24 research outputs found
Guaranteed optimal reachability control of reaction-diffusion equations using one-sided Lipschitz constants and model reduction
We show that, for any spatially discretized system of reaction-diffusion, the
approximate solution given by the explicit Euler time-discretization scheme
converges to the exact time-continuous solution, provided that diffusion
coefficient be sufficiently large. By "sufficiently large", we mean that the
diffusion coefficient value makes the one-sided Lipschitz constant of the
reaction-diffusion system negative. We apply this result to solve a finite
horizon control problem for a 1D reaction-diffusion example. We also explain
how to perform model reduction in order to improve the efficiency of the
method
Verified global optimization for estimating the parameters of nonlinear models
Nonlinear parameter estimation is usually achieved via the minimization of some possibly non-convex cost function. Interval analysis allows one to derive algorithms for the guaranteed characterization of the set of all global minimizers of such a cost function when an explicit expression for the output of the model is available or when this output is obtained via the numerical solution of a set of ordinary differential equations. However, cost functions involved in parameter estimation are usually challenging for interval techniques, if only because of multi-occurrences of the parameters in the formal expression of the cost. This paper addresses parameter estimation via the verified global optimization of quadratic cost functions. It introduces tools for the minimization of generic cost functions. When an explicit expression of the output of the parametric model is available, significant improvements may be obtained by a new box exclusion test and by careful manipulations of the quadratic cost function. When the model is described by ODEs, some of the techniques available in the previous case may still be employed, provided that sensitivity functions of the model output with respect to the parameters are available
Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs
Guaranteed Error Bounds on Approximate Model Abstractions Through Reachability Analysis
It is well known that exact notions of model abstraction and reduction
for dynamical systems may not be robust enough in practice because they are
highly sensitive to the specific choice of parameters. In this paper we consider this
problem for nonlinear ordinary differential equations (ODEs) with polynomial
derivatives. We introduce approximate differential equivalence as a more permissive
variant of a recently developed exact counterpart, allowing ODE variables
to be related even when they are governed by nearby derivatives. We develop
algorithms to (i) compute the largest approximate differential equivalence; (ii)
construct an approximate quotient model from the original one via an appropriate
parameter perturbation; and (iii) provide a formal certificate on the quality of
the approximation as an error bound, computed as an over-approximation of the
reachable set of the perturbed model. Finally, we apply approximate differential
equivalences to study the effect of parametric tolerances in models of symmetric
electric circuits
A generic and efficient Taylor series based continuation method using a quadratic recast of smooth nonlinear systems
International audienceThis paper is concerned with a Taylor series based continuation algorithm, ie, the so-called Asymptotic Numerical Method (ANM). It describes a generic continuation procedure that apply the ANM principle at best, that is to say, that presents a high level of genericity without paying the price of this genericity by low computing performances. The way to quadratically recast a system of equation is now part of the method itself, and the way to handle elementary transcendental function is detailed with great attention. A sparse tensorial formalism is introduced for the internal representation of the system, which, when combines with a block condensation technique, provides a good computational efficiency of the ANM. Three examples are developed to show the performance and the versatility of the implementation of the continuation tool. Its robustness and its accuracy are explored. Finally, the potentiality of this method for complex non linear finite element analysis is enlightened by treating 2D elasticity problem with geometrical nonlinearities