1,227 results
Search Results
102. A Mixture Model for Robust Point Matching under Multi-Layer Motion.
- Author
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Ma, Jiayi, Chen, Jun, Ming, Delie, and Tian, Jinwen
- Subjects
ALGORITHMS ,NONPARAMETRIC estimation ,BAYESIAN analysis ,HILBERT space ,APPROXIMATION theory ,DEFORMATIONS (Mechanics) ,APPLIED mathematics - Abstract
This paper proposes an efficient mixture model for establishing robust point correspondences between two sets of points under multi-layer motion. Our algorithm starts by creating a set of putative correspondences which can contain a number of false correspondences, or outliers, in addition to the true correspondences (inliers). Next we solve for correspondence by interpolating a set of spatial transformations on the putative correspondence set based on a mixture model, which involves estimating a consensus of inlier points whose matching follows a non-parametric geometrical constraint. We formulate this as a maximum a posteriori (MAP) estimation of a Bayesian model with hidden/latent variables indicating whether matches in the putative set are outliers or inliers. We impose non-parametric geometrical constraints on the correspondence, as a prior distribution, in a reproducing kernel Hilbert space (RKHS). MAP estimation is performed by the EM algorithm which by also estimating the variance of the prior model (initialized to a large value) is able to obtain good estimates very quickly (e.g., avoiding many of the local minima inherent in this formulation). We further provide a fast implementation based on sparse approximation which can achieve a significant speed-up without much performance degradation. We illustrate the proposed method on 2D and 3D real images for sparse feature correspondence, as well as a public available dataset for shape matching. The quantitative results demonstrate that our method is robust to non-rigid deformation and multi-layer/large discontinuous motion. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
103. Checking the optimality of entanglement witnesses: an application to structural physical approximations.
- Author
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Augusiak, R, Bae, J, Tura, J, and Lewenstein, M
- Subjects
QUANTUM entanglement ,APPROXIMATION theory ,QUANTUM theory ,STRUCTURAL mechanics ,MATHEMATICS - Abstract
In 2008, the conjecture that structural physical approximations (SPAs) to optimal entanglement witnesses are separable states (in general unnormalized) was posed. In an attempt to disprove it, Ha and Kye (2012 arXiv:1210.1088v3) proposed a decomposable entanglement witness, whose SPA is entangled, and argued that it is optimal. In this paper, which is based on a comment to the latter work (Augusiak et al 2013 arXiv:1304.2040v1), we show, both analytically and numerically, that this entanglement witness is not optimal, and as such it is not a counterexample to the conjecture. To this end, we make use of a method for checking the optimality of entanglement witnesses developed already in Lewenstein et al (2000 Phys. Rev. A 62 052310), however, hardly exploited so far in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
104. [InlineEquation not available: see fulltext.]-Approximation of signals (functions) belonging to weighted [InlineEquation not available: see fulltext.]-class by [InlineEquation not available: see fulltext.] summability method of conjugate series of its Fourier series.
- Author
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Mishra, Vishnu Narayan, Sonavane, Vaishali, and Mishra, Lakshmi Narayan
- Subjects
LIPSCHITZ spaces ,EQUATIONS ,APPROXIMATION theory ,MONOTONIC functions ,MATHEMATICS - Abstract
Recently, Lal (Appl. Math. Comput. 209:346-350, 2009) has determined the degree of approximation of a function belonging to Lip α and weighted [InlineEquation not available: see fulltext.]-classes using product [InlineEquation not available: see fulltext.] summability with non-increasing weights [InlineEquation not available: see fulltext.]. In this paper, we determine the degree of approximation of function [InlineEquation not available: see fulltext.], conjugate to a 2 π-periodic function f belonging to weighted [InlineEquation not available: see fulltext.]-class by dropping the monotonicity on the generating sequence [InlineEquation not available: see fulltext.] with a new (proper) set of conditions, which in turn generalizes the results of Mishra et al. (Bull. Math. Anal. Appl., 2013) on [InlineEquation not available: see fulltext.]-class and rectifies (removes) the errors of Mishra et al. (Mat. Vesn., 2013). Few examples and applications are also highlighted in this manuscript. MSC: Primary 42B05, 42B08, 40G05, 41A10. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
105. DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods.
- Author
-
Tanaka, Ken'ichiro, Okayama, Tomoaki, Matsuo, Takayasu, and Sugihara, Masaaki
- Subjects
NUMERICAL calculations ,SINC function ,EXPONENTIAL functions ,MATHEMATICS ,APPROXIMATION theory ,INTEGRALS - Abstract
In this paper, the theoretical convergence rate of the Sinc indefinite integration combined with the double-exponential (DE) transformation is given for a class of functions for which the single-exponential (SE) transformation is suitable. Although the DE transformation is considered as an enhanced version of the SE transformation for Sinc-related methods, the function space for which the DE transformation is suitable is smaller than that for SE, and therefore, there exist some examples such that the DE transformation is not better than the SE transformation. Even in such cases, however, some numerical observations in the literature suggest that there is almost no difference in the convergence rates of SE and DE. In fact, recently, the observations have been theoretically explained for two explicit approximation formulas: the Sinc quadrature and the Sinc approximation. The conclusion is that in such cases, the DE's rate is slightly lower, but almost the same as that of the SE. The contribution of this study is the derivation of the same conclusion for the Sinc indefinite integration. Numerical examples that support the theoretical result are also provided. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
106. Neural Approximation of Empirical Functions.
- Author
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ROJ, J.
