Computer Algebra in Scientific Computing

Name: Computer Algebra in Scientific Computing | 26th International Workshop, CASC 2024, Rennes, France, September 2-6, 2024, Proceedings
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26th International Workshop, CASC 2024, Rennes, France, September 2-6, 2024, Proceedings

François Boulier Chenqi Mou Timur M. Sadykov Evgenii V. Vorozhtsov(Editor)

Springer (Publisher)

Published on 20. August 2024

XIV, 394 pages

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978-3-031-69070-9 (ISBN)

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Content

Intro
Preface
Organization
Polynomial System Solving through Gröbner Bases in Scientific Computing (Invited Talk)
Contents
Recent Developments in Real Quantifier Elimination and Cylindrical Algebraic Decomposition (Extended Abstract of Invited Talk)*-12pt
1 Real Quantifier Elimination
2 Cylindrical Algebraic Decomposition
3 The Doubly Exponential Wall
4 CAD and Satisfiability Checking
4.1 SAT and SMT
4.2 CAD as Theory Solver
4.3 New CAD-Based Algorithms
4.4 SC2
5 CAD and Machine Learning
5.1 CAD Variable Ordering Choice
5.2 Future Progress from Explainable AI?
References
Advances in Elimination Theory for Algebraic Differential and Difference Equations
1 Introduction
2 Basic Notions in Differential Algebra
3 Differential Chow Forms and Differential Chow Varieties
3.1 Differential Chow Forms
3.2 Differential Chow Varieties
3.3 Algorithms and Further Work
4 Differential Resultants, Sparse Differential Resultants and Sparse Difference Resultants
4.1 Differential Resultants
4.2 Sparse Differential Resultants
4.3 Sparse Difference Resultants
5 Effective Nullstellensatz and Elimination for Algebraic Differential-Difference Equations
References
A Modular Algorithm to Compute the Resultant of Multivariate Polynomials over Algebraic Number Fields Presented with Multiple Extensions
1 Introduction
1.1 Computing over Q(1,.,n)
1.2 Organization of the Paper
2 Preliminaries
2.1 Mapping Q(1,.,n) to a Single Extension Q()
2.2 Resultants
2.3 Computing Resultants of Univariate Polynomials
3 The Modular Resultant Algorithm
3.1 Algorithm PRES
3.2 Algorithm MRES
4 Implementation and Benchmarks
5 Complexity
6 Failure Probability
6.1 Lc-Bad Primes and Evaluation Points
6.2 Det-Bad Primes
7 Conclusion
References
The Liouville Generator for Producing Integrable Expressions
1 Introduction
2 Existing Data Generation Methods
2.1 Deep Learning for Symbolic Mathematics
2.2 Generating Elementary Integrable Expressions
2.3 The Substitution Rule
3 Background Material for the New Method
3.1 Liouville's Theorem
3.2 Parallel Risch Algorithm
4 The LIOUVILLE Data Generation Method
4.1 The Main Idea
4.2 Design Choices
5 Discussion of the New Data Generation Method
5.1 Example
5.2 Benefits over FWD and BWD
5.3 Benefits over RISCH
5.4 The Effect of Normalisation
6 Conclusion and Future Work
References
Symbolic-Numeric Solving Boundary Value Problems: Collective Models of Atomic Nuclei
1 Introduction
2 Formulation of BVP and GTM and FEM Schemes
3 BVP for Five-Dimensional Quadrupole Hamiltonian
4 5D Harmonic Oscillator Model in Affine Coordinates
5 Benchmark Calculations of 154Gd in the RMF Model
6 Conclusions
References
On Formal Power Series Solutions of Regular Differential Chains
1 Introduction
2 Mathematical Background
3 The Prolongation Bounds
3.1 An Approximation Lemma
3.2 Examples
4 Basic Notions on Regular Chains
5 The Case of Series Depending on Parameters
6 Computation of the Nonnegative Integer Roots
