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approximation error estimates and inverse inequalities for B-splines of maximum smoothness

MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES 2016年第7期26卷 1411-1445页

作者： Takacs, Stefan Takacs, Thomas Johannes Kepler Univ Linz Dept Computat Math Altenberger Str 69 A-4020 Linz Austria Univ Pavia Dept Math Via Ferrata 5 I-27100 Pavia Italy

In this paper, we develop approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard Sobolev norms and semi-norms. The presented approximation error estimates do not depend on the polynomial degree of the splines but only on the grid size. We will see that the approximation lives in a subspace of the classical B-spline space. We show that for this subspace, there is an inverse inequality which is also independent of the polynomial degree. As the approximation error estimate and the inverse inequality show complementary behavior, the results shown in this paper can be used to construct fast iterative methods for solving problems arising from isogeometric discretizations of partial differential equations.

关键词： approximation error B-spline isogeometric analysis robustness

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approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging

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JOURNAL OF BIOMEDICAL OPTICS 2012年第9期17卷 96012-1页

作者： Heiskala, Juha Kolehmainen, Ville Tarvainen, Tanja Kaipio, Jari. P. Arridge, Simon R. UCL Dept Comp Sci London WC1E 6BT England Univ Eastern Finland Dept Appl Phys Kuopio 70211 Finland Univ Auckland Dept Math Auckland Mail Ctr Auckland 1142 New Zealand

Diffuse optical tomography can image the hemodynamic response to an activation in the human brain by measuring changes in optical absorption of near-infrared light. Since optodes placed on the scalp are used, the measurements are very sensitive to changes in optical attenuation in the scalp, making optical brain activation imaging susceptible to artifacts due to effects of systemic circulation and local circulation of the scalp. We propose to use the Bayesian approximation error approach to reduce these artifacts. The feasibility of the approach is evaluated using simulated brain activations. When a localized cortical activation occurs simultaneously with changes in the scalp blood flow, these changes can mask the cortical activity causing spurious artifacts. We show that the proposed approach is able to recover from these artifacts even when the nominal tissue properties are not well known. (C) 2012 Society of Photo-Optical Instrumentation Engineers (SPIE). [DOI: 10.1117/***.17.9.096012]

关键词： diffuse optical imaging approximation error Bayesian

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Bayesian approximation error Approach in Full-Wave Ultrasound Tomography

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IEEE TRANSACTIONS ON ULTRASONICS FERROELECTRICS AND FREQUENCY CONTROL 2014年第10期61卷 1627-1637页

作者： Koponen, Janne Huttunen, Tomi Tarvainen, Tanja Kaipio, Jari P. Univ Eastern Finland Dept Appl Phys Kuopio Finland UCL Dept Comp Sci London England Univ Auckland Dept Math Auckland New Zealand

In ultrasound tomography, the spatial distribution of the speed of sound in a region of interest is reconstructed based on transient measurements made around the object. The computation of the forward problem (the full-wave solution) within the object is a computationally intensive task and can often be prohibitive for practical purposes. The purpose of this paper is to investigate the feasibility of using approximate forward solvers and the partial recovery from the related errors by employing the Bayesian approximation error approach. In addition to discretization error, we also investigate whether the approach can be used to reduce the reconstruction errors that are due to the adoption of approximate absorbing boundary models. We carry out two numerical studies in which the objective is to reduce the computational times to around 3% of the time that would be required by a numerically accurate forward solver. The results show that the Bayesian approximation error approach improves the reconstructions.

关键词： Computational modeling Numerical models approximation error Bayes methods Inverse problems Mathematical model

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Neuro-adaptive control of mobile manipulators based on compensation of approximation error

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ELECTRONICS LETTERS 2002年第16期38卷 935-936页

作者： Lee, CY Eom, TD Lee, JJ Korea Adv Inst Sci & Technol Dept Elect Engn & Comp Sci Yusung Gu Taejon 305701 South Korea

A stable adaptive control method based on neural networks is proposed for mobile manipulators. Adaptive compensation of the approximation error of a neural network based on a priori knowledge on the dynamic form of th... 详细信息

关键词： manipulators mobile manipulators neural networks neurocontrollers Mobile robots neuro-adaptive control control system design error compensation system dynamics stable adaptive control method Self-adjusting control systems control system synthesis Manipulators globally stable method adaptive compensation stability approximation error Neurocontrol mobile robots Control system analysis and synthesis methods Stability in control theory adaptive control

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Estimating the approximation error when fixing unessential factors in global sensitivity analysis

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RELIABILITY ENGINEERING & SYSTEM SAFETY 2007年第7期92卷 957-960页

作者： Sobol, I. M. Tarantola, S. Gatelli, D. Kucherenko, S. S. Mauntz, W. Commiss European Communities Joint Res Ctr Inst Protect & Secur Citizen I-21020 Ispra VA Italy Russian Acad Sci Inst Math Modelling Moscow Russia Univ London Imperial Coll Sci & Technol London England Univ Dortmund Dept Chem & Biochem Engn Dortmund Germany

One of the major settings of global sensitivity analysis is that of fixing non-influential factors, in order to reduce the dimensionality of a model. However, this is often done without knowing the magnitude of the approximation error being produced. This paper presents a new theorem for the estimation of the average approximation error generated when fixing a group of non-influential factors. A simple function where analytical solutions are available is used to illustrate the theorem. The numerical estimation of small sensitivity indices is discussed. (C) 2006 Elsevier Ltd. All rights reserved.

