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Nonlocal operators are used in a wide variety of models and applications due to many natural phenomena being driven by nonlocal dynamics. Nonlocal operators are integral operators allowing for interactions between two distinct points in space. The nonlocal models investigated in this thesis involve kernels that are assumed to have a finite range of nonlocal interactions. Kernels of this type are used in nonlocal elasticity and convection-diffusion models as well as finance and image analysis. Also within the mathematical theory they arouse great interest, as they are asymptotically related to fractional and classical differential equations.
The results in this thesis can be grouped according to the following three aspects: modeling and analysis, discretization and optimization.
Mathematical models demonstrate their true usefulness when put into numerical practice. For computational purposes, it is important that the support of the kernel is clearly determined. Therefore nonlocal interactions are typically assumed to occur within an Euclidean ball of finite radius. In this thesis we consider more general interaction sets including norm induced balls as special cases and extend established results about well-posedness and asymptotic limits.
The discretization of integral equations is a challenging endeavor. Especially kernels which are truncated by Euclidean balls require carefully designed quadrature rules for the implementation of efficient finite element codes. In this thesis we investigate the computational benefits of polyhedral interaction sets as well as geometrically approximated interaction sets. In addition to that we outline the computational advantages of sufficiently structured problem settings.
Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface-dependent kernels. We derive the shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques.
The dissertation deals with methods to improve design-based and model-assisted estimation techniques for surveys in a finite population framework. The focus is on the development of the statistical methodology as well as their implementation by means of tailor-made numerical optimization strategies. In that regard, the developed methods aim at computing statistics for several potentially conflicting variables of interest at aggregated and disaggregated levels of the population on the basis of one single survey. The work can be divided into two main research questions, which are briefly explained in the following sections.
First, an optimal multivariate allocation method is developed taking into account several stratification levels. This approach results in a multi-objective optimization problem due to the simultaneous consideration of several variables of interest. In preparation for the numerical solution, several scalarization and standardization techniques are presented, which represent the different preferences of potential users. In addition, it is shown that by solving the problem scalarized with a weighted sum for all combinations of weights, the entire Pareto frontier of the original problem can be generated. By exploiting the special structure of the problem, the scalarized problems can be efficiently solved by a semismooth Newton method. In order to apply this numerical method to other scalarization techniques as well, an alternative approach is suggested, which traces the problem back to the weighted sum case. To address regional estimation quality requirements at multiple stratification levels, the potential use of upper bounds for regional variances is integrated into the method. In addition to restrictions on regional estimates, the method enables the consideration of box-constraints for the stratum-specific sample sizes, allowing minimum and maximum stratum-specific sampling fractions to be defined.
In addition to the allocation method, a generalized calibration method is developed, which is supposed to achieve coherent and efficient estimates at different stratification levels. The developed calibration method takes into account a very large number of benchmarks at different stratification levels, which may be obtained from different sources such as registers, paradata or other surveys using different estimation techniques. In order to incorporate the heterogeneous quality and the multitude of benchmarks, a relaxation of selected benchmarks is proposed. In that regard, predefined tolerances are assigned to problematic benchmarks at low aggregation levels in order to avoid an exact fulfillment. In addition, the generalized calibration method allows the use of box-constraints for the correction weights in order to avoid an extremely high variation of the weights. Furthermore, a variance estimation by means of a rescaling bootstrap is presented.
Both developed methods are analyzed and compared with existing methods in extensive simulation studies on the basis of a realistic synthetic data set of all households in Germany. Due to the similar requirements and objectives, both methods can be successively applied to a single survey in order to combine their efficiency advantages. In addition, both methods can be solved in a time-efficient manner using very comparable optimization approaches. These are based on transformations of the optimality conditions. The dimension of the resulting system of equations is ultimately independent of the dimension of the original problem, which enables the application even for very large problem instances.
Properties Evaluation of Composite Materials Based on Gypsum Plaster and Posidonia Oceanica Fibers
(2023)
Estimating the amount of material without significant losses at the end of hybrid casting is a problem addressed in this study. To minimize manufacturing costs and improve the accuracy of results, a correction factor (CF) was used in the formula to estimate the volume percent of the material in order to reduce material losses during the sample manufacturing stage, allowing for greater confidence between the approved blending plan and the results obtained. In this context, three material mixing schemes of different sizes and shapes (gypsum plaster, sand (0/2), gravel (2/4), and Posidonia oceanica fibers (PO)) were created to verify the efficiency of CF and more precisely study the physico-mechanical effects on the samples. The results show that the use of a CF can reduce mixing loss to almost 0%. The optimal compressive strength of the sample (S1B) with the lowest mixing loss was 7.50 MPa. Under optimal conditions, the addition of PO improves mix volume percent correction (negligible), flexural strength (5.45%), density (18%), and porosity (3.70%) compared with S1B. On the other hand, the addition of PO thermo-chemical treatment by NaOH increases the compressive strength (3.97%) compared with PO due to the removal of impurities on the fiber surface, as shown by scanning electron microscopy. We then determined the optimal mixture ratio (PO divided by a mixture of plaster, sand, and gravel), which equals 0.0321 because Tunisian gypsum contains small amounts of bassanite and calcite, as shown by the X-ray diffraction results.