- Subjects
ARTIFICIAL neural networks ,ARTIFICIAL intelligence ,MATHEMATICS ,APPROXIMATION theory ,LEARNING - Abstract
The paper presents the results of simulation studies of selected neural network structures used for non-linear function approximation based on a limited accuracy data. There was performed the analysis of the interdependence of the network structure and the size of the set of learning patterns. The approximation inaccuracy was expressed by the uncertainty interval width. The approximation properties of the neural method were compared with those of the piece-wise linear and polynomial: "cubic" and "spline" methods. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
107. On some vertical cohomologies of complex Finsler manifolds.
- Author
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Ida, Cristian
- Subjects
FINSLER spaces ,DIFFERENTIAL geometry ,MATHEMATICS ,APPROXIMATION theory ,GEOMETRY - Abstract
In this paper we study some vertical cohomologies of complex Finsler manifolds as vertical cohomology attached to a function and vertical Lichnerowicz cohomology. We also study a relative vertical cohomology attached to a function associated to a holomorphic Finsler subspace. [ABSTRACT FROM AUTHOR]
- Published
- 2013
108. Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation.
- Author
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Bambusi, Dario, Faou, Erwan, and Grébert, Benoît
- Subjects
SUBSPACE identification (Mathematics) ,SUBSPACES (Mathematics) ,APPROXIMATION theory ,EQUATIONS ,SYMPLECTIC spaces ,MATHEMATICS - Abstract
In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schrödinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schrödinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical ground state which is close in energy norm to the continuous ground state. Such result is valid under a CFL condition of the form $$\tau h^{-2}\le C$$ where $$\tau $$ and $$h$$ denote the time and space step size respectively. Furthermore we prove that if the initial datum is symmetric and close to the continuous ground state $$\eta $$ then the associated numerical solution remains close to the orbit of $$\eta ,\Gamma =\cup _\alpha \{e^{i\alpha }\eta \}$$, for very long times. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
109. A numerical scheme based on differential quadrature method to solve time dependent Burgers' equation.
- Author
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Mittal, R.C., Jiwari, Ram, and Sharma, Kapil K.
- Subjects
BOUNDARY value problems ,NUMERICAL solutions to nonlinear differential equations ,NUMERICAL analysis ,APPROXIMATION theory ,QUASILINEARIZATION - Abstract
Purpose – The purpose of this paper is to propose a numerical method to solve time dependent Burgers' equation with appropriate initial and boundary conditions. Design/methodology/approach – The presence of the nonlinearity in the problem leads to severe difficulties in the solution approximation. In construction of the numerical scheme, quasilinearization is used to tackle the nonlinearity of the problem which is followed by semi discretization for spatial direction using differential quadrature method (DQM). Semi discretization of the problem leads to a system of first order initial value problems which are followed by fully discretization using RK4 scheme. The method is analyzed for stability and convergence. Findings – The method is illustrated and compared with existing methods via numerical experiments and it is found that the proposed method gives better accuracy and is quite easy to implement. Originality/value – The new scheme is developed by using some numerical schemes. The scheme is analyzed for stability and convergence. In support of predicted theory some test examples are solved using the presented method. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
110. Numerical study of amplitude equations for SPDEs with degenerate forcing.
- Author
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Blömker, Dirk, Mohammed, WaelW., Nolde, Christian, and Wöhrl, Franz
- Subjects
NUMERICAL solutions to stochastic partial differential equations ,NUMERICAL analysis ,BURGERS' equation ,APPROXIMATION theory ,MATHEMATICS ,MATHEMATICAL analysis ,FORCING (Model theory) - Abstract
In this paper, we give a review on rigorous and numerical results for amplitude equations. We focus on the Swift-Hohenberg equation and the Burgers' equation in order to determine the quality of the approximation and the impact of degenerate noise on the approximating equation. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
111. Inhomogeneous approximation by coprime integers.
- Author
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Haynes, Alan
- Subjects
DIOPHANTINE approximation ,DIOPHANTINE analysis ,APPROXIMATION theory ,INTEGERS ,MATHEMATICS - Abstract
This paper addresses a problem recently raised by Laurent and Nogueira about inhomogeneous Diophantine approximation with coprime integers. A corollary of our main theorem is that for any irrational α ∈ R and for any γ ∈ R and ∈ > 0 there are infinitely many pairs of coprime integers m; n such that |nα - m - γ| ≤ 1/|n|
1-∈ . This improves upon previously known results, in which the exponent of approximation was at best 1/2. [ABSTRACT FROM AUTHOR]- Published
- 2012
112. On Randomized Approximation for Finding a Level Ideal of a Poset and the Generalized Median Stable Matchings.
- Author
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Kijima, Shuji and Nemoto, Toshio
- Subjects
ALGORITHMS ,APPROXIMATION theory ,INTEGERS ,MATHEMATICS ,POLYNOMIALS - Abstract
This study is concerned with finding a level ideal (LI) of a partially ordered set (poset). Given a finite poset P, the level of each element p ∈ P is defined as the number of ideals that do not include p, then the problem is to find the ith LI-the ideal consisting of elements whose levels are less than a given integer i. The concept of a level ideal is naturally derived from the generalized median stable matchings, introduced by Teo and Sethuraman [Teo, C. P., J. Sethuraman. 1998. The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4) 874-891] in the context of "fairness" of matchings in a stable marriage problem. Cheng [Cheng, C. T. 2010. Understanding the generalized median stable matchings. Algorithmica 58(1) 34-51] showed that finding the ith LI is #P-hard when i = θ (N), where N is the total number of ideals of P. This paper shows that finding the ith LI is #P-hard even if i = θ (N
1/c ), where c is an arbitrary constant at least one. Meanwhile, we present a polynomial time exact algorithm when i = O(logN)c' , where c' is an arbitrary positive constant. We also devise two randomized approximation schemes for the ideals of a poset, by using an oracle of an almost-uniform sampler. [ABSTRACT FROM AUTHOR]- Published
