Symbolic-Numeric computing has demonstrated many advantages over purely numeric computing in academic circles. However, many of the algorithms that scientists and engineers use in practice are not symbolic-numeric. In this talk we will focus on software that is being adopted by practitioners which incorporate symbolic-numeric aspects. ModelingToolkit.jl will be described in detail, along with its symbolic compiler passes which allow for automating index reduction of differential-algebraic equations (DAEs), alias elimination, and tearing of nonlinear systems to allow for efficient acausal modeling. The equivalence and differences between symbolic algorithms and compiler passes will be highlighted along with a discussion of how symbolic algorithms are being extended to apply directly to non-symbolic code. Applications in aerospace, HVAC design, and the optimal use of next generation battery chemistries will be showcased to demonstrate how symbolic-numeric computing is transforming the industrial use of computational mathematics.

In this talk, we discuss the certified homotopy tracking using interval arithmetic. While homotopy continuation is a widely recognized method for locating solutions to polynomial systems by tracing the homotopy path of solutions, its general correctness in path tracking is not guaranteed. We exploit interval arithmetic for conservative computation to track the given solution path in a certified way. The main idea is to trace the solution path using a series of interval boxes that uniquely enclose the path. The Krawczyk method is introduced to verify the existence and uniqueness of the solution path at each interval box. We discuss the algorithm, termination statement, and experimental results from a proof-of-concept implementation. It is a preliminary report based on a joint work with Tim Duff.

A mathematical model is called sparse if it can be represented by only a few non-zero terms. In computer algebra, the aim of sparse interpolation is to determine such a model from a small number of samples. Sparse techniques solve the problem statement from a number of samples proportional to the number of terms in the representation, rather than the number of available data points or available generating elements. Sparse representations reduce the complexity in several ways: data collection, algorithmic complexity and model complexity.

In this talk, we will introduce sparse polynomial interpolation and the interesting connections to exponential analysis, structured matrices, generalized eigenvalue computation and rational approximation. In the past few years, insight gained from the computer algebra community combined with methods developed by the numerical analysis community, has led to significant progress in several signal processing applications. The connection to tensor decomposition opens new possibilities to exploit sparsity in analysing tensor-structured datasets.

A symbolic-numeric method to solve a system of polynomial equations constructs in a first stage a homotopy connecting the system that must be solved to a system with similar structure that is easier to solve. In the second stage, numerical algorithms are applied to track the solution paths defined by the homotopy.  At a singular solution, the matrix of all partial derivatives is rank deficient.

In theory, using random complex numbers, singularities do not occur, except at the end of the path, when the system that must be solved has singular solutions.  In practice, an apriori step size control mechanism takes into account the curvature of the path and the distance to the nearest singularity, applying the ratio theorem of Fabry. Extrapolation methods are effective in locating and approximating the singularities at the end of a solution path.

This talk is based on joint work with Nathan Bliss (Linear Algebra and Applications 2018), with Simon Telen and Marc Van Barel (SISC 2020), and on ongoing work with Kylash Visvanathan (CASC 2022).

Abstractly, the IVP (Initial Value Problem) in ODEs is the problem of computing a solution to the ODE x' = f(t,x), with initial conditions x(0)=x_0. Here, x=(x_1,...,x_n) and time domain is [0,1]. The input data is (f,x_0). This is an important basic problem in applications of differential equations, especially in modeling.

Validated algorithms produce output with a priori (correctness) guarantees. They are ultimately based on interval methods. There are many validated versions of the IVP. E.g., instead of x_0, we are given an interval I_0 and we are interested in all solutions x with x(0) in I_0.  The output is an interval J that contains x(1) for all such x.

We will review the state-of-the-art, and the basic theory behind such validated solutions.  There are many open problems: (1) gaps in the validated theory, (2) better algorithms for a variety of validated IVP-like problems, (3) faster software.

Reconstructing a 3D scene from multiple images is an intrinsically geometric problem with multiple applications. I will introduce a line of work that aims to characterize the set of all valid algebraic constraints that relate any number of perspective cameras, 3D points, and their 2D projections. More formally, this framework involves the study of certain multigraded vanishing ideals. This leads to several new results, as well as new proofs of old results about the well-studied multiview ideal. For example, we give some perspective about a "folklore theorem" from geometric computer vision which roughly states: "all algebraic constraints on the 2D projections of 3D points can be obtained from those involving 2, 3, or 4 cameras." I will also discuss a complementary line of work focused on practical estimation methods. Incremental 3D reconstruction systems usually focus on estimating the relative orientation of two cameras. This in turn requires solving systems of algebraic equations with (very) special structure. I will describe recent progress extending the domain of such solvers to problems involving three or four cameras, including non-perspective cameras with lens distortion. The key players are numerical homotopy continuation methods, and the Galois/monodromy groups that capture their inherent complexity.

We consider the problem of complex root classification, i.e., finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients. Those polynomials in the coefficients are called discriminants for multiplicities. It is also known that discriminants can be obtained by using repeated parametric gcd’s. The resulting discriminants are usually nested determinants, that is, determinants of matrices whose entries are determinants, and so on. In this talk, we give a new type of discriminants which are not based on repeated gcd’s. The new discriminants are simpler in the sense that they are non-nested determinants and have smaller maximum degrees.

We will discuss an approach for estimating values of the parameters from data for systems of ODEs. This approach does not rely on optimization and, instead, uses differential algebra, interpolation of the data using rational functions, and multivariable polynomial system solving.