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Amir Hashemi (Ed.)We present Hermite polynomial interpolation algorithms that for a sparse univariate polynomial f with coefficients from a field compute the polynomial from fewer points than the classical algorithms. If the interpolating polynomial f has t terms, our algorithms, require argument/value triples (w^i, f(w^i), f'(w^i)) for i=0,...,t + ceiling( (t+1)/2 ) - 1, where w is randomly sampled and the probability of a correct output is determined from a degree bound for f. With f' we denote the derivative of f. Our algorithms generalize to multivariate polynomials, higher derivatives and sparsity with respect to Chebyshev polynomial bases. We have algorithms that can correct errors in the points by oversampling at a limited number of good values. If an upper bound B >= t for the number of terms is given, our algorithms use a randomly selected w and, with high probability, ceiling( t/2 ) + B triples, but then never return an incorrect output. The algorithms are based on Prony's sparse interpolation algorithm. While Prony's algorithm and its variants use fewer values, namely, 2t+1 and t+B values f(w^i), respectively, they need more arguments w^i. The situation mirrors that in algebraic error correcting codes, where the Reed-Solomon code requires fewer values than the multiplicity code, which is based on Hermite interpolation, but the Reed-Solomon code requires more distinct arguments. Our sparse Hermite interpolation algorithms can interpolate polynomials over finite fields and over the complex numbers, and from floating point data. Our Prony-based approach does not encounter the Birkhoff phenomenon of Hermite interpolation, when a gap in the derivative values causes multiple interpolants. We can interpolate from t+1 values of f and 2t-1 values of f'.more » « less
null (Ed.)A critical step in Raman spectroscopy is baseline correction. This procedure eliminates the background signals generated by residual Rayleigh scattering or fluorescence. Baseline correction procedures relying on asymmetric loss functions have been employed recently. They operate with a reduced penalty on positive spectral deviations that essentially push down the baseline estimates from invading Raman peak areas. However, their coupling with polynomial fitting may not be suitable over the whole spectral domain and can yield inconsistent baselines. Their requirement of the specification of a threshold and the non-convexity of the corresponding objective function further complicates the computation. Learning from their pros and cons, we have developed a novel baseline correction procedure called the iterative smoothing-splines with root error adjustment (ISREA) that has three distinct advantages. First, ISREA uses smoothing splines to estimate the baseline that are more flexible than polynomials and capable of capturing complicated trends over the whole spectral domain. Second, ISREA mimics the asymmetric square root loss and removes the need of a threshold. Finally, ISREA avoids the direct optimization of a non-convex loss function by iteratively updating prediction errors and refitting baselines. Through our extensive numerical experiments on a wide variety of spectra including simulated spectra, mineral spectra, and dialysate spectra, we show that ISREA is simple, fast, and can yield consistent and accurate baselines that preserve all the meaningful Raman peaks.more » « less
We develop a mixed formulation for incompressible hyperelastodynamics based on a continuum modeling framework recently developed in the work of Liu and Marsden and smooth generalizations of the Taylor‐Hood element based on nonuniform rational B‐splines (NURBSs). This continuum formulation draws a link between computational fluid dynamics and computational solid dynamics. This link inspires an energy stability estimate for the spatial discretization, which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity. The inf‐sup condition is utilized to provide a bound for the pressure field. The generalized‐
αmethod is applied for temporal discretization, and a nested block preconditioner is invoked for the solution procedure. The inf‐sup stability for different pairs of NURBS elements is elucidated through numerical assessment. The convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method. The numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials. Finally, a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties.
This paper presents an isogeometric collocation method for a computationally expedient random field discretization by means of the Karhunen‐Loève expansion. The method involves a collocation projection onto a finite‐dimensional subspace of continuous functions over a bounded domain, basis splines (B‐splines) and nonuniform rational B‐splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Similar to the existing Galerkin isogeometric method, the isogeometric collocation method preserves an exact geometrical representation of many commonly used physical or computational domains and exploits the regularity of isogeometric basis functions delivering globally smooth eigensolutions. However, in the collocation method, the construction of the system matrices for a
d‐dimensional eigenvalue problem asks for at most d‐dimensional domain integrations, as compared with 2 d‐dimensional integrations required in the Galerkin method. Therefore, the introduction of the collocation method for random field discretization offers a huge computational advantage over the existing Galerkin method. Three numerical examples, including a three‐dimensional random field discretization problem, illustrate the accuracy and convergence properties of the collocation method for obtaining eigensolutions.
null (Ed.)Multiplicity code decoders are based on Hermite polynomial interpolation with error correction. In order to have a unique Hermite interpolant one assumes that the field of scalars has characteristic 0 or >= k+1, where k is the maximum order of the derivatives in the list of values of the polynomial and its derivatives which are interpolated. For scalar fields of characteristic k+1, the minimum number of values for interpolating a polynomial of degree <= D is D+1+2E(k+1) when <= E of the values are erroneous. Here we give an error-correcting Hermite interpolation algorithm that can tolerate more errors, assuming that the characteristic of the scalar field is either 0 or >= D+1. Our algorithm requires (k+1)D + 1 - (k+1)k/2 + 2E values. As an example, we consider k = 2. If the error ratio (number of errors)/(number of evaluations) <= 0.16, our new algorithm requires ceiling( (4+7/17) D - (1+8 /17) ) values, while multiplicity decoding requires 25D+25 values. If the error ratio is <= 0.2, our algorithm requires 5D-2 evaluations over characteristic 0 or >= D+1, while multiplicity decoding for an error ratio 0.2 over fields of characteristic 3 is not possible for D >= 3. Our algorithm is based on Reed-Solomon interpolation without multiplicities, which becomes possible for Hermite interpolation because of the high redundancy necessary for error-correction.more » « less