Numerical Treatment of Inverse Problems in Differential and Integral Equations

In many scientific or engineering applications, where ordinary differen­ tial equation (OOE),partial differential equation (POE), or integral equation (IE) models are involved, numerical simulation is in common use for prediction, monitoring, or control purposes. In many cases, however, successful simulation of a process must be preceded by the solution of the so-called inverse problem, which is usually more complex: given meas­ ured data and an associated theoretical model, determine unknown para­ meters in that model (or unknown functions to be parametrized) in such a way that some measure of the "discrepancy" between data and model is minimal. The present volume deals with the numerical treatment of such inverse probelms in fields of application like chemistry (Chap. 2,3,4, 7,9), molecular biology (Chap. 22), physics (Chap. 8,11,20), geophysics (Chap. 10,19), astronomy (Chap. 5), reservoir simulation (Chap. 15,16), elctrocardiology (Chap. 14), computer tomography (Chap. 21), and control system design (Chap. 12,13). In the actual computational solution of inverse problems in these fields, the following typical difficulties arise: (1) The evaluation of the sen­ sitivity coefficients for the model. may be rather time and storage con­ suming. Nevertheless these coefficients are needed (a) to ensure (local) uniqueness of the solution, (b) to estimate the accuracy of the obtained approximation of the solution, (c) to speed up the iterative solution of nonlinear problems. (2) Often the inverse problems are ill-posed. To cope with this fact in the presence of noisy or incomplete data or inev­ itable discretization errors, regularization techniques are necessary.

Contenu

I: Inverse Initial Value Problems in Ordinary Differential Equations.- 1. Smooth Numerical Solutions of Ordinary Differential Equation Systems.- 2. Towards Parameter Identification for Large Chemical Reaction Systems.- 3. Identification of Rate Constants in Bistable Chemical Reactions.- 4. New Methods of Parameter Identification in Kinetics of Closed and Open Reaction Systems.- 5. On the Estimation of Small Perturbations in Ordinary Differential Equations.- II: Inverse Boundary and Eigenvalue Problems in Ordinary Differential Equations.- 6. Multiple Shooting Techniques Revisited.- 7. Recent Advances in Parameteridentification Techniques for O.D.E..- 8. Some Examples of Parameter Estimation by Multiple Shooting.- 9. Unrestricted Harmonic Balance III, Application to Running and Standing Chemical Waves.- 10. Inverse Eigenvalue Problems for the Mantle.- 11. Inverse Problems of Quantal Potential Scattering at Fixed Energy.- 12. An Inverse Eigenvalue Problem from Control Theory.- 13. Numerical Methods for Robust Eigenstructure Assignment in Control System Design.- III: Inverse Problems in Partial Differential Equations.- 14. Some Inverse Problems in Electrocardiology.- 15. Determination of Coefficients in Reservoir Simulation.- 16. Identification of Nonlinear Soil Physical Parameters from an Input-Output Experiment.- 17. On an Inverse Non-Linear Diffusion Problem.- 18. The Numerical Solution of a Non-Characteristic Cauchy Problem for a Parabolic Equation.- 19. The Inverse Problem in Geoelectrical Prospecting Assuming a Horizontally Layered Half-Space.- 20. Two Dimensional Velocity Inversion for Acoustic Waves with Incomplete Information.- IV: Fredholm Integral Equations of the First Kind.- 21. Exploiting the Ranges of Radon Transforms in Tomography.- 22. Regularization Techniques for Inverse Problems in Molecular Biology.- 23. A Comparison of Statistical Regularization and Fourier Extrapolation Methods for Numerical Deconvolution.- 24. Deconvolution of Gaussian and Other Kernels.- 25. Regularization by Least-Squares Collocation.- Contributors.