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: A new, second-order accurate, volume conservative, material-order-independent interface reconstruction method for multi-material flow simulations is presented, which improves interfaces by minimizing an objective function that smoothes interface normals while enforcing convexity and volume constraints for the pure material subcells.
41 citations
TL;DR: In this paper, the authors describe how to pose straight band initial value problems for lattice equations defined on arbitrary stencils, in finitely many directions, arriving at discrete Goursat problems and in the remaining directions, finding Cauchy problems.
Abstract: We describe how to pose straight band initial value problems for lattice equations defined on arbitrary stencils. In finitely many directions, we arrive at discrete Goursat problems and in the remaining directions we find Cauchy problems. Next, we consider (s1, s2)-periodic initial value problems. In the Goursat directions, the periodic solutions are generated by correspondences. In the Cauchy directions, assuming the equation to be multi-linear, the periodic solution can be obtained uniquely by iteration of a particularly simple mapping, whose dimension is a piecewise linear function of s1, s2.
41 citations
TL;DR: It is illustrated that the proposed PL relaying scheme can improve on sophisticated block Markov encoding when the source-relay link is ill-conditioned (relative to other links) and can work at rates close to those achieved by side-information encoding, but at a much lower complexity.
Abstract: The Gaussian, three-node relay channel with orthogonal receive components (i.e., the transmitted signals from the source and the relay do not interfere with each other) is investigated. For such channels, linear relaying is a suboptimal strategy in general. This is because a linear scheme merely repeats the received noisy signal and does not utilize the available degrees of freedom efficiently. At this background, nonlinear, symbol-wise (as opposed to block-wise) relaying strategies are developed to compensate for the shortcomings of the linear strategy. Optimal strategies are presented for two special cases of the general scenario, and it is shown that memoryless relaying can achieve the capacity. Furthermore, for the general Gaussian relay channel, a parametric piecewise linear (PL) mapping is proposed and analyzed. The achievable rates obtained by the PL mapping are computed numerically and optimized for a certain number of design parameters. It is concluded that optimized PL relaying always outperforms conventional instantaneous linear relaying (amplify-and-forward). It is also illustrated that the proposed PL relaying scheme can improve on sophisticated block Markov encoding (i.e., decode-and-forward) when the source-relay link is ill-conditioned (relative to other links). Furthermore, PL relaying can work at rates close to those achieved by side-information encoding (i.e., compress-and-forward), but at a much lower complexity.
41 citations
TL;DR: The analysis explores the (hidden) dual network flow structure of the appointment scheduling problem and thus greatly simplifies that of prior work, and finds the number of samples needed to compute a near optimal solution when only some independent samples of processing times are known.
Abstract: We consider the problem of determining the optimal schedules for a given sequence of jobs on a single processor. The objective is to minimize the expected total cost incurred by job waiting and processor idling, where the job processing times are random variables. It is known in the prior literature that if the processing times are integers and the costs are linear functions satisfying a mild condition, then the problem can be solved in a polynomial number of expected cost evaluations. In this work, we extend the result to piecewise linear cost functions, which include many useful objective functions in practice. Our analysis explores the (hidden) dual network flow structure of the appointment scheduling problem and thus greatly simplifies that of prior work. We also find the number of samples needed to compute a near optimal solution when only some independent samples of processing times are known.
41 citations
TL;DR: A constructive neural network with a piecewise linear or nonlinear local interpolation capability to approximate arbitrary continuous functions is proposed by introducing a space tessellation which is a covering of the Euclidean space by nonoverlapping hyperpolyhedral convex cells.
Abstract: This paper proposes a constructive neural network with a piecewise linear or nonlinear local interpolation capability to approximate arbitrary continuous functions. This neural network is devised by introducing a space tessellation which is a covering of the Euclidean space by nonoverlapping hyperpolyhedral convex cells. In the proposed neural network, a number of neural network granules (NNG's) are processed in parallel and repeated regularly with the same structures. Each NNG does a local mapping with an interpolation capability for a corresponding hyperpolyhedral convex cell in a tessellation. The plastic weights of the NNG can be calculated to implement the mapping for training data; consequently, this reduces training time and alleviates the difficulties of local minima in training. In addition, the interpolation capability of the NNG improves the generalization for the new data within the convex cell. The proposed network requires additional neurons for tessellation over the standard multilayer neural networks. This increases the network size but does not slow the retrieval response when implemented by parallel architecture. >
41 citations