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: It is proved that the average of derivatives of this sample performance function with a given initial state does converge, with probability one, to the derivative of the conditional mean value, given the same initial state.
Abstract: A stochastic system such as a queueing network can be specified by system parameters and a set of sequences of random variables that represents the randomness in the system. A “sample performance function” is a measure of system performance as a function of system parameters, for each realization of the random sequences. Although the average of N sample performance functions converges to the expected value of the performance with probability one when N goes to infinity, the average of the derivatives of these N sample performance functions with respect to a parameter may not converge to the derivative of the expected value. In this paper, we study a sample performance function of a closed Jackson queueing network; specifically, the time required by a server to serve a finite number of customers. We show that this sample performance function is a continuous, piecewise linear function of the mean service time. We prove that the average of derivatives of this sample performance function with a given initial ...
41 citations
TL;DR: This work extends results from Part I about estimating gradient errors elementwise a posteriori, given there for quadratic and higher elements, to the piecewise linear case and gives posteriori estimators for second derivatives on each element.
Abstract: We extend results from Part I about estimating gradient errors elementwise a posteriori, given there for quadratic and higher elements, to the piecewise linear case. The key to our new result is to consider certain technical estimates for differences in the error, e(x 1 )-e(x 2 ), rather than for e(x) itself. We also give a posteriori estimators for second derivatives on each element.
41 citations
TL;DR: In this article, the inverse boundary value problem of determining the potential q = 0 in the reduced wave equation was considered and a result of global Lipschitz stability was obtained in dimension n/geq 3 for potentials that are piecewise linear on a given partition of Euclidean space.
Abstract: We consider the inverse boundary value problem of determining the potential $q$ in the equation $\\Delta u + qu = 0$ in $\\Omega\\subset\\mathbb{R}^n$, from local Cauchy data. A result of global Lipschitz stability is obtained in dimension $n\\geq 3$ for potentials that are piecewise linear on a given partition of $\\Omega$. No sign, nor spectrum condition on $q$ is assumed, hence our treatment encompasses the reduced wave equation $\\Delta u + k^2c^{-2}u=0$ at fixed frequency $k$.
41 citations
01 Jan 1998
TL;DR: An explicit construction of compactly supported prewavelets on linear finite element spaces is introduced on non-uniform meshes on polyhedron domains and on boundaries of such domains, and the basis transformation from wavelet- to nodal basis (the wavelet transform) can be implemented more efficiently.
Abstract: In this paper, an explicit construction of compactly supported prewavelets on linear finite element spaces is introduced on non-uniform meshes on polyhedron domains and on boundaries of such domains. The obtained basas are stable in the Sobolev spaces Hr for |r| < 3/2. The only condition we need is that of uniform refinements. Compared to existing prewavelets bases on uniform meshes, with our construction the basis transformation from wavelet- to nodal basis (the wavelet transform) can be implemented more efficiently.
41 citations
TL;DR: In this paper, a new interface reconstruction method in 3D is presented, which involves a conservative level-contour reconstruction coupled with a cubic-Bezier interpolation, achieving second-order spatial and temporal accuracy.
Abstract: A new interface reconstruction method in 3D is presented. The method involves a conservative level-contour reconstruction coupled to a cubic-Bezier interpolation. The use of the proposed piecewise linear interface calculation (PLIC) reconstruction scheme coupled to a multidimensional time integration provides solutions of second-order spatial and temporal accuracy. The accuracy and efficiency of the proposed reconstruction algorithm are demonstrated through several tests, whose results are compared with those obtained with other recently proposed methods. An overall improvement in accuracy with respect to other recent methods has been achieved, along with a substantial reduction in the central processing unit time required.
41 citations