- 2012
- Full Text
- View/download PDF
113. POLYNOMIAL APPROXIMATIONS FOR CONTINUOUS LINEAR PROGRAMS.
- Author
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Bampou, Dimitra and Kuhn, Daniel
- Subjects
POLYNOMIALS ,APPROXIMATION theory ,LINEAR programming ,APPLIED mathematics ,MATHEMATICS - Abstract
Continuous linear programs have attracted considerable interest due to their potential for modeling manufacturing, scheduling, and routing problems. While efficient simplex-type algorithms have been developed for separated continuous linear programs, crude time discretization remains the method of choice for solving general (nonseparated) problenl instances. In this paper we propose a more generic approximation scheme for nonseparated continuous linear programs, where we approximate the functional decision variables (policies) by polynomial and piecewise polynomial decision rules. This restriction results in an upper bound on the original problem, which can be computed efficiently by solving a tractable semidefinite program. To estimate the approximation error, we also compute a lower bound by solving a dual continuous lineal program in (piecewise) polynomial decision rules. We establish the convergence of the primal and dual approximations under Stater-type constraint qualifications. We also highlight the potential of our method for optimizing large-scale multiclass queueing systems and dynamic Leontief models. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
114. Subdivision schemes and multi-resolution modelling for automated music synthesis and analysis.
- Author
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Hed, Sigalit, Gjerdingen, RobertO., and Levin, David
- Subjects
MUSICAL composition ,MATHEMATICS ,DATA analysis ,REGRESSION analysis ,PERTURBATION theory ,APPROXIMATION theory ,FUNCTIONAL analysis - Abstract
Subdivision schemes are special multi-resolution analysis (MRA) methods that have become prevalent in computer-aided geometric design. This paper draws useful analogies between the mathematics of subdivision schemes and the hierarchical structures of music compositions. Based on these analogies, we propose new methods for music synthesis and analysis through MRA, which provide a different perspective on music composition, representation and analysis. We demonstrate that the structure and recursive nature of the recently proposed subdivision models [S. Hed and D. Levin, Subdivision models for varying-resolution and generalized perturbations, Int. J. Comput. Math. 88(17) (2011), pp. 3709–3749; S. Hed and D. Levin, A ‘subdivision regression’ model for data analysis, 2012, in preparation] are well suited to the synthesis and analysis of monophonic and polyphonic musical patterns, doubtless due in large part to the strongly hierarchical nature of traditional musical structures. The analysis methods demonstrated enable the compression and decompression (reconstruction) of selected musical pieces and derive useful features of the pieces, laying groundwork for music classification. [ABSTRACT FROM PUBLISHER]
- Published
- 2012
- Full Text
- View/download PDF
115. Some best approximation formulas and inequalities for the Bateman's G-function.
- Author
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Hegazi, Ahmed, Mahmoud, Mansour, Talat, Ahmed, and Moustafa, Hesham
- Subjects
- *
APPROXIMATION theory , *FUNCTIONAL analysis , *MATHEMATICAL functions , *MATHEMATICAL constants , *MATHEMATICS - Abstract
In the paper, the authors established two best approximation formulas for the Bateman's G-function. Also, they studied the completely monotonicity of some functions involving G(x). Some new inequalities are deduced for the function and its derivative such as 1/2 ln [1 + 2x + a/x2 + 2x + 4/3 ] < G(x + 2) < 1/2 ln [ 1 + 2x + b/x2 + 2x + 4/3 ], x > 0 where a = 3 and b = e4-16/12 are the best possible constants. Our results improve some recent inequalities about the function G(x). [ABSTRACT FROM AUTHOR]
- Published
- 2019
116. Restriction Method for Approximating Square Roots.
- Author
-
Batiha, Iqbal M.
- Subjects
SQUARE root ,NUMERICAL roots ,APPROXIMATION theory ,RATIONAL numbers ,MATHEMATICS - Abstract
The aim of this paper is to present a new method called "Restriction Method for Approximating Square Roots", that helps students to find an approximate value for any square root of positive Rational Number with an easy and simple way. Also, we prove this method and we give some examples that enhance our method. [ABSTRACT FROM AUTHOR]
- Published
- 2011
117. An efficient data space conjugate gradient Occam's method for three-dimensional magnetotelluric inversion.
- Author
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Siripunvaraporn, Weerachai and Sarakorn, Weerachai
- Subjects
MAGNETOTELLURIC prospecting ,INVERSION (Geophysics) ,NUMERICAL analysis ,APPROXIMATION theory ,ELECTROMAGNETIC theory ,MATHEMATICS ,GEOPHYSICS ,CONJUGATE gradient methods - Abstract
SUMMARY In this paper, we start with the implementation of a data space conjugate gradient (DCG) method for 3-D magnetotelluric (MT) data. This code will be referred to as WSDCG3DMT. It is an extension of the 2-D method previously developed. Several experiments on both synthetic and real data sets show that WSDCG3DMT usually needs more computational time than the data space Occam's inversion (OCCAM) for which the corresponding code is referred to as WSINV3DMT. However, the memory requirement of WSDCG3DMT is only a fraction of that of WSINV3DMT. Based on the knowledge gained from several studies of both codes, we have created a new hybrid scheme called the DCG Occam's inversion (DCGOCC) and the corresponding code, WSDCGOCC3DMT, from combining aspects of the OCCAM and DCG methods. As with OCCAM, the DCGOCC method divides the inversion into two phases. In Phase I the misfit is brought down to a desired level. In Phase II unnecessary structures are smoothed out. Because its mathematical basis is of a similar form to that of DCG, its memory requirement is similarly low but more stable. However, DCGOCC is significantly faster than both methods. We demonstrate the computational performances with comparisons of all three methods with both synthetic and EXTECH field data sets. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
