scholarly journals Analysis of Tensor Approximation Schemes for Continuous Functions

2021 ◽  
Author(s):  
Michael Griebel ◽  
Helmut Harbrecht
Keyword(s):  
Sobolev Space ◽  
Continuous Functions ◽  
Approximation Schemes ◽  
Continuous Analogue ◽  
Tensor Approximation ◽  
The Cost

AbstractIn this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate dimension weights.

Automatic techniques for economic computer computation of continuous functions

SIMULATION ◽  
1968 ◽  
Vol 11 (1) ◽  
pp. 37-48 ◽  
Keyword(s):  
High Speed ◽  
Piecewise Linear ◽  
Continuous Functions ◽  
Computational Error ◽  
Mathematical Function ◽  
Chemical Plants ◽  
Approximation Techniques ◽  
Allowable Error ◽  
The Cost ◽  
Error Envelope

Methods are presented for the automatic preparation of functions of one or more variables for economical calculation by high-speed digital computers. The cost of calculation is considered according to the factors of number of functions, complexity, requirements for precision, and the frequency with which functions are to be calculated. Contrary to classic approaches, con sideration is not given to minimizing computational error for its own sake. On the contrary, the maximum allowable error may be sought in order to minimize computational costs. In this respect, each function is represented by an error envelope that specifies the required limits of computational precision. It is the error envelope rather than the function itself which is dealt with. The approximation techniques dealt with in this paper are limited to piecewise linear ap proximation of functions of one or two independent variables. Projects requiring the maintaining and computation of large quantities of continuous functions are fre quently to be found in industry and research; for example, in the simulation of real-time processes— aircraft flight and flight trainer simulations, simula tion for control and regulation of continuous pro cesses as in chemical plants, weather calculations, radiation studies, etc. In addition, computer service centers, providing computational services to many users, may extend the range and effectiveness of their mathematical function program library by the use of the economical com putational methods of this paper.

A Thermodynamic and Cost Analysis of Solar Syngas From the Zinc/Zinc-Oxide Cycle

2014 ◽  
Author(s):  
Julia Haltiwanger Nicodemus ◽  
Morgan McGuinness ◽  
Rijan Maharjan
Keyword(s):  
Cost Analysis ◽  
Plant Size ◽  
Continuous Functions ◽  
Production Cycle ◽  
Fuel Production ◽  
Ideal Case ◽  
Cost Component ◽  
Maximum Heat ◽  
Syngas Production ◽  
The Cost

We present a thermodynamic and cost analysis of synthesis gas (syngas) production by the Zn/ZnO solar thermochemical fuel production cycle. A mass, energy and entropy balance over each step of the Zn/ZnO syngas production cycle is presented. The production of CO and H2 is considered simultaneously across the range of possible stoichiometric combinations and the effects of irreversibilities due to both recombination in the quenching process following dissociation of ZnO and incomplete conversion in the fuel production step are explored. In the cost analysis, continuous functions for each cost component are presented, allowing estimated costs of syngas fuel produced at plants between 50 and 500MWth. For a solar concentration ratio of 10000, a dissociation temperature of 2300K, and a CO fraction in the syngas of 1/3, the maximum cycle efficiency is 39% for an ideal case in which there is no recombination in the quencher, complete conversion in the oxidizer, and maximum heat recovery. In a 100MWth plant, the cost to produce syngas would be $0.025/MJ for this ideal case. The effect of heat recuperation, recombination in the quencher, and incomplete conversion on efficiency and cost are explored. The effect of plant size and feedstock costs on the cost of solar syngas are also explored. The results underscore the importance improving quencher and oxidizer processes to reduce costs. However, even assuming the ideal case, the predicted cost of solar syngas is 5.5 times more expensive than natural gas on an energy basis. The process will therefore require incentive policies that support early implementation in order to become economically competitive.

scholarly journals An EPQ Model with Unit Production Cost and Set-Up Cost as Functions of Production Rate

