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14
Notes on triangular sets and triangulationdecomposition algorithms II: Differential Systems
 SYMBOLIC AND NUMERICAL SCIENTIFIC COMPUTING
, 2003
"... This is the second in a series of two tutorial articles devoted to triangulationdecomposition algorithms. The value of these notes resides in the uniform presentation of triangulationdecomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We em ..."
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Cited by 65 (8 self)
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This is the second in a series of two tutorial articles devoted to triangulationdecomposition algorithms. The value of these notes resides in the uniform presentation of triangulationdecomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. The present article deals with differential systems. It uses results presented in the first article on polynomial systems but can be read independently.
Factorizationfree decomposition algorithms in differential algebra
 Journal of Symbolic Computation
, 2000
"... Insight on the structure of differential ideals defined by coherent autoreduced set allows one to uncouple the differential and algebraic computations in a decomposition algorithm. Original results as well as concise new proofs of already presented theorems are exposed. As a consequence, an effectiv ..."
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Cited by 29 (0 self)
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Insight on the structure of differential ideals defined by coherent autoreduced set allows one to uncouple the differential and algebraic computations in a decomposition algorithm. Original results as well as concise new proofs of already presented theorems are exposed. As a consequence, an effective version of Ritt’s algorithm can be simply described. c ○ 2000 Academic Press 1.
Factorization Free Decomposition Algorithms In Differential Algebra
 Journal of Symbolic Computation
, 1999
"... . We present an effective version of Ritt's algorithm. We apply material of (Boulier et al. 1995) for which we give new concise proofs. We present original results in constructive algebra that makes the algorithm flexible and simple. 1. Introduction We present an algorithm to compute a represe ..."
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Cited by 23 (1 self)
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. We present an effective version of Ritt's algorithm. We apply material of (Boulier et al. 1995) for which we give new concise proofs. We present original results in constructive algebra that makes the algorithm flexible and simple. 1. Introduction We present an algorithm to compute a representation of the radical differential ideal generated by a system of differential equations (ordinary or partial). The method we use here involves characteristic set techniques 1 ; these were introduced by J.F. Ritt, the founder of differential algebra. The basic idea is to write the radical differential ideal generated by a finite set \Sigma of differential polynomials as an intersection of (radical) differential ideals that are uniquely defined by their characteristic sets. We will call these latter differential ideals components of f\Sigmag and we will call their intersection the characteristic decomposition of f\Sigmag. With a characteristic decomposition of f\Sigmag we can decide membership...
The Symbolic Integration of Exact PDEs,
 J. Symb. Comp.
, 2000
"... Abstract An algorithm is described which decides if a given polynomial differential expression ∆ of multivariate functions is exact, i.e. whether there exists a first integral P such that D x P = ∆ for any one x of a set of n variables and to provide the integral P . A generalization is given to al ..."
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Cited by 10 (6 self)
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Abstract An algorithm is described which decides if a given polynomial differential expression ∆ of multivariate functions is exact, i.e. whether there exists a first integral P such that D x P = ∆ for any one x of a set of n variables and to provide the integral P . A generalization is given to allow integration in the case that the exactness is prevented by terms which contain only functions of less than n independent variables. Motivation The common way to deal with problems that involve the solution of nonlinear differential equations is to try different ansätze which are either geometrically motivated or just chosen to simplify computations. Typical examples are the investigation of infinitesimal symmetries, the search for classes of integrating factors and related first integrals/conservation laws or the search for a variational principle equivalent to a given system of equations. In all these cases overdetermined systems of partial differential 1
Size Reduction and Partial Decoupling of Systems of Equations
 J. Symb. Comput
, 1999
"... A method is presented that reduces the number of terms of systems of linear equations (algebraic, ODEs or PDEs). As a byproduct these systems become partially decoupled. A variation of this method is applicable to nonlinear systems. Modifications to improve efficiency are given and examples are sho ..."
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Cited by 6 (6 self)
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A method is presented that reduces the number of terms of systems of linear equations (algebraic, ODEs or PDEs). As a byproduct these systems become partially decoupled. A variation of this method is applicable to nonlinear systems. Modifications to improve efficiency are given and examples are shown. This procedure can be used as a presimplification step of a computation of the radical of a differential ideal (pseudo differential Grobner basis). Algorithms for applying integrability conditions to a system of differential equations in a systematic way in order to generate simplified differential equations are implemented in a number of programs ([3, 4, 7, 8, 10, 11, 12] and more in [6]). Such calculations result in the radical or a (pseudo) differential Grobner Basis of the differential ideal generated by the original system. A common problem of these algorithms, and consequently their implementations, is an explosive expression swell. Optimizations like Buchbergers 2 nd criterion...
Probabilistic Algorithms for Computing Resolvent Representations of Regular Differential Ideals
, 2006
"... In a previous article [14], we proved the existence of resolvent representations for regular differential ideals. The present paper provides practical algorithms for computing such representations. We propose two different approaches. The first one uses differential characteristic decompositions whe ..."
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Cited by 2 (0 self)
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In a previous article [14], we proved the existence of resolvent representations for regular differential ideals. The present paper provides practical algorithms for computing such representations. We propose two different approaches. The first one uses differential characteristic decompositions whereas the second one proceeds by prolongation and algebraic elimination. Both constructions depend on the choice of a tuple over the differential base field and their success relies on the chosen tuple to be separating. The probabilistic aspect of the algorithms comes from this choice. To control it, we exhibit a family of tuples for which we can bound the probability that one of its element is separating.
State and Parameter Estimation of Partially Observed Linear Ordinary Differential Equations with Deterministic Optimal Control
"... Ordinary Differential Equations are a simple but powerful framework for modeling complex systems. Parameter estimation from times series can be done by Nonlinear Least Squares (or other classical approaches), but this can give unsatisfactory results because the inverse problem can be illposed, even ..."
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Ordinary Differential Equations are a simple but powerful framework for modeling complex systems. Parameter estimation from times series can be done by Nonlinear Least Squares (or other classical approaches), but this can give unsatisfactory results because the inverse problem can be illposed, even when the differential equation is linear. Following recent approaches that use approximate solutions of the ODE model, we propose a new method that converts parameter estimation into an optimal control problem: our objective is to determine a control and a parameter that are as close as possible to the data. We derive then a criterion that makes a balance between discrepancy with data and with the model, and we minimize it by using optimization in functions spaces: our approach is related to the socalled Deterministic Kalman Filtering, but different from the usual statistical Kalman filtering. We show the rootn consistency and asymptotic normality of the estimators for the parameter and for the states. Experiments in a toy model and in a real case shows that our approach is generally more accurate and more reliable than Nonlinear Least Squares and Generalized Smoothing, even in misspecified cases.