关键词： Sobol' method approximation error sensitivity analysis by groups sensitivity indices

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Advantages of Using Unweighted approximation error Measures for Model Fit Assessment

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PSYCHOMETRIKA 2023年第2期88卷 413-433页

作者： Lubbe, Dirk Brandenburg Med Sch Theodor Fontane Neuruppin Germany Brandenburg Med Sch Theodor Fontane Dept Psychol Sect Psychol Methods Alten Gymnasium 1-3 D-16816 Neuruppin Germany

Fit indices are highly frequently used for assessing the goodness of fit of latent variable models. Most prominent fit indices, such as the root-mean-square error of approximation (RMSEA) or the comparative fit index (CFI), are based on a noncentrality parameter estimate derived from the model fit statistic. While a noncentrality parameter estimate is well suited for quantifying the amount of systematic error, the complex weighting function involved in its calculation makes indices derived from it challenging to interpret. Moreover, noncentrality-parameter-based fit indices yield systematically different values, depending on the indicators' level of measurement. For instance, RMSEA and CFI yield more favorable fit indices for models with categorical as compared to metric variables under otherwise identical conditions. In the present article, approaches for obtaining an approximation discrepancy estimate that is independent from any specific weighting function are considered. From these unweighted approximation error estimates, fit indices analogous to RMSEA and CFI are calculated and their finite sample properties are investigated using simulation studies. The results illustrate that the new fit indices consistently estimate their true value which, in contrast to other fit indices, is the same value for metric and categorical variables. Advantages with respect to interpretability are discussed and cutoff criteria for the new indices are considered.

关键词： discrepancy function goodness of fit approximation error structural equation model

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ESTIMATING THE approximation error IN LEARNING THEORY

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ANALYSIS AND APPLICATIONS 2003年第1期1卷 17-41页

作者： Smale, Steve Zhou, Ding-Xuan City Univ Hong Kong Dept Math Kowloon Hong Kong Peoples R China

Let B be a Banach space and (H, parallel to center dot parallel to(H)) be a dense, imbedded subspace. For a is an element of B, its distance to the ball of H with radius R (denoted as I(a, R)) tends to zero when R tends to infinity. We are interested in the rate of this convergence. This approximation problem arose from the study of learning theory, where B is the L-2 space and H is a reproducing kernel Hilbert space. The class of elements having I(a, R) = O(R-r) with r > 0 is an interpolation space of the couple (B, H). The rate of convergence can often be realized by linear operators. In particular, this is the case when H is the range of a compact, symmetric, and strictly positive definite linear operator on a separable Hilbert space B. For the kernel approximation studied in learning theory, the rate depends on the regularity of the kernel function. This yields error estimates for the approximation by reproducing kernel Hilbert spaces. When the kernel is smooth, the convergence is slow and a logarithmic convergence rate is presented for analytic kernels in this paper. The purpose of our results is to provide some theoretical estimates, including the constants, for the approximation error required for the learning theory.

关键词： Learning theory approximation error reproducing kernel Hilbert space kernel machine learning interpolation space logarithmic rate of convergence

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approximation error of single hidden layer neural networks with fixed weights

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INFORMATION PROCESSING LETTERS 2024年 185卷

作者： Ismailov, Vugar E. Inst Math & Mech Baku Azerbaijan Khazar Univ Baku Azerbaijan

Neural networks with finitely many fixed weights have the universal approximation property under certain conditions on compact subsets of the ��-dimensional Euclidean space, where approximation process is considered. Such conditions were delineated in our paper [26]. But for many compact sets it is impossible to approximate multivariate functions with arbitrary precision and the question on estimation or efficient computation of approximation error arises. This paper provides an explicit formula for the approximation error of single hidden layer neural networks with two fixed weights.

关键词： Neural network approximation error Mean periodic function Path approximation algorithms

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Density approximation error Assessment and Compensation in Point-Mass Filter 25

Density Approximation Error Assessment and Compensation in P...

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25th International Conference of Information Fusion (FUSION)

作者： Matousek, J. Dunik, J. Straka, O. Blasch, E. Univ West Bohemia Dept Cyber Univ 8 Plzen 30614 Czech Republic Air Force Res Lab Rome NY 13441 USA

ISBN: (纸本)9781665489416

This paper deals with the state estimation of non-linear stochastic dynamic systems with an emphasis on a probability density function approximation used by point-mass filters. approximation error of the standard point-mass density is analysed and quantified, and a novel point-mass density approximation with inherent approximation error minimisation is developed. The properties of the proposed point-mass are theoretically analysed and numerically illustrated.

关键词： State estimation Point-mass filter Density approximation approximation error

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A SEMI-ANALYTICAL BIVARIATE GAUSSIAN MODEL OF THE approximation error IMPACT ON THE MIN-SUM LDPC DECODING ALGORITHM

A SEMI-ANALYTICAL BIVARIATE GAUSSIAN MODEL OF THE APPROXIMAT...

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IEEE Workshop on Signal Processing Systems (SiPS)

作者： Kanistras, Nikos Paliouras, Vassilis Univ Patras Dept Elect & Comp Engn Patras 26500 Greece

ISBN: (纸本)9781467362382

In this paper, a new theoretical model that describes the impact of the approximation error on the decisions taken by LDPC decoders is discussed. In particular, the theoretical model extends previous results and reconstructs the mechanism, by means of which the approximation error alters the decisions of the decoding algorithm, with respect to the decisions taken by the optimal decoding algorithm, namely Log Sum-Product. We focus on the most popular algorithm for LDPC decoding, namely Min-Sum and its also popular modifications, normalized and offset Min-Sum. The model is applied to all of these decoding algorithms, which are actually approximations of the Log Sum-Product. Moreover a method that exploits the output of the proposed model in order to estimate the decoding performance is also proposed. Finally, experimental results prove the validity of both the proposed model and the method, demonstrating the usefulness of this contribution towards achieving accurate decoding behavior prediction without relying on time-consuming simulations.

关键词： Low-density parity-check (LDPC) codes approximation error error correction coding min-sum decoding

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