published_or_final_version / Mathematics / Master / Master of Philosophy

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In this work we study the following problem on constrained approximation. Let f be a continuous mapping defined on a bounded domain with piecewise smooth boundary in R^n and taking values in R^n. What are necessary and sufficient conditions for f to be uniformly approximable by C^1-smooth mappings with nonnegative Jacobian? When the dimension is equal to one, this is just approximation by monotone smooth functions. Hence, the necessary and sufficient condition is: the function is monotone. On the other hand, for higher dimensions the description is not as clear. We give a simple necessary condition in terms of the topological degree of continuous mapping. We also give some sufficient conditions for dimension 2. It also turns out that if the dimension is greater than one, then there exist real-analytic mappings with nonnegative Jacobian that cannot be approximated by smooth mappings with positive Jacobian. In our study of the above mentioned question we use topological degree theory, Schoenflies-type extension theorems, and Stoilow's topological characterization of complex analytic functions.

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We consider the task of computing reliable numerical approximations of the solutions of elliptic equations and systems where the coefficients vary discontinuously, rapidly, and by large orders of magnitude. Such problems, which occur in diffusion and in linear elastic deformation of composite materials, have solutions with low regularity with the result that reliable numerical approximations can be found only in approximating spaces, invariably with high dimension, that can accurately represent the large and rapid changes occurring in the solution. The use of the Galerkin approach with such high dimensional approximating spaces often leads to very large scale discrete problems which at best can only be solved using efficient solvers. However, even then, their scale is sometimes so large that the Galerkin approach becomes impractical and alternative methods of approximation must be sought. In this thesis we adopt two approaches. We propose a new asymptotic method of approximation for problems of diffusion in materials with periodic structure. This approach uses Fourier series expansions and enables one to perform all computations on a periodic cell; this overcomes the difficulty caused by the rapid variation of the coefficients. In the one dimensional case we have constructed problems with discontinuous coefficients and computed the analytical expressions for their solutions and the proposed asymptotic approximations. The rates at which the given asymptotic approximations converge, as the period of the material decreases, are obtained through extensive computational tests which show that these rates are fundamentally dependent on the level of regularity of the right hand sides of the equations. In the two dimensional case we show how one can use the Galerkin method to approximate the solutions of the problems associated with the periodic cell. We construct problems with discontinuous coefficients and perform extensive computational tests which show that the asymptotic properties of the approximations are identical to those observed in the one dimensional case. However, the computational results show that the application of the Galerkin method of approximation introduces a discretization error which can obscure the precise asymptotic rate of convergence for low regularity right hand sides. For problems of two dimensional linear elasticity we are forced to consider an alternative approach. We use domain decomposition techniques that interface the subdomains with conjugate gradient methods and obtain algorithms which can be efficiently implemented on computers with parallel architectures. We construct the balancing preconditioner, M,, and show that it has the optimal conditioning property k(Mh(^-1)Sh) =< C(1 + log(H/h))^2 where Sh is the discretized Steklov—Poincaré operator, C> 0 is a constant which is independent of the magnitude of the material discontinuities, H is the maximum subdomain diameter, and h is the maximum finite element diameter. These properties of the preconditioning operator Mh allow one to use the computational power of a parallel computer to overcome the difficulties caused by the changing form of the solution of the problem. We have implemented this approach for a variety of problems of planar linear elasticity and, using different domain decompositions, approximating spaces, and materials, find that the algorithm is robust and scales with the dimension of the approximating space and the number of subdomains according to the condition number bound above and is unaffected by material discontinuities. In this we have proposed and implemented new inner product expressions which we use to modify the bilinear forms associated with problems over subdomains that have pure traction boundary conditions.

In this dissertation, we consider the problem of piecewise polynomial approximation of functions over sets of triangulations. Recently developed adaptive methods, where the hierarchy of triangulations is not fixed in advance and depends on the local properties of the function, have received considerable attention. The quick development of these adaptive methods has been due to the discovery of the wavelet transform in the 1960's, probably the best tool for image coding. Since the mid 80's, there have been many attempts to design `Second Generation' adaptive techniques that particularly take into account the geometry of edge singularities of an image. But it turned out that almost none of the proposed `Second Generation' approaches are competitive with wavelet coding. Nevertheless, there are instances that show deficiencies in the wavelet algorithms. The method suggested in this dissertation incorporates the geometric properties of convex sets in the construction of adaptive triangulations of an image. The proposed algorithm provides a nearly optimal order of approximation for cartoon images of convex sets, and is based on the idea that the location of the centroid of certain types of domains provides a sufficient amount of information to construct a 'good' approximation of the boundaries of those domains. Along with the theoretical analysis of the algorithm, a Matlab code has been developed and implemented on some simple cartoon images.

