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1. Hermite Rational Function Interpolation with Error Correction

   https://doi.org/10.1007/978-3-030-60026-6_19  

   Kaltofen, Erich L.; Pernet, Clement; Yang, Zhi-Hong (September 2020, Proc. Computer Algebra in Scientific Computing (CASC) 2020)

   We generalize Hermite interpolation with error correction, which is the methodology for multiplicity algebraic error correction codes, to Hermite interpolation of a rational function over a field K from function and function derivative values. We present an interpolation algorithm that can locate and correct <= E errors at distinct arguments y in K where at least one of the values or values of a derivative is incorrect. The upper bound E for the number of such y is input. Our algorithm sufficiently oversamples the rational function to guarantee a unique interpolant. We sample (f/g)^(j)(y[i]) for 0 <= j <= L[i], 1 <= i <= n, y[i] distinct, where (f/g)^(j) is the j-th derivative of the rational function f/g, f, g in K[x], GCD(f,g)=1, g <= 0, and where N = (L[1]+1)+...+(L[n]+1) >= C + D + 1 + 2(L[1]+1) + ... + 2(L[E]+1) where C is an upper bound for deg(f) and D an upper bound for deg(g), which are input to our algorithm. The arguments y[i] can be poles, which is truly or falsely indicated by a function value infinity with the corresponding L[i]=0. Our results remain valid for fields K of characteristic >= 1 + max L[i]. Our algorithm has the same asymptotic arithmetic complexity as that for classical Hermite interpolation, namely soft-O(N). For polynomials, that is, g=1, and a uniform derivative profile L[1] = ... = L[n], our algorithm specializes to the univariate multiplicity code decoder that is based on the 1986 Welch-Berlekamp algorithm. 

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2. Hermite Interpolation With Error Correction: Fields of Zero or Large Characteristic and Large Error Rate

   https://doi.org/10.1145/3452143.3465525  

   Kaltofen, Erich L.; Pernet, Clément; Yang, Zhi-Hong (July 2021, ISSAC '21: Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation)

   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. 

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3. Sparse Polynomial Hermite Interpolation

   https://doi.org/10.1145/3476446.3535501  

   Kaltofen, Erich L. (July 2022, ISSAC '22 Proc. 2022 ACM Internat. Symp. Symbolic Algebraic Comput.)

   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'. 

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4. Fast Barycentric-Based Evaluation Over Spectral/hp Elements

   https://doi.org/10.1007/s10915-021-01750-2  

   Laughton, Edward; Zala, Vidhi; Narayan, Akil; Kirby, Robert M.; Moxey, David (January 2022, Journal of Scientific Computing)

   Abstract As the use of spectral/hpelement methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendous attention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/hpelement methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/hpelement libraryNektar++by running a series of baseline evaluations against the ‘standard’ Lagrangian method, where an interpolation matrix is generated and matrix-multiplication applied to evaluate a point at a given location. We present results from a rigorous series of benchmarking tests for a variety of element shapes, polynomial orders and dimensions. We show that when the point of interest is to be repeatedly evaluated, the barycentric method performs at worst$$50\%$$ $50\%$slower, when compared to a cached matrix evaluation. However, when the point of interest changes repeatedly so that the interpolation matrix must be regenerated in the ‘standard’ approach, the barycentric method yields far greater performance, with a minimum speedup factor of$$7\times $$ $7\times$. Furthermore, when derivatives of the solution evaluation are also required, the barycentric method in general slightly outperforms the cached interpolation matrix method across all elements and orders, with an up to$$30\%$$ $30\%$speedup. Finally we investigate a real-world example of scalar transport using a non-conformal discontinuous Galerkin simulation, in which we observe around$$6\times $$ $6\times$speedup in computational time for the barycentric method compared to the matrix-based approach. We also explore the complexity of both interpolation methods and show that the barycentric interpolation method requires$${\mathcal {O}}(k)$$ $O\left(k\right)$storage compared to a best case space complexity of$${\mathcal {O}}(k^2)$$ $O\left(k^2\right)$for the Lagrangian interpolation matrix method. 

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5. Attribute-Aware RBFs: Interactive Visualization of Time Series Particle Volumes Using RT Core Range Queries

   https://doi.org/10.1109/TVCG.2023.3327366  

   Morrical, Nate; Zellmann, Stefan; Sahistan, Alper; Shriwise, Patrick; Pascucci, Valerio (January 2024, IEEE Transactions on Visualization and Computer Graphics)

   Smoothed-particle hydrodynamics (SPH) is a mesh-free method used to simulate volumetric media in fluids, astrophysics, and solid mechanics. Visualizing these simulations is problematic because these datasets often contain millions, if not billions of particles carrying physical attributes and moving over time. Radial basis functions (RBFs) are used to model particles, and overlapping particles are interpolated to reconstruct a high-quality volumetric field; however, this interpolation process is expensive and makes interactive visualization difficult. Existing RBF interpolation schemes do not account for color-mapped attributes and are instead constrained to visualizing just the density field. To address these challenges, we exploit ray tracing cores in modern GPU architectures to accelerate scalar field reconstruction. We use a novel RBF interpolation scheme to integrate per-particle colors and densities, and leverage GPU-parallel tree construction and refitting to quickly update the tree as the simulation animates over time or when the user manipulates particle radii. We also propose a Hilbert reordering scheme to cluster particles together at the leaves of the tree to reduce tree memory consumption. Finally, we reduce the noise of volumetric shadows by adopting a spatially temporal blue noise sampling scheme. Our method can provide a more detailed and interactive view of these large, volumetric, time-series particle datasets than traditional methods, leading to new insights into these physics simulations. 

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