7 Implementation
References
A Dataset for Suggesting Variable Orderings for Cylindrical Algebraic Decompositions
1 Introduction
2 Generation of the Dataset
2.1 The CAD Implementation to Use
2.2 The Initial Setting for Generating the Examples
2.3 Statistical Information of the Generated Datasets
2.4 Organization of the Datasets
3 Performance of the Heuristic Methods on the Datasets
4 Training and Testing on the Datasets
4.1 Feature Engineering
4.2 Training of the Models
4.3 Testing of the Models
5 Conclusion and Future Work
References
Stability Analysis of a Differential Model for Quasi-Periodic Plasma Perturbations
1 Introduction
2 KCC Theory and Jacobi Stability of Dynamical Systems
3 Linear Stability Analysis of Model (1)
4 Jacobi Stability Analysis of Model (1)
4.1 The Nonlinear and Berwald Connections, and the KCC Invariants
4.2 Jacobi Stability of the Fixed Points
5 Dynamics of Deviation Vector in Model (1)
5.1 Dynamics of the Deviation Vector Near E0
5.2 Dynamics of the Deviation Vector Near E+ and E-
5.3 The Curvature of the Deviation Vector
6 Conclusions
References
Counting the Integer Points of Parametric Polytopes: A Maple Implementation
1 Introduction
2 Generating Functions of Rational Cones and Polytopes
3 Quasi-polynomials
4 Generic Case Discussion
5 Counting the Integer Points
5.1 Counting the Integer Points of a Non-parametric Polytope
5.2 Counting the Integer Points of a Parametric Polytope
6 Experimentation
References
Algebraic Representations for Faster Predictions in Convolutional Neural Networks
1 Introduction
1.1 Machine Learning and Algebraic Geometry
1.2 Contributions and Structure
2 CNN Setup
3 Skip Connections in Linear Networks
3.1 Transformation Matrices for Resampling and Padding
3.2 Skip Connections
4 Removing Skip Connections with a Homotopy
5 Computational Experiments
5.1 Equipment, Software, and Data
5.2 Pre-computing CNNs
5.3 Practical Experiment for Removing Skip Connections
6 Future Directions
7 Conclusion
References
Computing a Basis of the Set of Isogenies Between Two Supersingular Elliptic Curves
1 Introduction
2 Mathematical Background
2.1 Elliptic Curves over Finite Fields
2.2 Properties of Supersingular Elliptic Curves
3 Generators of Hom(E, E') for Supersingular Elliptic Curves E, E'
4 Computing a Basis of Hom(E, E') for Supersingular Elliptic Curves E, E'
4.1 Details and Implementation of Each Step
4.2 Numerical Examples
5 Conclusion
References
On the Radical of a Polynomial Ideal with Parameters
1 Introduction
2 Comprehensive Gröbner Systems
2.1 Preliminaries
2.2 Comprehensive Gröbner Systems
3 Tools for Parametric Ideals
3.1 Dimensions of a Parametric Ideal
3.2 Squarefree Part of a Univariate Polynomial with Parameters
3.3 Intersection of Parametric Ideals
3.4 Least Common Multiples of Parametric Polynomials
3.5 Saturation for a Parametric Ideal
4 Parametric Radical System
5 Zero Dimensional Case
6 Key Result
7 Non-zero Dimensional Case
References
Contribution to Integral Elimination
1 Introduction
2 Background
2.1 Integral Algebra
2.2 Ordering the Integral Monomials
2.3 Basic Rewriting Rules
3 Extended Rewriting Rules and Reduction
3.1 Reduced-Product Rule
3.2 Reduced-Power Rule
3.3 Reduction
4 Critical Pairs
5 Exponentials
5.1 Algorithms find_A_A0_G_Fand update_exp
6 Integral Elimination Prototype
7 Examples
7.1 Intra-Host Model of Malaria (taken from ch13malaria)
7.2 SIWR Model Cholera (taken from ch13dong2022differential)
References
Sparse Tensors and Subdivision Methods for Finding the Zero Set of Polynomial Equations