118. Variable precision rough set model over two universes and its properties.
- Author
-
Yonghong Shen and Faxing Wang
- Subjects
SET theory ,MATHEMATICAL variables ,APPROXIMATION theory ,PRECISION (Information retrieval) ,MATHEMATICS - Abstract
The extension of rough set model is an important research direction in the rough set theory. In this paper, based on the rough set model over two universes, we firstly propose the variable precision rough set model (VPRS-model) over two universes using the inclsion degree. Meantime, the concepts of the reverse lower and upper approximation operators are presented. Afterwards, the properties of the approximation operators are studied. Finally, the approximation operators with two parameters are introduced as a generalization of the VPRS-model over two universes, and the related conclusions are discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
119. APPROXIMATION OF NEGATIVE PLURISUBHARMONIC FUNCTIONS WITH GIVEN BOUNDARY VALUES.
- Author
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HED, LISA
- Subjects
PLURISUBHARMONIC functions ,APPROXIMATION theory ,BOUNDARY element methods ,MATHEMATICS ,BOUNDARY value problems - Abstract
In this paper, we study the approximation of negative plurisubharmonic functions with given boundary values. We want to approximate a plurisubharmonic function by an increasing sequence of plurisubharmonic functions defined on strictly larger domains. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
120. Error bounds for approximation in Chebyshev points.
- Author
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Shuhuang Xiang, Xiaojun Chen, and Haiyong Wang
- Subjects
CHEBYSHEV polynomials ,NUMERICAL analysis ,APPROXIMATION theory ,DIFFERENTIAL equations ,INTEGRAL equations ,MATHEMATICS - Abstract
This paper improves error bounds for Gauss, Clenshaw–Curtis and Fejér’s first quadrature by using new error estimates for polynomial interpolation in Chebyshev points. We also derive convergence rates of Chebyshev interpolation polynomials of the first and second kind for numerical evaluation of highly oscillatory integrals. Preliminary numerical results show that the improved error bounds are reasonably sharp. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
121. Identifiability and Minimality in Rational Models.
- Author
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Pestano-Gabino, Celina, González-Concepción, Concepción, and Gil-Fariña, María Candelaria
- Subjects
- *
ALGEBRAIC functions , *MULTIVARIATE analysis , *APPROXIMATION theory , *CANONICAL correlation (Statistics) , *ALGORITHMS , *AUTOREGRESSION (Statistics) , *VECTOR algebra , *MATRICES (Mathematics) , *MATHEMATICS - Abstract
This paper uses key algebraic relationships between matrix Padé approximation and certain multivariate time series models. These relationships help us to obtain relevant results for solving the problems of identifiability and exchangeability in several models. We develop a new generalization of the corner method and apply it to the multivariate case. One advantage of the procedure is the presentation of the results in easily interpretable tables. We define new canonical representations. The paper also contains additional theoretical results improving on formulations of the corresponding algorithm that will assist us. The technique is illustrated in Vectorial Autoregressive Moving Average models by using a theoretical example. [ABSTRACT FROM AUTHOR]
- Published
- 2010
122. Korovkin type approximation theorem for functions of two variables in statistical sense.
- Author
-
Dirik, Fadime and Demirci, Kamil
- Subjects
APPROXIMATION theory ,FUNCTIONAL analysis ,BERNSTEIN polynomials ,LINEAR operators ,MATHEMATICAL analysis ,MATHEMATICS - Abstract
In this paper, using the concept of A-statistical convergence for double sequences, we investigate a Korovkin-type approximation theorem for sequences of positive linear operator on the space of all continuous real valued functions defined on any compact subset of the real two-dimensional space. Then we display an application which shows that our new result is stronger than its classical version. We also obtain a Voronovskaya-type theorem and some differential properties for sequences of positive linear operators constructed by means of the Bernstein polynomials of two variables. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
123. O(√log n) APPROXIMATION TO SPARSEST CUT IN ō(n2) TIME.
- Author
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Arora, Sanjeev, Hazan, Elad, and Kale, Satyen
- Subjects
APPROXIMATION theory ,MARKOV processes ,ALGORITHMS ,FOUNDATIONS of arithmetic ,MATHEMATICS - Abstract
This paper shows how to compute O(√log n)-approximations to the Sparsest Cut and Balanced Separator problems in Õ(n²) time, thus improving upon the recent algorithm of Arora, Rao, and Vazirani [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 222-231]. Their algorithm uses semidefinite programming and requires Õ (n
9.5 ) time. Our algorithm relies on efficiently finding expander flows in the graph and does not solve semidefinite programs. The existence of expander flows was also established by Arora, Rao, and Vazirani [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 222-231]. [ABSTRACT FROM AUTHOR]- Published
- 2010
- Full Text
- View/download PDF
124. Tractability of Multivariate Approximation over a Weighted Unanchored Sobolev Space.
- Author
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Werschulz, Arthur G. and Woźniakowski, H.
- Subjects
SOBOLEV spaces ,TENSOR products ,APPROXIMATION theory ,MATHEMATICS ,LINEAR algebra - Abstract
We study d-variate L
2 -approximation for a weighted unanchored Sobolev space having smoothness m≥1. This space is equipped with an unusual norm which is, however, equivalent to the norm of the d-fold tensor product of the standard Sobolev space. One might hope that the problem should become easier as its smoothness increases. This is true for our problem if we are only concerned with asymptotic analysis: the nth minimal error is of order n−( m− δ) for any δ>0. However, it is unclear how long we need to wait before this asymptotic behavior kicks in. How does this waiting period depend on d and m? It is easy to prove that no matter how the weights are chosen, the waiting period is at least md , even if the error demand ε is arbitrarily close to 1. Hence, for m≥2, this waiting period is exponential in d, so that the problem suffers from the curse of dimensionality and is intractable. In other words, the fact that the asymptotic behavior improves with m is irrelevant when d is large. So we will be unable to vanquish the curse of dimensionality unless m=1, i.e., unless the smoothness is minimal. In this paper, we prove the more difficult fact that our problem can be tractable if m=1. That is, we can find an ε-approximation using polynomially-many (in d and ε−1 ) information operations, even if only function values are permitted. When m=1, it is even possible for the problem to be strongly tractable, i.e., we can find an ε-approximation using polynomially-many (in ε−1 ) information operations, independently of d. These positive results hold when the weights of the Sobolev space decay sufficiently quickly or are bounded finite-order weights, i.e., the d-variate functions we wish to approximate can be decomposed as sums of functions depending on at most ω variables, where ω is independent of d. [ABSTRACT FROM AUTHOR]- Published
