\input zb-basic \input zb-ioport \iteman{io-port 02243098} \itemau{Klimke, W. Andreas} \itemti{Uncertainty modeling using fuzzy arithmetic and sparse grids.} \itemso{Industriemathematik und Angewandte Mathematik. Aachen: Shaker Verlag; Stuttgart: Univ. Stuttgart, Institut f\"ur Angewandte Analysis und Numerische Simulation (Diss.) (ISBN 3-8322-4766-1/pbk). xii, 124~p. EUR~45.80 (2006).} \itemab Summary: In today's thorough understanding of physical, economic, and other complex systems, developing mathematical models and performing numerical simulations play a key role. With increasing capabilities of modern computers, the models are becoming more sophisticated and realistic. This not only has raised our expectations of solving practical problems more and more accurately, but has also resulted in the need to gather much more detailed data. However, one observes that an often large part of the required data cannot be obtainned precisely, but rather exhibits some degree of uncertainty. Various sources of uncertainty are encountered in this context. Physical systems, for instance, are based on model parameters that are usually obtained via measurements, which cannot be exact by nature. It is thus recognized that quantifying, analyzing, and controlling uncertainty is essential to more reliable and more realistic results in modeling and computer simulation. Unfortunately, methods to treat uncertainty require extensive computational resources. Monte Carlo methods, for example, require a large number of samples to achieve good results. Obtaining not only probabilistic but possibilistic results (i.e., the worst-case bounds of the results for a given set of uncertain parameters) requires the solution of global optimization problems, where each uncertain parameter becomes an independent variable of the problem. Solving a multivariate global optimization is mostly an NP-hard problem in which the complexity often grows exponentially with the problem dimension. Therefore, specifically problems with a large number of uncertain parameters, i.e., high-dimensional problems, pose an enormous challenge to current research. In this thesis, the uncertain parameters are modeled using fuzzy set theory, which represents a well-studied approach to uncertainty management that places emphasis on gradations of possibility rather than on randomness and probability, such as stochastic methods. Computing with fuzzy sets is performed with the so-called extension principle that can extend any real-valued function to a function of fuzzy numbers. As the main contribution of this work, it is shown how sparse grid interpolation can be used in the fuzzy set-based uncertainty modeling process. The most important property of sparse grid interpolation methods is the fact that they scale well to higher-dimensional problems. Increasingly accurate, error-controlled interpolants are constructed adaptively from mere model evaluations up to a desired estimated accuracy or up to a maximum number of function evaluations. These interpolants are then employed as surrogate functions in the computation of the extension principle, which requires multiple global optimization problems to be solved. Applications in dynamic systems modeling and vibration engineering illustrate the wide potential of the novel approach. \itemrv{~} \itemcc{} \itemut{Monte Carlo methods; complexity; fuzzy set theory; uncertainty modeling; sparse grid interpolation; global optimization} \itemli{http://www.shaker.de/de/content/catalogue/index.asp?lang=de&ID=8&ISBN=978-3-8322-4766-9} \end