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Finite difference methods, such as the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial differential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in… (More)
SUMMARY For the approximate solution of ill-posed inverse problems, the formulation of a regularization functional involves two separate decisions: the choice of the residual minimizer and the choice of the regularizor. In this paper, the Kullback–Leibler functional is used for both. The resulting regularization method can solve problems for which the… (More)
The goal of this chapter is to describe in simple terms how the use of ordinary differential equation (ODE) modeling, in conjunction with experimentation, can be utilized to improve our understanding of the dynamics of gene silencing and virus resistance in plants.
When deriving rates of convergence for the approximations generated by the application of Tikhonov regularization to ill–posed operator equations , assumptions must be made about the nature of the stabilization (i.e., the choice of the seminorm in the Tikhonov regularization) and the regularity of the least squares solutions which one looks for. In fact, it… (More)
Because of their causal structure, (convolution) Volterra integral equations arise as models in a variety of real-world situations including rheological stress-strain analysis, population dynamics and insurance risk prediction. In such practical situations, often only an approximation for the kernel is available. Consequently, a key aspect in the analysis… (More)
In the modelling of genetic signalling, communication and switching (GSCS), there is a need to identify the various mechanistic models, which nature has discovered, in terms of simple positional information rules (Wolpert (1969a)). The discovery of such simple rules, however, is a highly non-trivial process; in part, because of the complexity of the… (More)