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Lower Bounds for Norms of Inverses of Interpolation Matrices for Radial Basis Functions
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, November 1994, Pages 287-306
Lower Bounds for Norms of Inverses of Interpolation Matrices for Radial Basis Functions
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Univ Gottingen, Inst Numer & Angew Math, Lotzestr 16 18, W 3400 Gottingen, Germany
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Interpolation of scattered data at distinct points xI,..., xn ∈ Rd by linear combinations of translates Φ(||x - xj||2) of a radial basis function Φ : R≥ 0 → R requires the solution of a linear system with the n by n distance matrix A ? (Φ(||xi - xj||2). Recent results of Ball, Narcowich and Ward, using Laplace transform methods, provide upper bounds for ||A-1||2, while Ball, Sivakumar, and Ward constructed examples with regularly spaced points to get special lower bounds. This paper proves general lower bounds by application of results of classical approximation theory. The bounds increase with the smoothness of Φ. In most cases, they leave no more than a factor of n-2 to be gained by optimization of data placement, starting from regularly distributed data. This follows from comparison with results of Ball, Baxter, Sivakumar, and Ward for points on scaled integer lattices and supports the hypothesis that regularly spaced data are near-optimal, as far as the condition of the matrix A is concerned.
Remember meWavelet analysis of condensed phase molecular dynamics
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, 5 November 1993, Pages 362-366
Wavelet analysis of condensed phase molecular dynamics
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Department of Chemistry, University of California, Irvine, CA , USAResearch Unit in Advance Computing, University of California, Irvine, CA , USA
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A novel application of wavelet analysis to the characterization of many-body dynamics is described. In this approach, the velocities of particles in a molecular dynamics simulation are decomposed into contributions from a hierarchy of time scales. A scale spectrum is defined, in analogy with conventional spectral densities, which gives the effective average “temperature” of the resulting scales of motion. The method is applied to simulations of solid and liquid argon, and the signatures of liquid—solid phase transition in the results of the wavelet analysis are investigated.
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