1 Introduction
1.1 Amortized Evaluation on a Grid of Boxes
1.2 Amortized Evaluation on a Sparse Subset of a Grid
1.3 Notations
1.4 Main Result
2 Evaluating Polynomials with Compressed Sparse Fibers
2.1 Sparse Tensor Data Structure
2.2 Evaluation Algorithm
3 Applications
3.1 The Fast Fourier Transform Revisited
3.2 Subdivision Algorithm
4 Experiments
4.1 Random Polynomials
4.2 Polynomials Coming from Applications
References
Merging Adjacent Cells During Single Cell Construction
1 Introduction
2 Preliminaries
2.1 Cylindrical Algebraic Decomposition, McCallum's Projection, and Levelwise Single Cell Construction
3 Detecting ``Irrelevant'' Roots
4 Local Delineability and Modified Projection
5 Weak Orderings and (Half-)closed Intervals
6 Experimental Results
7 Conclusion
References
Computing Clipped Products
1 Introduction
2 Preliminaries
3 Straightforward Methods
4 Clipped Polynomial Products
5 Clipped Integer Multiplication
6 Combining Methods
7 Further Thoughts and Conclusions
References
Fast Integral Bases Computation
1 Introduction
2 OM Algorithm and Local Integral Basis
2.1 Local Integral Basis and OM Factorisation
2.2 Precision of the OM Algorithm
3 Triangular p-Integral Bases
3.1 Reduced Triangular p-integral Bases
3.2 The MaxMin Algorithm
3.3 Global Integral Bases
4 Bases of Fractional Ideals
4.1 Fractional Ideals
4.2 p-Bases of Fractional Ideals
4.3 Improvements via S-basis
4.4 Global Triangular Bases of Fractional Ideals
5 Complexity and Proofs of the Main Results
6 An Illustrative Example
References
On Rational Recursion for Holonomic Sequences
1 Introduction
2 Problem Statement in Difference Algebra
3 Lower-Degree Holonomic Difference Polynomials
4 Proof and Algorithms
5 Concluding Remarks
References
GPU Accelerated Newton for Taylor Series Solutions of Polynomial Homotopies in Multiple Double Precision
1 Introduction
1.1 Problem Statement
1.2 Multiprecision Arithmetic
1.3 Numerical Condition of Taylor Series
2 Linearized Series and Newton's Method
3 Columns of Monomials
4 Staggered Computations
5 Accelerating Newton's Method
5.1 Arithmetic Intensity of Convolutions
5.2 Complex Vectorized Arithmetic
5.3 Accelerated Least Squares
5.4 Accelerated Updates and Residuals
6 Staging Multiple Doubles
6.1 Inlining of Arithmetical Kernels
6.2 Shared Memory and Registers
7 Computational Results
7.1 Graphics Processing Units
7.2 Proportions of Kernel Times
7.3 Performance of Convolutions
7.4 Doubling the Precisions
7.5 Experiments on the RTX 2080 and 4080
8 Conclusions
References
New Three- and Five-Stage Symplectic Schemes in the Forest-Ruth Family
1 Introduction
2 The Family of the Forest-Ruth Symplectic Schemes
3 The Three-Stage Forest-Ruth Schemes
4 The Five-Stage Forest-Ruth Schemes
5 Examples of Numerical Computations
5.1 Harmonic Oscillator
5.2 Kepler's Problem
6 Conclusions
References
Merging Multiple Algorithms for Computing Comprehensive Gröbner Systems Using Parallel Processing
1 Introduction
2 Preliminaries
3 Two Algorithms for Computing Comprehensive Gröbner Systems
3.1 Kapur-Sun-Wang Algorithm
3.2 Nabeshima Algorithm
4 Zero-Dimensional Cases
4.1 Gröbner Basis in a Polynomial Ring Over a Commutative von Neumann Regular Ring
4.2 Zero Dimensional Prime Ideal
4.3 Standard Way
5 Merging Comprehensive Gröbner System Algorithms Using Parallel Processing
References
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Computer Algebra in Scientific Computing

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