
Hybrid Methods for Nonlinear Least Squares Problems
Lukšan, Ladislav ; Matonoha, Ctirad ; Vlček, Jan
This contribution contains a description and analysis of effective methods for minimization of the nonlinear least squares function F(x) = (1=2)fT (x)f(x), where x ∈ Rn and f ∈ Rm, together with extensive computational tests and comparisons of the introduced methods. All hybrid methods are described in detail and their global convergence is proved in a unified way. Some proofs concerning trust region methods, which are difficult to find in the literature, are also added. In particular, the report contains an analysis of a new simple hybrid method with Jacobian corrections (Section 8) and an investigation of the simple hybrid method for sparse least squares problems proposed previously in [33] (Section 14).
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On the Optimal Initial Conditions for an Inverse Problem of Model Parameter Estimation  a Complementarity Principle
Matonoha, Ctirad ; Papáček, Š.
This contribution represents an extension of our earlier studies on the paradigmatic example of the inverse problem of the diffusion parameter estimation from spatiotemporal measurements of fluorescent particle concentration, see [6, 1, 3, 4, 5]. More precisely, we continue to look for an optimal bleaching pattern used in FRAP (Fluorescence Recovery After Photobleaching), being the initial condition of the Fickian diffusion equation maximizing a sensitivity measure. As follows, we define an optimization problem and we show the special feature (socalled complementarity principle) of the optimal binaryvalued initial conditions.

 
 

Powers of interval matrices
Říha, David ; Hartman, David (advisor) ; Matonoha, Ctirad (referee)
The aim of this thesis is to analyse methods of how to calculate the interval enclosure of interval matrix powers, investigate special cases where exponentiation is easier than in the general case and those methods implement to software MATLAB. In the thesis will be introduced two algorithms for calculations of interval enclosure of general interval matrix. First uses spectral decomposition, thus the decomposition based on eigenvalues and eigenvectors and the second one is based on well known binary exponentiation. Special cases include for example nonnegative interval matrices or cube power of diagonally interval matrices. For researched methods, the theory on which they are built, are explained and the methods themselves are described both verbally and by code. At the end is done the testing of quality for the interval enclosures and time complexity of calculations.

 
 

On the Optimization of Initial Conditions for a Model Parameter Estimation
Matonoha, Ctirad ; Papáček, Š. ; Kindermann, S.
The design of an experiment, e.g., the setting of initial conditions, strongly influences the accuracy of the process of determining model parameters from data. The key concept relies on the analysis of the sensitivity of the measured output with respect to the model parameters. Based on this approach we optimize an experimental design factor, the initial condition for an inverse problem of a model parameter estimation. Our approach, although case independent, is illustrated at the FRAP (Fluorescence Recovery After Photobleaching) experimental technique. The core idea resides in the maximization of a sensitivity measure, which depends on the initial condition. Numerical experiments show that the discretized optimal initial condition attains only two values. The number of jumps between these values is inversely proportional to the value of a diffusion coefficient D (characterizing the biophysical and numerical process). The smaller value of D is, the larger number of jumps occurs.

 
 