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Faster Groebner bases for Lie derivatives of OdE systems via monomial orderings 49

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Faster Groebner bases for Lie derivatives of ODE systems via...

49th International Symposium on Symbolic and Algebraic Computation, ISSAC 2024

作者： Bessonov, Mariya Ilmer, Ilia Konstantinova, Tatiana Ovchinnikov, Alexey Pogudin, Gleb Soto, Pedro Department of Mathematics Cuny Nyc College of Technology New YorkNY United States Ph.D. Program in Computer Science Cuny Graduate Center New YorkNY United States Department of Mathematics Cuny Queens College New YorkNY United States Department of Mathematics Cuny Queens College United States Ph.D. Programs in Mathematics and Computer Science Cuny Graduate Center New YorkNY United States Lix Cnrs École Polytechnique Institute Polytechnique de Paris Paris France

ISBN: (纸本)9798400706967

Symbolic computation for systems of differential equations is often computationally expensive. Many practical differential models have a form of polynomial or rational OdE system with specified outputs. A basic symbolic approach to analyze these models is to compute and then symbolically process the polynomial system obtained by sufficiently many Lie derivatives of the output functions with respect to the vector field given by the OdE system. In this paper, we present a method for speeding up Gröbner basis computation for such a class of polynomial systems by using specific monomial ordering, including weights for the variables, coming from the structure of the OdE model. We provide empirical results that show improvement across different symbolic computing frameworks and apply the method to speed up structural identifiability analysis of OdE models. © 2024 ACM.

关键词： Polynomials

Subdivision and Schröder's Polynomial Root-Finders Combined

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Subdivision and Schröder's Polynomial Root-Finders Combined

International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)

作者： Victor Y. Pan Department of Computer Science Lehman College of the City University of New York Bronx NY USA Ph.D. Programs in Mathematics and Computer Science The Graduate Center of the City University of New York New York NY USA

ISBN: (数字)9798331532833

ISBN: (纸本)9798331532840

关键词： Scientific computing Polynomials Classification algorithms

Algorithm for globally identifiable reparametrizations of OdEs

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arXiv 2024年

作者： Falkensteiner, Sebastian Ovchinnikov, Alexey Sendra, J. Rafael Max Planck Institute for Mathematics in The Sciences Leipzig Germany Department of Mathematics CUNY Queens College Ph.D. Programs in Mathematics and Computer Science CUNY Graduate Center New York United States Department of Quantitative Methods CUNEF-University Spain

Structural global parameter identifiability indicates whether one can determine a parameter’s value in an OdE model from given inputs and outputs. If a given model has parameters for which there is exactly one value, such parameters are called globally identifiable. Given an OdE model involving not globally identifiable parameters, first we transform the system into one with locally identifiable parameters. As a main contribution of this paper, then we present a procedure for replacing, if possible, the OdE model with an equivalent one that has globally identifiable parameters. We first derive this as an algorithm for one-dimensional OdE models and then reuse this approach for higher-dimensional *** Codes 93C15, 93B25, 93B30, 34A55, 14E08, 14M20, 14Q20, 12H05, 92B05 © 2024, CC BY-NC-SA.

关键词： Observability

ROBUST PARAMETER ESTIMATION FOR RATIONAL ORdINARY dIFFERENTIAL EQUATIONS

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arXiv 2023年

作者： Bassik, Oren Berman, Yosef Go, Soo Hong, Hoon Ilmer, Ilia Ovchinnikov, Alexey Rackauckas, Chris Soto, Pedro Yap, Chee Ph.D. Program in Mathematics CUNY Graduate Center New YorkNY United States Ph.D. Program in Computer Science CUNY Graduate Center New YorkNY United States Department of Mathematics North Carolina State University RaleighNC United States Department of Mathematics CUNY Queens College United States Ph.D. Programs in Mathematics and Computer Science CUNY Graduate Center New YorkNY United States Department of Mathematics Massachusetts Institute of Technology CambridgeMA United States Mathematical Institute University of Oxford Oxford United Kingdom Department of Computer Science Courant Institute of Mathematical Science New York University New YorkNY United States