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Behrouz Afshar-Nadjafi
Keyword(s):  
Production Rate ◽  
Production Cost ◽  
Continuous Functions ◽  
Optimal Production ◽  
Economic Production Quantity ◽  
Economic Production ◽  
Epq Model ◽  
Production Quantity ◽  
Set Up ◽  
The Cost

Extensive research has been devoted to economic production quantity (EPQ) problem. However, no attention has been paid to problems where unit production and set-up costs must be considered as functions of production rate. In this paper, we address the problem of determining the optimal production quantity and rate of production in which unit production and set-up costs are assumed to be continuous functions of production rate. Based on the traditional economic production quantity (EPQ) formula, the cost function associated with this model is proved to be nonconvex and a procedure is proposed to solve this problem. Finally, utility of the model is presented using some numerical examples and the results are analyzed.

scholarly journals Inclusion of Hajłasz – Sobolev class Mpα(X) into  the space of continuous functions in the critical case

2020 ◽  
pp. 6-12
Author(s):  
Sergey A. Bondarev
Keyword(s):  
Sobolev Space ◽  
Measure Space ◽  
Critical Case ◽  
Sobolev Class ◽  
General Structure ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Euclidean Case ◽  
Uniformly Perfect ◽  
Perfect Space

Let (X, d, µ) be a doubling metric measure space with doubling dimension γ, i. e. for any balls B(x, R) and B(x, r), r < R, following inequality holds µ(B(x, R)) ≤ aµ (R/r)γµ(B(x, r)) for some positive constants γ and aµ. Hajłasz – Sobolev space Mpα(X) can be defined upon such general structure. In the Euclidean case Hajłasz – Sobolev space coincides with classical Sobolev space when p > 1, α = 1. In this article we discuss inclusion of functions from Hajłasz – Sobolev space Mpα(X) into the space of continuous functions for p ≤ 1 in the  critical case γ = α p. More precisely, it is shown that any function from Hajłasz – Sobolev class Mpα(X), 0 < p ≤ 1, α > 0, has a continuous representative in case of uniformly perfect space (X, d, µ).

scholarly journals Singular value decomposition versus sparse grids: refined complexity estimates

2018 ◽  
Vol 39 (4) ◽  
pp. 1652-1671 ◽  
Author(s):  
Michael Griebel ◽  
Helmut Harbrecht
Keyword(s):  
Singular Value Decomposition ◽  
Sparse Grids ◽  
Singular Value ◽  
Sparse Grid ◽  
Approximation Schemes ◽  
Approximation Of Functions ◽  
Grid Approximation ◽  
Numer Anal ◽  
Value Decomposition ◽  
The Cost

Abstract We compare the cost complexities of two approximation schemes for functions that live on the product domain $\varOmega _1\times \varOmega _2$ of sufficiently smooth domains $\varOmega _1\subset \mathbb{R}^{n_1}$ and $\varOmega _2\subset \mathbb{R}^{n_2}$, namely the singular value / Karhunen–Lòeve decomposition and the sparse grid representation. We assume that appropriate finite element methods with associated orders $r_1$ and $r_2$ of accuracy are given on the domains $\varOmega _1$ and $\varOmega _2$, respectively. This setting reflects practical needs, since often black-box solvers are used in numerical simulation, which restrict the freedom in the choice of the underlying discretization. We compare the cost complexities of the associated singular value decomposition and the associated sparse grid approximation. It turns out that, in this situation, the approximation by the sparse grid is always equal or superior to the approximation by the singular value decomposition. The results in this article improve and generalize those from the study by Griebel & Harbrecht (2014, Approximation of bi-variate functions. Singular value decomposition versus sparse grids. IMA J. Numer. Anal., 34, 28–54). Especially, we consider the approximation of functions from generalized isotropic and anisotropic Sobolev spaces.

