approximators, under-, and over-estimators involve shift variables at the breakpoints allowing for the computation of an optimal piecewise linear, continuous approximator, under-, and over-estimator. We develop three non-convex optimization models: two yield the minimal number of breakpoints, and another in which, for a fixed number of breakpoints, the breakpoints are placed such that the maximal deviation is minimized. Alternatively, we use two heuristics which compute the breakpoints subsequently, solving small non-convex problems. We present computational results for 10 univariate functions. Our approach computes breakpoint systems with up to one order of magnitude less breakpoints compared to an equidistant approach.
42 citations
01 Jan 2012
TL;DR: A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved by a new technique based on an appropriate discretization of the objective function.
Abstract: Semilinear elliptic optimal control problems involving the L¹ norm of the control in the objective are considered. A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved. These are obtained by a new technique based on an appropriate discretization of the objective function. Numerical experiments confirm the convergence rates.
42 citations