- 2009
- Full Text
- View/download PDF
125. CLOSEDNESS OF THE SOLUTION MAP FOR PARAMETRIC VECTOR EQUILIBRIUM PROBLEMS.
- Author
-
Salamon, Júlia
- Subjects
VECTOR analysis ,MATHEMATICS ,QUATERNIONS ,CALCULUS of tensors ,EQUILIBRIUM ,STABILITY (Mechanics) ,DIFFERENTIAL geometry ,INTERPOLATION ,APPROXIMATION theory - Abstract
The objective of this paper is to study the parametric vector equilibrium problems governed by vector topologically pseudomonotone maps. The main result gives sufficient conditions for closedness of the solution map defined on the set of parameters. [ABSTRACT FROM AUTHOR]
- Published
- 2009
126. Spline functions in chemistry: approximation of surfaces over triangle domains.
- Author
-
Láng–Lázi, M., Heszberger, J., Molnár-Jobbágy, M., and Viczián, G.
- Subjects
SPLINE theory ,APPROXIMATION theory ,INTERPOLATION ,FUNCTIONAL analysis ,MATHEMATICS ,MATHEMATICAL programming - Abstract
Chemists often come across triangle domains - usually with the basic simplex in 2. A smooth surface is needed for approximating the chemical properties between the measured data for solving some model (differential) equations numerically. Our research group has been working on approximating ternary chemical surfaces of two special fields by smooth functions (vapour - Liquid equilibrium data and explosion-limit surfaces of ternary gas systems). A mathematical solution was given in both fields by special spline surfaces, and for visualization, our own software (TRIGON) was used. In this paper the summarized chemical background of each problem is provided, the mathematical solutions, the newest theoretical developments and their results are discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
127. On Nonobtuse Simplicial Partitions.
- Author
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Brandts, Jan, Korotov, Sergey, Křížek, Michal, and Šolc, Jakub
- Subjects
PARTITIONS (Mathematics) ,POLYNOMIALS ,FINITE element method ,MATHEMATICS ,NUMERICAL analysis ,APPROXIMATION theory - Abstract
This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute sad nonobtuse simplices in other areas of mathematics. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
128. On the theorem of M Golomb.
- Author
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Ismailov, Vugar E.
- Subjects
APPROXIMATION theory ,DUALITY theory (Mathematics) ,BOLTS & nuts ,ORTHOGONAL polynomials ,MATHEMATICS - Abstract
Let X
1 , … , Xn be compact spaces and X = X1 x…xXn . Consider the approximation of a function f ϵ C(X) by sums g1 (x1 )+…+gn (xn ), where gi ϵ C(Xi ), i = 1, … , n. In [8], Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of X, called 'projection cycles'. However, his proof had a gap, which was pointed out by Marshall and O'Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb's formula holds in a stronger form. [ABSTRACT FROM AUTHOR]- Published
- 2009
- Full Text
- View/download PDF
129. SOLUTION OF THE HURWITZ PROBLEM FOR LAURENT POLYNOMIALS.
- Author
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PAKOVICH, F.
- Subjects
POLYNOMIALS ,ALGEBRA ,APPROXIMATION theory ,BERNOULLI polynomials ,RANDOM polynomials ,MATHEMATICS ,MATHEMATICAL analysis - Abstract
We investigate the following existence problem for rational functions: for a given collection Π of partitions of a number n to define whether there exists a rational function f of degree n for which Π is the branch datum. An important particular case when the answer is known is the one when the collection Π contains a partition consisting of a single element (in this case, the corresponding rational function is equivalent to a polynomial). In this paper, we provide a solution in the case when Π contains a partition consisting of two elements. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
130. BOUNDS FOR THE RATE OF STRONG APPROXIMATION IN THE MULTIDIMENSIONAL INVARIANCE PRINCIPLE.
- Author
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Götze, F. and Zaitsev, A. Yu.
- Subjects
GAUSSIAN processes ,DISTRIBUTION (Probability theory) ,APPROXIMATION theory ,RANDOM variables ,SYMMETRY (Physics) ,MATHEMATICS - Abstract
The goal of this paper is to derive consequences of the result of Zaitsev [Theory Probab. Appl., 45 (2001), pp. 624-642; 46 (2002), pp. 490-514; 676-698]. We establish bounds for the rate of strong Gaussian approximation of sums of independent R
d -valued random vectors ξj having finite moments E∥ξj ∥γ , γ ≧ 2. A multidimensional version of the results of Sakhanenko [Trudy Inst. Mat., 5 (1985), pp. 27-44 (in Russian)] is obtained. [ABSTRACT FROM AUTHOR]- Published
- 2009
- Full Text
- View/download PDF
131. Long Time Estimates in the Mean Field Limit.
- Author
-
Caglioti, E., Rousset, F., and Otto, F.