scholarly journals $L^{\Phi }-L^{\infty }$\ Inequalities and Applications

2015 ◽  
Vol 7 (2) ◽  
pp. 201 ◽  
Author(s):  
Tiziano Granucci
Keyword(s):  
Sobolev Space ◽  
Continuous Functions ◽  
Integral Functionals ◽  
Scalar Integral

In this paper we prove some $L^{\Phi }-L^{\Phi }$ and $L^{\Phi }-L^{\infty }$inequalities for quasi-minima of scalar integral functionals defined inOrlicz-Sobolev space $W^{1}L^{\Phi }\left( \Omega \right) $, where $\Phi $\is a N-function and $\Phi \in \triangle _{2}$. Moreover, if $\Phi \in\triangle ^{^{\prime }}$ or if $\Phi \in \triangle _{2}\cap \nabla _{2}$, weprove that quasi-minima are H\"{o}lder continuous functions.

A Thermodynamic and Cost Analysis of Solar Syngas From the Zn/ZnO Cycle

2014 ◽  
Vol 137 (1) ◽  
Author(s):  
Julia Haltiwanger Nicodemus ◽  
Morgan McGuinness ◽  
Rijan Maharjan
Keyword(s):  
Cost Analysis ◽  
Plant Size ◽  
Continuous Functions ◽  
Production Cycle ◽  
Fuel Production ◽  
Ideal Case ◽  
Cost Component ◽  
Maximum Heat ◽  
Syngas Production ◽  
The Cost

We present a thermodynamic and cost analysis of synthesis gas (syngas) production by the Zn/ZnO solar thermochemical fuel production cycle. A mass, energy, and entropy balance over each step of the Zn/ZnO syngas production cycle is presented. The production of CO and H2 is considered simultaneously across the range of possible stoichiometric combinations, and the effects of irreversibilities due to both recombination in the quenching process following dissociation of ZnO and incomplete conversion in the fuel production step are explored. In the cost analysis, continuous functions for each cost component are presented, allowing estimated costs of syngas fuel produced at plants between 50 and 500 MWth. For a solar concentration ratio of 10,000, a dissociation temperature of 2300 K, and a CO fraction in the syngas of 1/3, the maximum cycle efficiency is 39% for an ideal case in which there is no recombination in the quencher, complete conversion in the oxidizer, and maximum heat recovery. In a 100 MWth plant, the cost to produce syngas would be $0.025/MJ for this ideal case. The effect of heat recuperation, recombination in the quencher, and incomplete conversion on efficiency and cost are explored. The effects of plant size and feedstock costs on the cost of solar syngas are also explored. The results underscore the importance improving quencher and oxidizer processes to reduce costs. However, even assuming the ideal case, the predicted cost of solar syngas is 5.5 times more expensive than natural gas on an energy basis. The process will therefore require incentive policies that support early implementation in order to become economically competitive.

The IBM-PC in electron microscopy

1996 ◽  
Vol 54 ◽  
pp. 612-613
Author(s):  
James F. Mancuso
Keyword(s):  
Head Start ◽  
Image Acquisition ◽  
Cost Effective ◽  
Financial Applications ◽  
Number Of Factors ◽  
Instrument Control ◽  
The Cost ◽  
Training And Support ◽  
Control Image ◽  
Ibm Pc

IBM PC compatible computers are widely used in microscopy for applications ranging from control to image acquisition and analysis. The choice of IBM-PC based systems over competing computer platforms can be based on technical merit alone or on a number of factors relating to economics, availability of peripherals, management dictum, or simple personal preference.IBM-PC got a strong “head start” by first dominating clerical, document processing and financial applications. The use of these computers spilled into the laboratory where the DOS based IBM-PC replaced mini-computers. Compared to minicomputer, the PC provided a more for cost-effective platform for applications in numerical analysis, engineering and design, instrument control, image acquisition and image processing. In addition, the sitewide use of a common PC platform could reduce the cost of training and support services relative to cases where many different computer platforms were used. This could be especially true for the microscopists who must use computers in both the laboratory and the office.

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