Modern decision making in the digital age is highly driven by the massive amount of
data collected from different technologies and thus affects both individuals as well as
economic businesses. The benefit of using these data and turning them into knowledge
requires appropriate statistical models that describe the underlying observations well.
Imposing a certain parametric statistical model goes along with the need of finding
optimal parameters such that the model describes the data best. This often results in
challenging mathematical optimization problems with respect to the model’s parameters
which potentially involve covariance matrices. Positive definiteness of covariance matrices
is required for many advanced statistical models and these constraints must be imposed
for standard Euclidean nonlinear optimization methods which often results in a high
computational effort. As Riemannian optimization techniques proved efficient to handle
difficult matrix-valued geometric constraints, we consider optimization over the manifold
of positive definite matrices to estimate parameters of statistical models. The statistical
models treated in this thesis assume that the underlying data sets used for parameter
fitting have a clustering structure which results in complex optimization problems. This
motivates to use the intrinsic geometric structure of the parameter space. In this thesis,
we analyze the appropriateness of Riemannian optimization over the manifold of positive
definite matrices on two advanced statistical models. We establish important problem-
specific Riemannian characteristics of the two problems and demonstrate the importance
of exploiting the Riemannian geometry of covariance matrices based on numerical studies.
Data used for the purpose of machine learning are often erroneous. In this thesis, p-quasinorms (p<1) are employed as loss functions in order to increase the robustness of training algorithms for artificial neural networks. Numerical issues arising from these loss functions are addressed via enhanced optimization algorithms (proximal point methods; Frank-Wolfe methods) based on the (non-monotonic) Armijo-rule. Numerical experiments comprising 1100 test problems confirm the effectiveness of the approach. Depending on the parametrization, an average reduction of the absolute residuals of up to 64.6% is achieved (aggregated over 100 test problems).
Copositive programming is concerned with the problem of optimizing a linear function over the copositive cone, or its dual, the completely positive cone. It is an active field of research and has received a growing amount of attention in recent years. This is because many combinatorial as well as quadratic problems can be formulated as copositive optimization problems. The complexity of these problems is then moved entirely to the cone constraint, showing that general copositive programs are hard to solve. A better understanding of the copositive and the completely positive cone can therefore help in solving (certain classes of) quadratic problems. In this thesis, several aspects of copositive programming are considered. We start by studying the problem of computing the projection of a given matrix onto the copositive and the completely positive cone. These projections can be used to compute factorizations of completely positive matrices. As a second application, we use them to construct cutting planes to separate a matrix from the completely positive cone. Besides the cuts based on copositive projections, we will study another approach to separate a triangle-free doubly nonnegative matrix from the completely positive cone. A special focus is on copositive and completely positive programs that arise as reformulations of quadratic optimization problems. Among those we start by studying the standard quadratic optimization problem. We will show that for several classes of objective functions, the relaxation resulting from replacing the copositive or the completely positive cone in the conic reformulation by a tractable cone is exact. Based on these results, we develop two algorithms for solving standard quadratic optimization problems and discuss numerical results. The methods presented cannot immediately be adapted to general quadratic optimization problems. This is illustrated with examples.
Krylov subspace methods are often used to solve large-scale linear equations arising from optimization problems involving partial differential equations (PDEs). Appropriate preconditioning is vital for designing efficient iterative solvers of this type. This research consists of two parts. In the first part, we compare two different kinds of preconditioners for a conjugate gradient (CG) solver attacking one partial integro-differential equation (PIDE) in finance, both theoretically and numerically. An analysis on mesh independence and rate of convergence of the CG solver is included. The knowledge of preconditioning the PIDE is applied to a relevant optimization problem. The second part aims at developing a new preconditioning technique by embedding reduced order models of nonlinear PDEs, which are generated by proper orthogonal decomposition (POD), into deflated Krylov subspace algorithms in solving corresponding optimization problems. Numerical results are reported for a series of test problems.