We present a new approach for estimating parameters in rational OdE models from given (measured) time series data. In typical existing approaches, an initial guess for the parameter values is made from a given search interval. Then, in a loop, the corresponding outputs are computed by solving the OdE numerically, followed by computing the error from the given time series data. If the error is small, the loop terminates and the parameter values are returned. Otherwise, heuristics/theories are used to possibly improve the guess and continue the loop. These approaches tend to be non-robust in the sense that their accuracy depend on the search interval and the true parameter values;furthermore, they cannot handle the case where the parameters are locally identifiable. In this paper, we propose a new approach, which does not suffer from the above non-robustness. In particular, it does not require making good initial guesses for the parameter values or specifying search intervals. Instead, it uses differential algebra, interpolation of the data using rational functions, and multivariate polynomial system solving. We also compare the performance of the resulting software with several other estimation software packages. © 2023, CC BY-NC-SA.

关键词： Parameter estimation

differentially closed fields and universality on a cone

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arXiv 2023年

作者： Miller, Russell Dept. of Mathematics Queens College 65-30 Kissena Blvd. QueensNY11367 United States Ph.D. Programs in Mathematics & Computer Science Graduate Center City University of New York 365 FIfth Avenue New YorkNY10016 United States

The class of all countable differentially closed differential fields K of characteristic 0 was shown by Marker and the author in [21] to be "one jump away" from universal for spectra of structures: for every nontrivial countable structure M, there is some K whose spectrum is the preimage under jump of the spectrum of M, and conversely, for every K, there is such an M. We show that the missing jump can be accounted for by adding to the signature of differential fields a predicate describing a certain algebraic transcendence property. The ensuing universality results for differentially closed fields in the new signature include not only spectra of structures, but also many properties related to computable categoricity. However, these latter universality results hold only on the cone above a specific Σ01 oracle set, whose decidability status remains unknown. Moreover, differentially closed fields simply fail flat-out to be universal for automorphism groups, even non-effectively. We also include a small erratum (Remark 3.2) to [21].MSC Codes 03d45, 03C57, 12H05 Copyright © 2023, The Authors. All rights reserved.

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Symbolic-numeric algorithm for parameter estimation in discrete-time models with exp

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arXiv 2024年

作者： Berman, Yosef Forrest, Joshua Grote, Matthew Ovchinnikov, Alexey Rueda, Sonia Ph.D. Program in Mathematics CUNY Graduate Center New York United States Ph.D. Program in Computer Science CUNY Graduate Center New York United States Department of Mathematics CUNY Queens College Queens United States Departamento de Matemática Aplicada E.T.S. Arquitectura Universidad Politécnica de Madrid Spain School of Mathematics and Statistics University of Melbourne VIC Australia

determining unknown parameter values in dynamic models is crucial for accurate analysis of the dynamics across the different scientific disciplines. discrete-time dynamic models are widely used to model biological processes, but it is often difficult to determine these parameters. In this paper, we propose a robust symbolic-numeric approach for parameter estimation in discrete-time models that involve non-algebraic functions such as exp. We illustrate the performance (precision) of our approach by applying our approach to the flour beetle (LPA) model, an archetypal discrete-time model in biology. Unlike optimization-based methods, our algorithm guarantees to find all solutions of the parameter values given time-series data for the measured *** Codes 92B05, 68W30, 14Q20, 39A60, 13P15 © 2024, CC BY-NC-SA.

关键词： Parameter estimation

New Progress in Classic Area: Polynomial Root-squaring and Root-finding

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arXiv 2022年

作者： Pan, Victor Y. Department of Computer Science Lehman College The City University of New York BronxNY10468 United States Ph.D. Programs in Mathematics and Computer Science The Graduate Center The City University of New York New YorkNY10036 United States

