id: 02243098
dt: b
an: 02243098
au: Klimke, W. Andreas
ti: Uncertainty modeling using fuzzy arithmetic and sparse grids.
so: Industriemathematik und Angewandte Mathematik. Aachen: Shaker Verlag;
    Stuttgart: Univ. Stuttgart, Institut für Angewandte Analysis und
    Numerische Simulation (Diss.) (ISBN 3-8322-4766-1/pbk). xii, 124~p.
    EUR~45.80 (2006).
py: 2006
pu: Aachen: Shaker Verlag; Stuttgart: Univ. Stuttgart, Institut für Angewandte
    Analysis und Numerische Simulation (Diss.)
la: EN
cc: 
ut: Monte Carlo methods; complexity; fuzzy set theory; uncertainty modeling;
    sparse grid interpolation; global optimization
ci: 
li: http://www.shaker.de/de/content/catalogue/index.asp?lang=de&ID=8&ISBN=978-3-8322-4766-9
ab: 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.
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