Multivariate approximation: theory and applications. An overview

Boor, Carl de; Gout, Christian; Kunoth, Angela; Rabut, Christophe

doi:10.1007/s11075-008-9190-y

Cited by 5 publications

(5 citation statements)

References 20 publications

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“…Interpolating scattered points in many dimensions is a difficult problem and an active subject of recent mathematical research, see e.g. the books by Wendland [328], Lai and Schumaker [329] and Fasshauer [330], the review by Fasshauer [331] and the overview by de Boor et al [332].…”

Section: Multi-dimensional Surface Buildingmentioning

confidence: 99%

Theoretical methods for small-molecule ro-vibrational spectroscopy

Lodi¹,

Tennyson²

2010

J. Phys. B: At. Mol. Opt. Phys.

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The solution of the first principle equations of quantum mechanics provides an increasingly accurate and predictive approach for solving problems involving atoms and small molecules. A general introduction to the methods used for the ab initio calculation of rotational–vibrational spectra of small molecules is presented, with a strong focus on triatomic systems. The use of multi-reference electronic structure methods to compute molecular potential-energy and dipole-moment surfaces is discussed. Issues related to the construction of such surfaces and the inclusion of corrections due to relativistic and non-Born–Oppenheimer effects are reviewed. The derivation of exact, internal-coordinate nuclear-motion-effective Hamiltonians and their solution using a discrete-variable representation are discussed. Sample results for the water molecules are used throughout the tutorial to illustrate the theoretical and numerical issues in such calculations.

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Section: Multi-dimensional Surface Buildingmentioning

confidence: 99%

Theoretical methods for small-molecule ro-vibrational spectroscopy

Lodi¹,

Tennyson²

2010

J. Phys. B: At. Mol. Opt. Phys.

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“…These modified techniques are used to solve a class of ordinary and partial differential equations where the source function is exponential, trigonometric, or hyperbolic functions rather than the polynomial one. The approximation of functions by polynomials is extremely important as different scientific experiments rely on them, such as the study of statistics in population dynamics [28], temperatures, and also in the approximation theory [7]. Moreover, polynomials are the best mathematical techniques to approximate as they can be characterized, figured, separated, and incorporated effortlessly.…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatment of Volterra-Fredholm Integro- Differential Equations of Fractional Order and its Convergence Analysis

PANDA,

MOHAPATRA

2024

KgJMat

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This work deals with semi-analytical and numerical methods to solve a class of fractional order Volterra-Fredholm integro-diﬀerential equations. First, a semi-analytical method is proposed using the Chebyshev and Bernstein polynomials in the Adomian decomposition method. The uniqueness of the solution and convergence of the method are proved. Further, we solve the model using a numerical scheme comparing the L1 scheme for the fractional order derivative in combination with appropriate quadrature rules for the integral parts. Numerical experiments are done by the proposed methods to show their eﬃciency through a few tabular data and plots. Some comparisons with the existing results show that the proposed methods are highly productive and reliable.

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“…Ritter (2000) contains a very detailed survey of various random process and field approximation problems. For an extensive studies of approximation problems in deterministic setting, we refer to, e.g., Nikolskii (1975);de Boor et al (2008); Kuo et al (2009).…”

Section: Introductionmentioning

confidence: 99%

Piecewise-Multilinear Interpolation of a Random Field

Abramowicz

Seleznjev

2013

Advances in Applied Probability

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We consider a multivariate piecewise linear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured by the integrated mean square error. Multivariate piecewise linear interpolator is defined by N field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field in mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large N . Moreover, for certain classes of continuous and continuously differentiable fields we provide the upper bound for the approximation accuracy in the uniform mean square norm.

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Multivariate approximation: theory and applications. An overview

Cited by 5 publications

References 20 publications

Theoretical methods for small-molecule ro-vibrational spectroscopy

Theoretical methods for small-molecule ro-vibrational spectroscopy

Numerical Treatment of Volterra-Fredholm Integro- Differential Equations of Fractional Order and its Convergence Analysis

Piecewise-Multilinear Interpolation of a Random Field

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