- Subjects
EQUATIONS ,APPROXIMATION theory ,EULER characteristic ,ESTIMATION theory ,MATHEMATICS - Abstract
In this paper, we prove a stability result for measure perturbations of some class of stationary distributions of a Vlasov equation. We use this result to prove that the N particles approximation of these stationary distributions is uniformly valid on a time scale of order N
1/8 , which is much longer than the usual log N scale. We also prove similar results for the approximation of the two-dimensional Euler equation by the vortex blob method. [ABSTRACT FROM AUTHOR]- Published
- 2008
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132. Linear Interpolation of the Functions with Three Variable Values with Simple Nodes.
- Author
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Dinu, Tănase
- Subjects
INTERPOLATION ,POLYNOMIALS ,NUMERICAL analysis ,MATHEMATICAL functions ,MATHEMATICS ,LINEAR algebra ,NUMERICAL integration ,APPROXIMATION theory - Abstract
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- 2008
133. Asymptotics of Studentized U-type processes for changepoint problems.
- Author
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Csörgõ, M., Szyszkowicz, B., and Wang, Q.
- Subjects
APPROXIMATION theory ,U-statistics ,DISTRIBUTION (Probability theory) ,KERNEL functions ,COMPLEX variables ,MATHEMATICAL models ,MATHEMATICS - Abstract
This paper investigates weighted approximations for Studentized U-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition E| h( X
1 , X2 )|2 < ∞ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence. [ABSTRACT FROM AUTHOR]- Published
- 2008
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134. On N-termed approximations in H s -norms with respect to the Haar system.
- Author
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Oswald, P.
- Subjects
HAAR system (Mathematics) ,APPROXIMATION theory ,RECTANGLES ,INTEGRAL equations ,SMOOTHNESS of functions ,MATHEMATICS - Abstract
In [9], we proved numerically that spaces generated by linear combinations of some two-dimensional Haar functions exhibit unexpectedly nice orders of approximation for solutions of the single-layer potential equation in a rectangle. This phenomenon is closely related, on the one hand, to the properties of the approximation method of hyperbolic crosses and on the other to the existence of a strong singularity for solutions of such boundary integral equations. In the present paper, we establish several results on the approximation for the hyperbolic crosses and on the best N-term approximations by linear combinations of Haar functions in the H
s -norms, −1 < s < 1/2; this provides a theoretical base for our numerical research. To the author's best knowledge, the negative smoothness case s < 0 was not studied earlier. [ABSTRACT FROM AUTHOR]- Published
- 2008
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135. Exponential Stability for Time-Delay Systems with Interval Time-Varying Delays and Nonlinear Perturbations.
- Author
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Kwon, O. M. and Park, J. H.
- Subjects
LYAPUNOV exponents ,DIFFERENTIAL equations ,EXPONENTIAL functions ,PERTURBATION theory ,APPROXIMATION theory ,MATRIX inequalities ,MATHEMATICAL inequalities ,TIME delay systems ,MATHEMATICS - Abstract
In this paper, the problem of an exponential stability for time-delay systems with interval time-varying delays and nonlinear perturbations is investigated. Based on the Lyapunov method, a new delay-dependent criterion for exponential stability is established in terms of LMI (linear matrix inequalities). Numerical examples are carried out to support the effectiveness of our results. [ABSTRACT FROM AUTHOR]
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- 2008
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136. ERROR ESTIMATES FOR POLYNOMIAL KRYLOV APPROXIMATIONS TO MATRIX FUNCTIONS.
- Author
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Diele, Fasma, Moret, Igor, and Ragni, Stefania
- Subjects
MATHEMATICS ,MATHEMATICAL analysis ,APPROXIMATION theory ,POLYNOMIALS ,MATRICES (Mathematics) ,ERROR analysis in mathematics - Abstract
In this paper we are interested in the polynomial Krylov approximations for the computation of φ(A)v, where A is a square matrix, v represents a given vector, and φ is a suitable function which can be employed in modern integrators for differential problems. Our aim consists of proposing and analyzing innovative a posteriori error estimates which allow a good control of the approximation procedure. The effectiveness of the results we provide is tested on some numerical examples of interest. [ABSTRACT FROM AUTHOR]
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- 2008
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137. HOW TO MAKE SIMPLER GMRES AND GCR MORE STABLE.
- Author
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Jiránek, Pavel, Rozložník, Miroslav, and Gutknech, Martin H.
- Subjects
MATHEMATICS ,MATHEMATICAL analysis ,NUMERICAL analysis ,LINEAR systems ,COORDINATES ,APPROXIMATION theory - Abstract
In this paper we analyze the numerical behavior of several minimum residual methods which are mathematically equivalent to the GMRES method. Two main approaches are compared: one that computes the approximate solution in terms of a Krylov space basis from an upper triangular linear system for the coordinates, and one where the approximate solutions are updated with a simple recursion formula. We show that a different choice of the basis can significantly influence the numerical behavior of the resulting implementation. While Simpler GMRES and ORTHODIR are less stable due to the ill-conditioning of the basis used, the residual basis is well-conditioned as long as we have a reasonable residual norm decrease. These results lead to a new implementation, which is conditionally backward stable, and they explain the experimentally observed fact that the GCR method delivers very accurate approximate solutions when it converges fast enough without stagnation. [ABSTRACT FROM AUTHOR]
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- 2008
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138. COMPACTLY SUPPORTED SYMMETRIC C∞ WAVELETS WITH SPECTRAL APPROXIMATION ORDER.
- Author
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BIN HAN and ZUOWEI SHEN
- Subjects
WAVELETS (Mathematics) ,APPROXIMATION theory ,MATHEMATICAL symmetry ,MATHEMATICS ,HARMONIC analysis (Mathematics) - Abstract
In this paper, we obtain symmetric C
∞ real-valued tight wavelet frames in L2 (ℝ) with compact support and the spectral frame approximation order. Furthermore, we present a family of symmetric compactly supported C∞ orthonormal complex wavelets in L2 (ℝ). A complete analysis of nonstationary tight wavelet frames and orthonormal wavelet bases in L2 (ℝ) is given. [ABSTRACT FROM AUTHOR]- Published
- 2008
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139. On Polyhedral Projection and Parametric Programming.
