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The special volume offers a global guide to new concepts and
approaches concerning the following topics: reduced basis methods,
proper orthogonal decomposition, proper generalized decomposition,
approximation theory related to model reduction, learning theory
and compressed sensing, stochastic and high-dimensional problems,
system-theoretic methods, nonlinear model reduction, reduction of
coupled problems/multiphysics, optimization and optimal control,
state estimation and control, reduced order models and domain
decomposition methods, Krylov-subspace and interpolatory methods,
and applications to real industrial and complex problems. The book
represents the state of the art in the development of reduced order
methods. It contains contributions from internationally respected
experts, guaranteeing a wide range of expertise and topics.
Further, it reflects an important effor t, carried out over the
last 12 years, to build a growing research community in this field.
Though not a textbook, some of the chapters can be used as
reference materials or lecture notes for classes and tutorials
(doctoral schools, master classes).
This textbook introduces the study of partial differential
equations using both analytical and numerical methods. By
intertwining the two complementary approaches, the authors create
an ideal foundation for further study. Motivating examples from the
physical sciences, engineering, and economics complete this
integrated approach. A showcase of models begins the book,
demonstrating how PDEs arise in practical problems that involve
heat, vibration, fluid flow, and financial markets. Several
important characterizing properties are used to classify
mathematical similarities, then elementary methods are used to
solve examples of hyperbolic, elliptic, and parabolic equations.
From here, an accessible introduction to Hilbert spaces and the
spectral theorem lay the foundation for advanced methods. Sobolev
spaces are presented first in dimension one, before being extended
to arbitrary dimension for the study of elliptic equations. An
extensive chapter on numerical methods focuses on finite difference
and finite element methods. Computer-aided calculation with Maple
(TM) completes the book. Throughout, three fundamental examples are
studied with different tools: Poisson's equation, the heat
equation, and the wave equation on Euclidean domains. The
Black-Scholes equation from mathematical finance is one of several
opportunities for extension. Partial Differential Equations offers
an innovative introduction for students new to the area. Analytical
and numerical tools combine with modeling to form a versatile
toolbox for further study in pure or applied mathematics.
Illuminating illustrations and engaging exercises accompany the
text throughout. Courses in real analysis and linear algebra at the
upper-undergraduate level are assumed.
This research monograph addresses recent developments of wavelet concepts in the context of large scale numerical simulation. It offers a systematic attempt to exploit the sophistication of wavelets as a numerical tool by adapting wavelet bases to the problem at hand. This includes both the construction of wavelets on fairly general domains and the adaptation of wavelet bases to the particular structure of function spaces governing certain variational problems. Those key features of wavelets that make them a powerful tool in numerical analysis and simulation are clearly pointed out. The particular constructions are guided by the ultimate goal to ensure the key features also for general domains and problem classes. All constructions are illustrated by figures and examples are given.
Dieses Lehrbuch gibt eine Einfuhrung in die partiellen
Differenzialgleichungen. Wir beginnen mit einigen ganz konkreten
Beispielen aus den Natur-, Ingenieur und Wirtschaftswissenschaften.
Danach werden elementare Loesungsmethoden dargestellt, z.B. fur die
Black-Scholes-Gleichung aus der Finanzmathematik. Schliesslich wird
die analytische Untersuchung grosser Klassen von partiellen
Differenzialgleichungen dargestellt, wobei Hilbert-Raum-Methoden im
Mittelpunkt stehen. Numerische Verfahren werden eingefuhrt und mit
konkreten Beispielen behandelt. Zu jedem Kapitel finden sich
UEbungsaufgaben, mit deren Hilfe der Stoff eingeubt und vertieft
werden kann. Dieses Buch richtet sich an Studierende im Bachelor
oder im ersten Master-Jahr sowohl in der (Wirtschafts-)Mathematik
als auch in den Studiengangen Informatik, Physik und
Ingenieurwissenschaften. Die 2. Auflage ist vollstandig
durchgesehen, an vielen Stellen didaktisch weiter optimiert und um
die Beschreibung variationeller Methoden in Raum und Zeit fur
zeitabhangige Probleme erganzt. Stimme zur ersten Auflage Auf
dieses Lehrbuch haben wir gewartet. Prof. Dr. Andreas Kleinert in
zbMATH
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