C. De Boor and A. Ron, Computational aspects of polynomial interpolation in several variables, Math. Comp., 58, (1992), 705-727.  Home/Search   Document Not in Database   Summary   Related Articles   Check

..... The nine runs of DFO produced di#erent results because the DFO algorithm includes a stochastic decision. The only di#erence between the two implementations of SMF is the choice of approximating families: one choice interpolated known function values with variable order multivariate polynomials [12] whose degrees were increased as more function values were obtained; the other choice interpolated known function values by kriging. The latter family of approximations, which we also used for the helicopter rotor blade design problem, is discussed in Section 4. In each implementation, the initial ....

....optimization problem. Because we do not want to make a priori assumptions about the structure of f , we require a large, flexible class of functions from which surrogates can be selected. Plausible families of approximating functions include neural networks and low degree interpolating polynomials [12]. In 2.3 we gave evidence that the SMF can use di#erent families of approximation. We have opted for the family of functions defined by the kriging procedures discussed in the DACE literature. The kriging parameterization, defined by means and covariances of function values, is more intuitive ....

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C. De Boor and A. Ron, Computational aspects of polynomial interpolation in several variables, Mathematics of Computation, 58 (1992), pp. 705--727.

....set of distinct points X = fx 1 ; xN g, P l (X ) is an interpolation space which is degree reducing. That is, for any q 2 Pi d the interpolant p 2 P l (X ) defined by p (x j ) q (x j ) satisfies deg p deg q. The issue of an algorithmic construction of P l has been considered in [15,8] introducing and using the technique of Gauss elimination by segments. We will compare this technique to work in Computational Algebra in section 6. We also remark that the least interpolation approach can be extended to ideal interpolation schemes [5] which might also be understood as a very ....

.... on an infinite matrix, it always works on a finite matrix only: as shown in [105] the set A of multiindices constructed above always satisfies A ae n ff 2 N d : jffj # Theta Gamma 1 o : A different approach towards elimination by segments in the Vandermonde matrix has been taken in [15], collecting all monomials of the same total degree 30 M. Gasca and T. Sauer Polynomial interpolation into one object and mutually orthogonalizing them to obtain uniqueness. The result of this elimination process is then the least interpolation space described above. We will, however, omit the ....

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C. de Boor and A. Ron. Computational aspects of polynomial interpolation in several variables. Math. Comp., 58 (1992), 705--727.

....either, because any quadratic which is a multiple of the equation of the circle can be added to the interpolant without affecting (3.2) One therefore sees that some geometric conditions on Y must be added to the conditions (3. 2) to ensure existence and uniqueness of the quadratic interpolant (see [11] or [28] for more detail) The purpose of the next two sections will be to relate these conditions to the more general question of exploring along suitable directions during the optimization. In the case of our second example, we must have that the interpolation points do not lie on any quadratic ....

C. De Boor and A. Ron. Computational aspects of polynomial interpolation in several variables. Mathematics of Computation, 58(198):705--727, 1992.

....models of the Navier Stokes equations. The fidelity of a physical model can vary depending on something as simple as a grid spacing parameter, or as complex as a turbulence model formulation in a Navier Stokes equation solver. Examples of the functional type of model are polynomial interpolation [15, 20, 59], splines[13] neural nets and other type of curve fits (particularly response surface models [9] 43] and least squares models (linear or nonlinear) 45] 5 Hybrids that have characteristics of both types can be used. For example, one could take a simplified version of the equations describing ....

Carl deBoor and A. Ron. Computational aspects of polynomial interpolation in several variables. Mathematics of Computation, 58(198):705--727, 1992.

....to choose such a correct interpolating space in dependence on the pointset. A particular choice of such a polynomial space ## for given # has recently been proposed in [BR90] a list of its many properties has been o#ered and proved in [BR90 92] its computational aspects have been detailed in [BR91], and its generalization, from interpolation at a set of n points in IR d to interpolation at n arbitrary linearly independent linear functionals on the space # = #(IR d ) of all polynomials on IR d , has been treated in much detail in [BR92] The present short note o#ers some discussion ....

....obvious exception to this is the case when ## = # k : the collection of polynomials of total degree # k. Thus, only for this case does one obtain from I # a ready multivariate divided di#erence. Unless the ordering (# 1 , # 2 , # n ) is carefully chosen (e.g. as in the algorithm in [BR91]) there is no reason for the corresponding sequence (deg p #1 , deg p #n ) to be nondecreasing. In particular, deg p #,x may well be smaller than deg I # f . For example, if x is not in the a#ne hull of #, then deg p #,x = 1. This means that the order of the interpolation error, i.e. ....

C. de Boor and Amos Ron (199x), "Computational aspects of polynomial interpolation in several variables", Math. Comp. xx, xxx--xxx.

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C. De Boor and A. Ron, Computational aspects of polynomial interpolation in several variables, Math. Comp., 58, (1992), 705-727.

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de Boor, C., and Ron, A., Computational aspects of polynomial interpolation in several variables, Math. Comp. 58 (1992), 705--727.

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C. DE BOOR AND A. RON, Computational aspects of polynomial interpolation in several variables, Math. Comp., 58, 1992, pp. 705-727.

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C. de Boor and A. Ron. Computational aspects of polynomial interpolation in several variables. Math. Comp., 58 (1992), 705--727.

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