An Algorithm for Converting Nonlinear Differential Equations to Integral Equations with an Application to Parameter Estimation from Noisy Data

“…Consider l the higher order derivative of y in (2). ∆R(y, u) denotes the functional determinant composed of the {m k (y, u)} 1≤k≤q and given by the Wronskian [4] (1) . .…”

Section: Problem Formulationmentioning

confidence: 99%

“…Theorem 1: Assume that the functional determinant ∆R(y, u) is not identically equal to zero 1 . Consider P * a connected subset of U P .…”

Section: Problem Formulationmentioning

confidence: 99%

“…From a model and an elimination order, the algebra elimination theory permits to eliminate specific variables as unknown ones in favor of inputs and outputs leading to differential polynomials called input-output polynomials ( [2], [5]). The way to exploit them has resulted in several papers ( [1], [3], [6], [14], [19], [22], [20]). Their use goes from identifiability analysis of the model to parameter estimation.…”

Section: Introductionmentioning

confidence: 99%

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Improvements in bounded error parameter estimation using distribution theory

Verdière

Jauberthie

Travé-Massuyès

2018

2018 European Control Conference (ECC)

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A bounded error estimation procedure based on integro-differential polynomials linking the inputs, the outputs and the parameters of the model is presented in this paper. These polynomials are obtained from differential algebra tools given input-output polynomials. The use of the distribution theory permits to obtain new relations in which the order of derivatives of the model outputs are smaller. This method is applied on the water tank example and the results are compared with the classical method based on the simple use of inputoutput polynomials. As it will be seen, this method significantly improves the parameter estimation results.

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“…The first algorithmic approach to the problem of integrating expressions with unspecified functions was proposed in [20], and independently for differential polynomials in [21]. This was generalized recently to integro-differential polynomials [22], [23], to differential fields [24], [25], and to fractions of differential polynomials [26], [27].…”

Section: B Integrationmentioning

confidence: 99%

Symbolic Derivation of Mean-Field PDEs from Lattice-Based Models

Koutschan

Ranetbauer

Regensburger

et al. 2015

2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)

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Abstract-Transportation processes, which play a prominent role in the life and social sciences, are typically described by discrete models on lattices. For studying their dynamics a continuous formulation of the problem via partial differential equations (PDE) is employed. In this paper we propose a symbolic computation approach to derive mean-field PDEs from a latticebased model. We start with the microscopic equations, which state the probability to find a particle at a given lattice site. Then the PDEs are formally derived by Taylor expansions of the probability densities and by passing to an appropriate limit as the time steps and the distances between lattice sites tend to zero. We present an implementation in a computer algebra system that performs this transition for a general class of models. In order to rewrite the mean-field PDEs in a conservative formulation, we adapt and implement symbolic integration methods that can handle unspecified functions in several variables. To illustrate our approach, we consider an application in crowd motion analysis where the dynamics of bidirectional flows are studied. However, the presented approach can be applied to various transportation processes of multiple species with variable size in any dimension, for example, to confirm several proposed mean-field models for cell motility.

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An Algorithm for Converting Nonlinear Differential Equations to Integral Equations with an Application to Parameter Estimation from Noisy Data

Cited by 8 publications

References 16 publications

Symbolic-Numeric Methods for Nonlinear Integro-Differential Modeling

Symbolic-Numeric Methods for Nonlinear Integro-Differential Modeling

Improvements in bounded error parameter estimation using distribution theory

Symbolic Derivation of Mean-Field PDEs from Lattice-Based Models

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