Related papers: Numerical algebraic geometry for model selection a…
We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a…
Polynomial dynamical systems are widely used to model and study real phenomena. In biochemistry, they are the preferred choice for modelling the concentration of chemical species in reaction networks with mass-action kinetics. These systems…
It is known that statistical model selection as well as identification of dynamical equations from available data are both very challenging tasks. Physical systems behave according to their underlying dynamical equations which, in turn, can…
The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…
We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.
Biochemical networks are used in computational biology, to model the static and dynamical details of systems involved in cell signaling, metabolism, and regulation of gene expression. Parametric and structural uncertainty, as well as…
Geometry optimization is an important part of both computational materials and surface science because it is the path to finding ground state atomic structures and reaction pathways. These properties are used in the estimation of…
We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on…
Complex analyses involving multiple, dependent random quantities often lead to graphical models - a set of nodes denoting variables of interest, and corresponding edges denoting statistical interactions between nodes. To develop statistical…
In many areas of applied mathematics and statistics, it is a fundamental problem to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the objective function…
We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both…
Variance parameter estimation in linear mixed models is a challenge for many classical nonlinear optimization algorithms due to the positive-definiteness constraint of the random effects covariance matrix. We take a completely novel view on…
Geometric programming problem is a powerful tool for solving some special type non-linear programming problems. It has a wide range of applications in optimization and engineering for solving some complex optimization problems. Many…
Any mass action network gives rise to a parameterised family of polynomial equations whose positive solutions are the positive equilibria of the network. Here, we consider alternative systems of equations, whose solutions are in smooth,…
The article gives an overview of the parameter numerical continuation methodology applied to setpoint control and parameter identification of nonlinear systems. The control problems for affine systems as well as general (nonaffine)…
We revisit the classical problem of optimal experimental design (OED) under a new mathematical model grounded in a geometric motivation. Specifically, we introduce models based on elementary symmetric polynomials; these polynomials capture…
Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters. In this paper, for…
Most statistical software packages implement numerical strategies for computation of maximum likelihood estimates in random effects models. Little is known, however, about the algebraic complexity of this problem. For the one-way layout…
The clear understanding of the non-convex landscape of neural network is a complex incomplete problem. This paper studies the landscape of linear (residual) network, the simplified version of the nonlinear network. By treating the gradient…
One of the most challenging and frequently arising problems in many areas of science is to find solutions of a system of multivariate nonlinear equations. There are several numerical methods that can find many (or all if the system is small…