In this work we study the following problem on constrained approximation. Let f be a continuous mapping defined on a bounded domain with piecewise smooth boundary in R^n and taking values in R^n. What are necessary and sufficient conditions for f to be uniformly approximable by C^1-smooth mappings with nonnegative Jacobian? When the dimension is equal to one, this is just approximation by monotone smooth functions. Hence, the necessary and sufficient condition is: the function is monotone. On the other hand, for higher dimensions the description is not as clear. We give a simple necessary condition in terms of the topological degree of continuous mapping. We also give some sufficient conditions for dimension 2. It also turns out that if the dimension is greater than one, then there exist real-analytic mappings with nonnegative Jacobian that cannot be approximated by smooth mappings with positive Jacobian. In our study of the above mentioned question we use topological degree theory, Schoenflies-type extension theorems, and Stoilow's topological characterization of complex analytic functions.

In modern day engineering, the designer has become more and more dependent on computer simulation. Oftentimes, computational cost and convergence accuracy accompany these simulations to reach global solutions for engineering design problems causes traditional optimization techniques to perform poorly. To overcome these issues nontraditional optimization algorithms based region elimination and space exploration are introduced. Approximation models, which are also known as metamodels or surrogate models, are used to explore and give more information about the design space that needs to be explored. Usually the approximation models are constructed in the promising regions where global solutions are expected to exist. The approximation models imitate the original expensive function, black-box function, and contribute towards getting comparably acceptable solutions with fewer resources and at low computation cost. The primary contributions of this dissertation are associated with the development of new methods for exploring the design space for large scale computer simulations. Primarily, the proposed design space exploration procedure uses a hierarchical partitioning method to help mitigate the curse of dimensionality often associated with the analysis of large scale systems. The research presented in this dissertation focuses on introducing new optimization algorithms based on metamodeling techniques that alleviate the burden of the computation cost associated with complex engineering design problems. Three new global optimization algorithms were introduced in this dissertation, Approximated Unimodal Region Elimination (AUMRE), Space Exploration and Unimodal Region Elimination (SEUMRE), and Mixed Surrogate Space Exploration (MSSE) for computation intensive and black-box engineering design optimization problems. In these algorithms, the design space was divided into many subspaces and the search was focused on the most promising regions to reach global solutions with the resources available and with less computation cost. Metamodeling techniques such as Response Surface Method (RSM), Radial Basis Function (RBF), and Kriging (KRG) are introduced and used in this work. RSM has been used because of its advantages such as being easy to construct, understand and implement. Also due to its smoothing capability, it allows quick convergence of noisy functions in the optimization. RBF has the advantage of smoothing data and interpolating them. KRG metamodels can provide accurate predictions of highly nonlinear or irregular behaviours. These features in metamodeling techniques have contributed largely towards obtaining comparably accurate global solutions besides reducing the computation cost and resources. Many multi-objective optimization algorithms, specifically those used for engineering problems and applications involve expensive fitness evaluations. In this dissertation, a new multi-objective global optimization algorithm for black-box functions is also introduced and tested on benchmark test problems and real life engineering applications. Finally, the new proposed global optimization algorithms were tested using benchmark global optimization test problems to reveal their pros and cons. A comparison with other well known and recently introduced global optimization algorithms were carried out to highlight the proposed methods’ advantages and strength points. In addition, a number of practical examples of global optimization in industrial designs were used and optimized to further test these new algorithms. These practical examples include the design optimization of automotive Magnetorheological Brake Design and the design optimization of two-mode hybrid powertrains for new hybrid vehicles. It is shown that the proposed optimization algorithms based on metamodeling techniques comparably provide global solutions with the added benefits of fewer function calls and the ability to efficiently visualize the design space. / Graduate

Thesis (Ph. D.)--Oregon State University, 2009. / Printout. Includes bibliographical references (leaves 62-64). Also available on the World Wide Web.

Zugl.: Passau, Univ., Diss., 2007