- Author
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Jones, C. N., Kerrigan, E. C., and Maciejowski, J. M.
- Subjects
POLYHEDRAL functions ,LINEAR programming ,ALGEBRAIC functions ,POLYNOMIALS ,ALGORITHMS ,DYNAMIC programming ,FUNCTIONAL analysis ,MATHEMATICS ,APPROXIMATION theory - Abstract
This paper brings together two fundamental topics: polyhedral projection and parametric linear programming. First, it is shown that, given a parametric linear program (PLP), a polyhedron exists whose projection provides the solution to the PLP. Second, the converse is tackled and it is shown how to formulate a PLP whose solution is the projection of an appropriately defined polyhedron described as the intersection of a finite number of halfspaces. The input to one operation can be converted to an input of the other operation and the resulting output can be converted back to the desired form in polynomial time-this implies that algorithms for computing projections or methods for solving parametric linear programs can be applied to either problem class. [ABSTRACT FROM AUTHOR]
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- 2008
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140. Robust Reliable Control for Uncertain Time-Delay Systems with IQC Performance.
- Author
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Lien, C. H. and Yu, K. W.
- Subjects
INTEGRALS ,MATRICES (Mathematics) ,MATRIX inequalities ,PERTURBATION theory ,APPROXIMATION theory ,FUNCTIONAL analysis ,MATHEMATICS - Abstract
A robust reliable control with integral quadratic constraint (IQC) performance for a class of uncertain systems with state and input delays is considered in this paper. Two classes of failure situations for sensor or actuator are studied. In the first class, a delay-dependent criterion for time-delay systems without perturbations is proposed to design the reliable control with IQC performance. Next, a criterion for uncertain time-delay systems with parameter uncertainties is obtained via simple derivations. The linear matrix inequality (LMI) approach is used to design a robust reliable state feedback control with IQC performance. In the second class, a reliable control with IQC performance is also provided from he previous method. A numerical example is given to illustrate the effectiveness of the procedure. [ABSTRACT FROM AUTHOR]
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- 2008
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141. A generalized Stieltjes criterion for the complete indeterminacy of interpolation problems.
- Author
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Dyukarev, Yu. M.
- Subjects
MOMENT problems (Mathematics) ,STIELTJES integrals ,STIELTJES transform ,INTERPOLATION ,APPROXIMATION theory ,NUMERICAL analysis ,MATHEMATICS - Abstract
The main result of this paper is a generalized Stieltjes criterion for the complete indeterminacy of interpolation problems in the Stieltjes class. This criterion is a generalization to limit interpolation problems of the classical Stieltjes criterion for the indeterminacy of moment problems. It is stated in terms of the Stieltjes parameters M
j and Nj . We obtain explicit formulas for the Stieltjes parameters. General constructions are illustrated by examples of the Stieltjes moment problem and the Nevanlinna-Pick problem in the Stieltjes class. [ABSTRACT FROM AUTHOR]- Published
- 2008
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142. Rough Rule Extracting From Various Conditions: Incremental and Approximate Approaches for Inconsistent Data.
- Author
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Liu, Yong, Xu, Congfu, Zhang, Qiong, and Pan, Yunhe
- Subjects
ROUGH sets ,APPROXIMATION theory ,ALGORITHMS ,SET theory ,MATHEMATICAL analysis ,MATHEMATICS - Abstract
Rough rule extraction refers to the rule induction method by using rough set theory. Although rough set theory is a powerful mathematical tool in dealing with vagueness and uncertainty in data sets, it is lack of effective rule extracting approach under complex conditions. This paper proposes several algorithms to perform rough rule extraction from data sets with different properties. Firstly, in order to obtain uncertainty rules from inconsistent data, we introduce the concept of confidence factor into the rule extracting process. Then, an improved incremental rule extracting algorithm is proposed based on the analysis of the incremental data categories. Finally, above algorithms are further extended to perform approximate rule extraction from huge data sets. Preliminary experiment results are encouraging. [ABSTRACT FROM AUTHOR]
- Published
- 2008
143. LOWER-RANK TENSOR APPROXIMATION AND MULTIWAY FILTERING.
- Author
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Muti, Damien, Bourennane, Salah, and Marot, Julien
- Subjects
APPROXIMATION theory ,TENSOR products ,TENSOR algebra ,MATRICES (Mathematics) ,MATHEMATICAL analysis ,MATHEMATICS - Abstract
This paper presents some recent filtering methods based on the lower-rank tensor approximation approach for denoising tensor signals. In this approach, multicomponent data are represented by tensors, that is, multiway arrays, and the presented tensor filtering methods rely on multilinear algebra. First, the classical channel-by-channel SVD-based filtering method is overviewed. Then, an extension of the classical matrix filtering method is presented. It is based on the lower rank- (K
1 , … , KN ) truncation of the higher order SVD which performs a multimode principal component analysis (PCA) and is implicitly developed for an additive white Gaussian noise. Two tensor filtering methods recently developed by the authors are also overviewed. The first method consists of an improvement of the multimode PCA-based tensor filtering in the case of an additive correlated Gaussian noise. This improvement is specially done thanks to the fourth order cumulant slice matrix. The second method consists of an extension of Wiener filtering for data tensors. The performances and comparative results between all these tensor filtering methods are presented for the cases of noise reduction in color images, multispectral images, and multicomponent seismic data. [ABSTRACT FROM AUTHOR]- Published
- 2008
- Full Text
- View/download PDF
144. Variations for the Q- and H-factors in the polar decomposition.
- Author
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Chen, Xiao-Shan and Li, Wen
- Subjects
PERTURBATION theory ,HERMITIAN structures ,MATRICES (Mathematics) ,APPROXIMATION theory ,DECOMPOSITION method ,MATHEMATICS - Abstract
In this paper, the variations of both subunitary polar factors and Hermitian positive semidefinite polar factors in the polar decomposition are studied. New perturbation bounds of both polar factors are given without the restriction that A and its perturbed matrix à have the same rank. These improve recent results. [ABSTRACT FROM AUTHOR]
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- 2008
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145. Rough approximations in a general approximation space and their fundamental properties.
