Numerische Methoden zur Behandlung hochdimensionaler Aufgaben der Parameteridentifizierung. (Numerical methods for the treatment of high- dimensional parameter identification problems.). (German) Zbl 0639.65036

Schlöder, Johannes P.

Bonn. Math. Schr. 187, 150 p. (1988).

Problems of parameter estimation in ordinary differential equations, where data from different experimental situations (the dynamical behaviour, stable an \(y(a)=\alpha\) is equivalent to the integral equation \[ y(t)=\alpha +\int^{t}_{a}f(s,y(s))ds,\quad t\in [a,b]. \] The author formulates and analyzes integrand-approximation formulas, each of which can be written in the form \[ (2)\quad y_{\tau}(t)=\alpha +\int^{t}_{a}A_{\tau}[f(.,y_{\tau}(\cdot))](s)ds,\quad t\in [a,b], \] for the numerical solutions of the initial value problem (1). \(A_{\tau}[f(\cdot,y_{\tau}(\cdot))]\) (s) in (2) is an approximation to \(f(s,y_{\tau}(s))\), and the index \(\tau\) is associated with the particular stepsize and order sequence used in the numerical integration. Here \(A_{\tau}\) is a linear operator. The formulation (2) enables the author to apply many of the theorems for differential equations and the techniques of functional analysis to study the convergence of \(y_{\tau}\) to y.

Reviewer: N.Parhi

MSC:

65K10	Numerical optimization and variational techniques
65L10	Numerical solution of boundary value problems involving ordinary differential equations
34A55	Inverse problems involving ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
93B30	System identification

Keywords:

high-dimensional parameter identification problems; parameter estimation; convergence