arXiv subjects

Alexey Ovchinnikov

Publications and source records attributed to Alexey Ovchinnikov.

At least 37 records · Page 2Linked to original sources

arXiv2021

Algorithms yield upper bounds in differential algebra

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including for example, difference fields). We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

math.AC↗

arXiv2021

Multi-experiment parameter identifiability of ODEs and model theory

Structural identifiability is a property of an ODE model with parameters that allows for the parameters to be determined from continuous noise-free data. This is a natural prerequisite for practical identifiability. Conducting multiple independent experiments could make more parameters or functions of parameters identifiable, which is a desirable property to have. How many experiments are sufficient? In the present paper, we provide an algorithm to determine the exact number of experiments for multi-experiment local identifiability and obtain an upper bound that is off at most by one for the number of experiments for multi-experiment global identifiability. Interestingly, the main theoretical ingredient of the algorithm has been discovered and proved using model theory (in the sense of mathematical logic). We hope that this unexpected connection will stimulate interactions between applied algebra and model theory, and we provide a short introduction to model theory in the context of parameter identifiability. As another related application of model theory in this area, we construct a nonlinear ODE system with one output such that single-experiment and multiple-experiment identifiability are different for the system. This contrasts with recent results about single-output linear systems. We also present a Monte Carlo randomized version of the algorithm with a polynomial arithmetic complexity. Implementation of the algorithm is provided and its performance is demonstrated on several examples. The source code is available at https://github.com/pogudingleb/ExperimentsBound.

math.AG↗

arXiv2021

Web-based Structural Identifiability Analyzer

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://maple.cloud/app/6509768948056064.

cs.MS↗

arXiv2021

Computing all identifiable functions of parameters for ODE models

Parameter identifiability is a structural property of an ODE model for recovering the values of parameters from the data (i.e., from the input and output variables). This property is a prerequisite for meaningful parameter identification in practice. In the presence of nonidentifiability, it is important to find all functions of the parameters that are identifiable. The existing algorithms check whether a given function of parameters is identifiable or, under the solvability condition, find all identifiable functions. However, this solvability condition is not always satisfied, which presents a challenge. Our first main result is an algorithm that computes all identifiable functions without any additional assumptions, which is the first such algorithm as far as we know. Our second main result concerns the identifiability from multiple experiments (with generically different inputs and initial conditions among the experiments). For this problem, we prove that the set of functions identifiable from multiple experiments is what would actually be computed by input-output equation-based algorithms (whether or not the solvability condition is fulfilled), which was not known before. We give an algorithm that not only finds these functions but also provides an upper bound for the number of experiments to be performed to identify these functions. We provide an implementation of the presented algorithms.

eess.SY↗

arXiv2020

Parameter identifiability and input-output equations

Structural parameter identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. One of the standard approaches to assessing this problem is via input-output equations and, in particular, characteristic sets of differential ideals. The precise relation between identifiability and input-output identifiability is subtle. The goal of this note is to clarify this relation. The main results are: 1) identifiability implies input-output identifiability; 2) these notions coincide if the model does not have rational first integrals; 3) the field of input-output identifiable functions is generated by the coefficients of a "minimal" characteristic set of the corresponding differential ideal. We expect that some of these facts may be known to the experts in the area, but we are not aware of any articles in which these facts are stated precisely and rigorously proved.

math.AG↗

arXiv2020

CLUE: Exact maximal reduction of kinetic models by constrained lumping of differential equations

Motivation: Detailed mechanistic models of biological processes can pose significant challenges for analysis and parameter estimations due to the large number of equations used to track the dynamics of all distinct configurations in which each involved biochemical species can be found. Model reduction can help tame such complexity by providing a lower-dimensional model in which each macro-variable can be directly related to the original variables. Results: We present CLUE, an algorithm for exact model reduction of systems of polynomial differential equations by constrained linear lumping. It computes the smallest dimensional reduction as a linear mapping of the state space such that the reduced model preserves the dynamics of user-specified linear combinations of the original variables. Even though CLUE works with nonlinear differential equations, it is based on linear algebra tools, which makes it applicable to high-dimensional models. Using case studies from the literature, we show how CLUE can substantially lower model dimensionality and help extract biologically intelligible insights from the reduction. Availability: An implementation of the algorithm and relevant resources to replicate the experiments herein reported are freely available for download at https://github.com/pogudingleb/CLUE. Supplementary information: enclosed.

q-bio.MN↗

arXiv2020

Bounds for elimination of unknowns in systems of differential-algebraic equations

Elimination of unknowns in systems of equations, starting with Gaussian elimination, is a problem of general interest. The problem of finding an a priori upper bound for the number of differentiations in elimination of unknowns in a system of differential-algebraic equations (DAEs) is an important challenge, going back to Ritt (1932). The first characterization of this via an asymptotic analysis is due to Grigoriev's result (1989) on quantifier elimination in differential fields, but the challenge still remained. In this paper, we present a new bound, which is a major improvement over the previously known results. We also present a new lower bound, which shows asymptotic tightness of our upper bound in low dimensions, which are frequently occurring in applications. Finally, we discuss applications of our results to designing new algorithms for elimination of unknowns in systems of DAEs.

math.AC↗

arXiv2019

Effective difference elimination and Nullstellensatz

We prove effective Nullstellensatz and elimination theorems for difference equations in sequence rings. More precisely, we compute an explicit function of geometric quantities associated to a system of difference equations (and these geometric quantities may themselves be bounded by a function of the number of variables, the order of the equations, and the degrees of the equations) so that for any system of difference equations in variables $\mathbf{x} = (x_1, \ldots, x_m)$ and $\mathbf{u} = (u_1, \ldots, u_r)$, if these equations have any nontrivial consequences in the $\mathbf{x}$ variables, then such a consequence may be seen algebraically considering transforms up to the order of our bound. Specializing to the case of $m = 0$, we obtain an effective method to test whether a given system of difference equations is consistent.