- Author
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Davvaz, B. and Mahdavipour, M.
- Subjects
SET theory ,APPROXIMATION theory ,FUNCTIONAL analysis ,MATHEMATICAL logic ,MATHEMATICS - Abstract
This paper presents a framework for the study of generalizing the standard notion of rough set approximation space. We propose new definitions of lower and upper approximations, which are basic concepts of the rough set theory. These definitions follow naturally from a particular property on the universe and are called general lower and upper approximations. Properties of general approximations are investigated, and their connections are examined. [ABSTRACT FROM AUTHOR]
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- 2008
- Full Text
- View/download PDF
146. Levitin–Polyak well-posedness in generalized vector variational inequality problem with functional constraints.
- Author
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Zui Xu, Zhu, D. L., and Huang, X. X.
- Subjects
VECTOR analysis ,MATHEMATICS ,APPROXIMATION theory ,FUNCTIONAL analysis ,MATHEMATICAL mappings ,MATHEMATICAL functions - Abstract
In this paper, we study Levitin–Polyak type well-posedness for generalized vector variational inequality problems with abstract and functional constraints. Various criteria and characterizations for these types of well-posednesses are given. [ABSTRACT FROM AUTHOR]
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- 2008
- Full Text
- View/download PDF
147. Bounded solutions of families of systems of differential equations and their approximations.
- Author
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Dzhumabaev, D. S.
- Subjects
DIFFERENTIAL equations ,EQUATIONS ,MATHEMATICS ,MATRICES (Mathematics) ,APPROXIMATION theory ,FUNCTIONAL analysis - Abstract
The paper considers the problem of finding a bounded solution of a one-parametric family of systems of ordinary differential equations. Using the parametrization method, the author proves necessary and sufficient conditions for the existence of a unique solution of the problem considered that is bounded on the whole axis in terms of a two-sided, infinite block-band matrix composed with respect to integrals over intervals of length h > 0 of the matrix of the system of differential equations. Also, the author constructs a family of two-point boundary-value problems on a finite interval that approximates the problem of finding the bounded solution and finds an interconnection between the correct solvability of the initial and approximating problems. [ABSTRACT FROM AUTHOR]
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- 2008
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148. MULTIVALUED STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS VIA BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS.
- Author
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BOUFOUSSI, B. and MRHARDY, N.
- Subjects
STOCHASTIC difference equations ,APPROXIMATION theory ,PARTIAL differential equations ,CONVEX functions ,THERMAL conductivity ,MATHEMATICS - Abstract
In this paper, we establish by means of Yosida approximation, the existence and uniqueness of the solution of a backward doubly stochastic differential equation whose coefficient contains the subdifferential of a convex function. We will use this result to prove the existence of stochastic viscosity solution for some multivalued parabolic stochastic partial differential equation. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
149. A modified trust region method with Beale's PCG technique for optimization.
- Author
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Wenyu Sun, Liusheng Hou, and Chuangying Dang
- Subjects
MATHEMATICAL optimization ,MATHEMATICAL analysis ,MATHEMATICS ,CONJUGATE gradient methods ,APPROXIMATION theory ,NUMERICAL solutions to equations - Abstract
It is well-known that the conjugate gradient method is widely used for solving large scale optimization problems. In this paper a modified trust-region method with Beale's Preconditioned Conjugate Gradient (BPCG) technique is developed for solving unconstrained optimization problems. The modified version adopts an adaptive rule and retains some useful information when an unsuccessful iteration occurs, and therefore improves the efficiency of the method. The behavior and the convergence properties are discussed. Some numerical experiments are reported. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
150. FAST RATES FOR ESTIMATION ERROR AND ORACLE INEQUALITIES FOR MODEL SELECTION.
- Author
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Peter L. Bartlett
- Subjects
MATHEMATICAL models ,APPROXIMATION theory ,ESTIMATION theory ,MATHEMATICAL sequences ,ERROR analysis in mathematics ,MATHEMATICS - Abstract
We consider complexity penalization methods for model selection. These methods aim to choose a model to optimally trade off estimation and approximation errors by minimizing the sum of an empirical risk term and a complexity penalty. It is well known that if we use a bound on the maximal deviation between empirical and true risks as a complexity penalty, then the risk of our choice is no more than the approximation error plus twice the complexity penalty. There are many cases, however, where complexity penalties like this give loose upper bounds on the estimation error. In particular, if we choose a function from a suitably simple convex function class with a strictly convex loss function, then the estimation error (the difference between the risk of the empirical risk minimizer and the minimal risk in the class) approaches zero at a faster rate than the maximal deviation between empirical and true risks. In this paper, we address the question of whether it is possible to design a complexity penalized model selection method for these situations. We show that, provided the sequence of models is ordered by inclusion, in these cases we can use tight upper bounds on estimation error as a complexity penalty. Surprisingly, this is the case even in situations when the difference between the empirical risk and true risk (and indeed the error of any estimate of the approximation error) decreases much more slowly than the complexity penalty. We give an oracle inequality showing that the resulting model selection method chooses a function with risk no more than the approximation error plus a constant times the complexity penalty.We gratefully acknowledge the support of the NSF under award DMS-0434383. Thanks also to three anonymous reviewers for useful comments that improved the presentation. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
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