The dLG root-squaring iterations, due to dandelin 1826 and rediscovered by Lobachevsky 1834 and Gräffe 1837, have been the main approach to root-finding for a univariate polynomial p(x) in the 19th century and beyond, but not so nowadays because these iterations are prone to severe numerical stability problems. Trying to avoid these problems we have found simple but novel reduction of the iterations applied for Newton's inverse ratio-p'(x)/p(x) to approximation of the power sums of the zeros of p(x) and its reverse polynomial. The resulting polynomial root-finders can be devised and performed independently of dLG iterations-based on Newton's identities or Cauchy integrals. In the former case the computation involve a set of leading or tailing coefficients of an input polynomial. In the latter case we must scale the variable and increase the arithmetic computational cost to ensure numerical stability. Nevertheless the cost is still manageable, at least for fast root-refinement, and the algorithms can be applied to a black box polynomial p(x)-given by a black box for the evaluation of the ratio p'(x)/p(x) rather than by its coefficients. This enables important computational benefits, including (i) efficient recursive as well as concurrent approximation of a set of zeros of p(x) or even all of its zeros, (ii) acceleration where an input polynomial can be evaluated fast, and (iii) extension to approximation of the eigen-values of a matrix or a polynomial matrix, being efficient if the matrix can be inverted fast, e.g., is data sparse. We also recall our recent fast algorithms for approximation of the root radii, that is, the distances to the roots from the origin or any complex value to the zeros of p(x), and propose to apply it for fast black box initialization of polynomial rootfinding by means of functional iterations such as Newton's, Ehrlich's, and Weierstrass's. Copyright © 2022, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

Fast Approximation of Polynomial Zeros and Matrix Eigenvalues

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arXiv 2022年

作者： Pan, Victor Y. Go, Soo Luan, Qi Zhao, Liang Department of Computer Science Lehman College of CUNY The City University of New York NY10468 United States Ph.D. Programs in Mathematics and Computer Science The Graduate Center of the City University of New York New YorkNY10036 United States

Given a black box (oracle) for the evaluation of a univariate polynomial p(x) of a degree d, we seek its zeros, that is, the roots of the equation p(x) = 0. At FOCS 2016 Louis and Vempala approximated a largest zero of such a real-rooted polynomial within 1/2b, by performing at NR cost of the evaluation of Newton’s ratio pp′((xx)) at O(b log(d)) points x. They readily extended this root-finder to record fast approximation of a largest eigenvalue of a symmetric matrix under the Boolean complexity model. We apply distinct approach and techniques to obtain more general results at the same computational cost. Let a square on the complex plane be reasonably well isolated from the external roots of p(x) = 0 and let it contain at most m roots counted with multiplicities, for a fixed m ≤ d. Then we approximate all internal roots at NR cost O(m3b log(d)) and extend this to approximation of all eigenvalues of a matrix in such a square at record Boolean cost. For any fixed pair of a real γ ≥ 1 and an integer v ≥ 1, our alternative Las Vegas randomized root-finder runs at NR cost in O(m2γvb log(d)), and outputs errors with a probability at most 1/γv. We prove that O(log(d)) bit-precision is sufficient in our root-finders provided that oracle evaluates p(x) with relative error of order log(d) at points x sufficiently isolated from the zeros of p(x). For a constant m we match the cost bounds of Louis and Vempala for approximation of roots and eigenvalues but relax the restrictions on input and output. Our use of Cauchy integrals and randomization are pioneering and nontrivial as well as our fast approximation of the root radii (that is, the distances to the roots) from any fixed complex point. © 2022, CC BY.

关键词： Polynomial approximation

Web-based structural identifiability analyzer

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arXiv 2021年

作者： Ilmer, Ilia Ovchinnikov, Alexey Pogudin, Gleb Ph.D. Program in Computer Science CUNY Graduate Center New York United States Department of Mathematics CUNY Queens College Ph.D. Programs in Mathematics and Computer Science CUNY Graduate Center New York United States LIX CNRS École Polytechnique Institute Polytechnique de Paris France

Parameter identifiability describes whether, for a given differential model, one can determine parameter values from model equations. Knowing global or local identifiability properties allows construction of better practical experiments to identify parameters from experimental data. In this work, we present a web-based software tool that allows to answer specific identifiability queries. Concretely, our toolbox can determine identifiability of individual parameters of the model and also provide all functions of parameters that are identifiable (also called identifiable combinations) from single or multiple experiments. The program is freely available at https://***/app/6509768948056064. © 2021, CC BY-SA.

关键词： Websites