math.AG↗

arXiv2018

Elimination of unknowns for systems of algebraic differential-difference equations

We establish effective elimination theorems for differential-difference equations. Specifically, we find a computable function $B(r,s)$ of the natural number parameters $r$ and $s$ so that for any system of algebraic differential-difference equations in the variables $\mathbf{x} = x_1, \ldots, x_q$ and $\mathbf{y} = y_1, \ldots, y_r$ each of which has order and degree in $\mathbf{y}$ bounded by $s$ over a differential-difference field, there is a non-trivial consequence of this system involving just the $\mathbf{x}$ variables if and only if such a consequence may be constructed algebraically by applying no more than $B(r,s)$ iterations of the basic difference and derivation operators to the equations in the system. We relate this finiteness theorem to the problem of finding solutions to such systems of differential-difference equations in rings of functions showing that a system of differential-difference equations over $\mathbb{C}$ is algebraically consistent if and only if it has solutions in a certain ring of germs of meromorphic functions.

math.AC↗

arXiv2018

SIAN: software for structural identifiability analysis of ODE models

Biological processes are often modeled by ordinary differential equations with unknown parameters. The unknown parameters are usually estimated from experimental data. In some cases, due to the structure of the model, this estimation problem does not have a unique solution even in the case of continuous noise-free data. It is therefore desirable to check the uniqueness a priori before carrying out actual experiments. We present a new software SIAN (Structural Identifiability ANalyser) that does this. Our software can tackle problems that could not be tackled by previously developed packages.

cs.SC↗

arXiv2018

Commuting planar polynomial vector fields for conservative Newton systems

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. Let $f \in K[x]$, where $K$ is a field of characteristic zero, and $d$ the derivation that corresponds to the differential equation $\ddot x = f(x)$ in a standard way. Let also $H$ be the Hamiltonian polynomial for $d$, that is $H=\frac{1}{2}y^2-\int{f(x)dx}$. It is known that the set of all polynomial derivations that commute with $d$ forms a $K[H]$-module $M_d$. In this paper, we show that, for every such $d$, the module $M_d$ is of rank $1$ if and only if $\text{deg}\; f\geqslant 2$. For example, the classical elliptic equation $\ddot x = 6x^2+a$, where $a \in \mathbb{C}$, falls into this category.

math.DS↗

arXiv2018

Corrigendum: 'On bounds for the effective differential Nullstellensatz, arXiv:1508.07508'

We correct a small gap found in the authors' paper 'On bounds for the effective differential Nullstellensatz' (J Algebra 449:1-21, 2016). This gap is due to an inequality that does not generally hold. However, under one additional assumption, it does hold. In this note, we provide a detailed proof of this. We then point out that this assumption is satisfied in all instances in which the inequality was used.

math.AC↗

arXiv2017

Triviality of differential Galois cohomologies of linear differential algebraic groups

We show that the triviality of the differential Galois cohomologies over a partial differential field K of a linear differential algebraic group is equivalent to K being algebraically, Picard-Vessiot, and linearly differentially closed. This former is also known to be equivalent to the uniqueness up to an isomorphism of a Picard-Vessiot extension of a linear differential equation with parameters.

math.AG↗

arXiv2016

New order bounds in differential elimination algorithms

We present a new upper bound for the orders of derivatives in the Rosenfeld-Groebner algorithm. This algorithm computes a regular decomposition of a radical differential ideal in the ring of differential polynomials over a differential field of characteristic zero with an arbitrary number of commuting derivations. This decomposition can then be used to test for membership in the given radical differential ideal. In particular, this algorithm allows us to determine whether a system of polynomial PDEs is consistent. Previously, the only known order upper bound was given by Golubitsky, Kondratieva, Moreno Maza, and Ovchinnikov for the case of a single derivation. We achieve our bound by associating to the algorithm antichain sequences whose lengths can be bounded using the results of Leon Sanchez and Ovchinnikov.

math.AC↗

arXiv2016

Calculating Galois groups of third order linear differential equations with parameters

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel's equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In the present paper, we give the first known algorithm that calculates the differential Galois group of a third order parameterized linear differential equation.

math.RT↗

arXiv2016

Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

math.AC↗

arXiv2015

On bounds for the effective differential Nullstellensatz

Understanding bounds for the effective differential Nullstellensatz is a central problem in differential algebraic geometry. Recently, several bounds have been obtained using Dicksonian and antichains sequences (with a given growth rate). In the present paper, we make these bounds more explicit and, therefore, more applicable to understanding the computational complexity of the problem, which is essential to designing more efficient algorithms.

math.AC↗

arXiv2014

New effective differential Nullstellensatz

We show new upper and lower bounds for the effective differential Nullstellensatz for differential fields of characteristic zero with several commuting derivations. Seidenberg was the first to address this problem in 1956, without giving a complete solution. The first explicit bounds appeared in 2009 in a paper by Golubitsky, Kondratieva, Szanto, and Ovchinnikov, with the upper bound expressed in terms of the Ackermann function. D'Alfonso, Jeronimo, and Solernó, using novel ideas, obtained in 2014 a new bound if restricted to the case of one derivation and constant coefficients. To obtain the bound in the present paper without this restriction, we extend this approach and use the new methods of Freitag and León Sánchez and of Pierce from 2014, which represent a model-theoretic approach to differential algebraic geometry.

math.AC↗