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This diploma thesis deals with the issue of tourism in the microregion Kahan. Theoretical part of the thesis described the issue of tourism, forms of tourism and the state tourism policy. In another part of the diploma thesis was mentioned the basic characteristic of the microregion, the localization and realization conditions and the publicity of the microregion. In addition, a questionnaire was drawn up, in order to find out the visitors preferences and the swot analysis was created. Afterwards, were defined the proposals that were based on results of swot analysis and on results of questionnaire and that lead to additional development of tourism in microregion.

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Iterative Krylov subspace methods have proven to be efficient tools for solving linear systems of equations. In the context of ill-posed inverse problems, they tend to exhibit semiconvergence behavior making it difficult detect ``inverted noise" and stop iterations before solutions become contaminated. Regularization methods such as spectral filtering methods use the singular value decomposition (SVD) and are effective at filtering inverted noise from solutions, but are computationally prohibitive on large problems. Hybrid methods apply regularization techniques to the smaller ``projected problem" that is inherent to iterative Krylov methods at each iteration, thereby overcoming the semiconvergence behavior. Commonly, the Golub-Kahan bidiagonalization is used to construct a set of orthonormal basis vectors that span the Krylov subspaces from which solutions will be chosen, but seeking a solution in the orthonormal basis generated by the Arnoldi process (which is fundamental to the popular iterative method GMRES) has been of renewed interest recently. We discuss some of the positive and negative aspects of each process and use example problems to examine some qualities of the bases they produce. Computing optimal solutions in a given basis gives some insight into the performance of the corresponding iterative methods and how hybrid methods can contribute. / Master of Science

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Spectral methods have become increasingly popular in designing fast algorithms for modern highdimensional datasets. This thesis looks at several problems in which spectral methods play a central role. In some cases, we also show that such procedures have essentially the best performance among all randomised polynomial time algorithms by exhibiting statistical and computational trade-offs in those problems. In the first chapter, we prove a useful variant of the well-known Davis{Kahan theorem, which is a spectral perturbation result that allows us to bound of the distance between population eigenspaces and their sample versions. We then propose a semi-definite programming algorithm for the sparse principal component analysis (PCA) problem, and analyse its theoretical performance using the perturbation bounds we derived earlier. It turns out that the parameter regime in which our estimator is consistent is strictly smaller than the consistency regime of a minimax optimal (yet computationally intractable) estimator. We show through reduction from a well-known hard problem in computational complexity theory that the difference in consistency regimes is unavoidable for any randomised polynomial time estimator, hence revealing subtle statistical and computational trade-offs in this problem. Such computational trade-offs also exist in the problem of restricted isometry certification. Certifiers for restricted isometry properties can be used to construct design matrices for sparse linear regression problems. Similar to the sparse PCA problem, we show that there is also an intrinsic gap between the class of matrices certifiable using unrestricted algorithms and using polynomial time algorithms. Finally, we consider the problem of high-dimensional changepoint estimation, where we estimate the time of change in the mean of a high-dimensional time series with piecewise constant mean structure. Motivated by real world applications, we assume that changes only occur in a sparse subset of all coordinates. We apply a variant of the semi-definite programming algorithm in sparse PCA to aggregate the signals across different coordinates in a near optimal way so as to estimate the changepoint location as accurately as possible. Our statistical procedure shows superior performance compared to existing methods in this problem.

Title: Regularization Techniques Based on the Least Squares Method Author: Marie Michenková Department: Department of Numerical Mathematics Supervisor: RNDr. Iveta Hnětynková, Ph.D. Abstract: In this thesis we consider a linear inverse problem Ax ≈ b, where A is a linear operator with smoothing property and b represents an observation vector polluted by unknown noise. It was shown in [Hnětynková, Plešinger, Strakoš, 2009] that high-frequency noise reveals during the Golub-Kahan iterative bidiagonalization in the left bidiagonalization vectors. We propose a method that identifies the iteration with maximal noise revealing and reduces a portion of high-frequency noise in the data by subtracting the corresponding (properly scaled) left bidiagonalization vector from b. This method is tested for different types of noise. Further, Hnětynková, Plešinger, and Strakoš provided an estimator of the noise level in the data. We propose a modification of this estimator based on the knowledge of the point of noise revealing. Keywords: ill-posed problems, regularization, Golub-Kahan iterative bidiagonalization, noise revealing, noise estimate, denoising 1

In this work we study the solution of linear approximation problems with multiple observations. Particulary we focus on the total least squares method, which belogs to the class of ortogonaly invariant problems. For these problems we describe the so called core reduction. The aim is to reduce dimesions of the problem while preserving the solution, if it exists. We present two ways of constructing core problems. One is based on the singular value decomposition and the other uses the generalized Golub-Kahan iterative bidiago- nalization. Further we investigate properties of the core problem and of the methods for its construction. Finally we preform numerical experiments in the Matlab enviroment in order to test the reliability of the discussed algorithms. 1

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For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems and medical imaging problems such as greenhouse gas tracking and computational tomography reconstruction. This dissertation describes advancements in computational tools for solving large-scale inverse problems and for uncertainty quantification. Oftentimes, inverse problems are ill-posed and large-scale. Iterative projection methods have dramatically reduced the computational costs of solving large-scale inverse problems, and regularization methods have been critical in obtaining stable estimations by applying prior information of unknowns via Bayesian inference. However, by combining iterative projection methods and variational regularization methods, hybrid projection approaches, in particular generalized hybrid methods, create a powerful framework that can maximize the benefits of each method. In this dissertation, we describe various advancements and extensions of hybrid projection methods that we developed to address three recent open problems. First, we develop hybrid projection methods that incorporate mixed Gaussian priors, where we seek more sophisticated estimations where the unknowns can be treated as random variables from a mixture of distributions. Second, we describe hybrid projection methods for mean estimation in a hierarchical Bayesian approach. By including more than one prior covariance matrix (e.g., mixed Gaussian priors) or estimating unknowns and hyper-parameters simultaneously (e.g., hierarchical Gaussian priors), we show that better estimations can be obtained. Third, we develop computational tools for a respirometry system that incorporate various regularization methods for both linear and nonlinear respirometry inversions. For the nonlinear systems, blind deconvolution methods are developed and prior knowledge of nonlinear parameters are used to reduce the dimension of the nonlinear systems. Simulated and real-data experiments of the respirometry problems are provided. This dissertation provides advanced tools for computational inversion and uncertainty quantification. / Doctor of Philosophy / For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems such as greenhouse gas tracking where the problem of estimating the amount of added or removed greenhouse gas at the atmosphere gets more difficult. The number of observations has been increased with improvements in measurement technologies (e.g., satellite). Therefore, the inverse problems become large-scale and they are computationally hard to solve. Another example of an inverse problem arises in tomography, where the goal is to examine materials deep underground (e.g., to look for gas or oil) or reconstruct an image of the interior of the human body from exterior measurements (e.g., to look for tumors). For tomography applications, there are typically fewer measurements than unknowns, which results in non-unique solutions. In this dissertation, we treat unknowns as random variables with prior probability distributions in order to compensate for a deficiency in measurements. We consider various additional assumptions on the prior distribution and develop efficient and robust numerical methods for solving inverse problems and for performing uncertainty quantification. We apply our developed methods to many numerical applications such as greenhouse gas tracking, seismic tomography, spherical tomography problems, and the estimation of CO2 of living organisms.