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Table Of Integrals Series And Products

The organization of behavior: A neuropsychological theory

Poisson surface reconstruction creates watertight surfaces from oriented point sets. In this work we extend the technique to explicitly incorporate the points as interpolation constraints. The extension can be interpreted as a generalization of the underlying mathematical framework to a screened Poisson equation. In contrast to other image and geometry processing techniques, the screening term is defined over a sparse set of points rather than over the full domain. We show that these sparse constraints can nonetheless be integrated efficiently. Because the modified linear system retains the same finite-element discretization, the sparsity structure is unchanged, and the system can still be solved using a multigrid approach. Moreover we present several algorithmic improvements that together reduce the time complexity of the solver to linear in the number of points, thereby enabling faster, higher-quality surface reconstructions.

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Screened poisson surface reconstruction

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

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Orthogonal Polynomials of Several Variables

We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241--282; A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461--1488; A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720--742].
The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications.
At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them central-upwind schemes.
The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton--Jacobi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navier--Stokes equations. The incompressibility condition in the latter equations allows us to treat them both in their conservative and transport form. We apply to these problems the central-upwind schemes, developed separately for each of them, and compute the corresponding numerical solutions.

Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations

The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic PDEs Abstractly, it can be viewed as determining a “good” n-dime

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Convergence Rates for Greedy Algorithms in Reduced Basis Methods

A family of Godunov-type central-upwind schemes for the Saint-Venant system of shallow water equations has been first introduced in (A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36, 397-425, 2002). Depending on the reconstruction step, the second-order versions of the schemes there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simultaneously. Here, we introduce an improved second-order central-upwind scheme which, unlike its forerun- ners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of one- and two-dimensional examples.

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A second-order well-balanced positivity preserving central-upwind scheme for the saint-venant system ∗

This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural networks approximate functions with classical smoothness to the same accuracy as classical linear methods of approximation, e.g. approximation by polynomials or by piecewise polynomials on prescribed partitions. However, approximation by neural networks depending on n parameters is a form of nonlinear approximation and as such should be compared with other nonlinear methods such as variable knot splines or n-term approximation from dictionaries. The performance of neural networks in targeted applications such as machine learning indicate that they actually possess even greater approximation power than these traditional methods of nonlinear approximation. The main results of this article prove that this is indeed the case. This is done by exhibiting large classes of functions which can be efficiently captured by neural networks where classical nonlinear methods fall short of the task. The present article purposefully limits itself to studying the approximation of univariate functions by ReLU networks. Many generalizations to functions of several variables and other activation functions can be envisioned. However, even in this simplest of settings considered here, a theory that completely quantifies the approximation power of neural networks is still lacking.

Nonlinear Approximation and (Deep) ReLU Networks.

Given a Banach space X and one of its compact sets $\mathcal{F}$
 , we consider the problem of finding a good n-dimensional space X
 n
 ⊂X which can be used to approximate the elements of $\mathcal{F}$
 . The best possible error we can achieve for such an approximation is given by the Kolmogorov width $d_{n}(\mathcal{F})_{X}$
 . However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Model. Math. Anal. Numer. 46:595–603, 2012) in the case $X=\mathcal{H}$
 is a Hilbert space. The results of Buffa et al. (Model. Math. Anal. Numer. 46:595–603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457–1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces.

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Guergana Petrova

Papers

Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations

Convergence Rates for Greedy Algorithms in Reduced Basis Methods

A second-order well-balanced positivity preserving central-upwind scheme for the saint-venant system ∗

Nonlinear Approximation and (Deep) ReLU Networks.

Greedy Algorithms for Reduced Bases in Banach Spaces