Partial Differential Equations
An Introduction to Analytical and Numerical Methods
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Open AccessA space–time variational method for optimal control problems: well-posedness, stability and numerical solution
We consider an optimal control problem constrained by a parabolic partial differential equation with Robin boundary conditions. We use a space–time variational formulation in Lebesgue–Bochner spaces yielding a...
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Maple , or why computers can sometimes help
Maple is a program developed and distributed by the company Maplesoft (also in cooperation with universities and research institutes). It is a compute...
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Modeling, or where do differential equations come from
Partial differential equations describe numerous phenomena in nature, technology, medicine, economics, ... In this first chapter we shall describe the derivation of the partial differential equations associate...
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Elementary methods
In this chapter we will derive explicit solutions for a number of partial differential equations. The equations we will be considering here represent a number of important models such as those of the vibrating...
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Sobolev spaces and boundary value problems in dimension one
Sobolev spaces on an interval have special appeal: they consist of functions which can be written as indefinite integrals of integrable functions. Numerous properties and rules of calculation, such as the inte...
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Neumann and Robin boundary conditions
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The Reduced Basis Method in Space and Time: Challenges, Limits and Perspectives
The simulation and optimization of several real-world industrial problems involve parameters, e.g. unknown constants, design parameters, controls etc. In many cases, these parameters are subject to uncertainties....
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Numerical methods
Thus far we have met and modeled a number of partial differential equations, considered elementary solution methods for them and finally developed mathematical theories to handle them. Everything up until now ...
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Classification and characteristics
In the preceding chapter we encountered a whole series of partial differential equations arising from natural processes as well as economic models. Given the huge variety of such equations, it should not be su...
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Hilbert spaces
The goal of this chapter is to give an elementary but comprehensive introduction to the theory of Hilbert spaces, where we wish to highlight the many-faceted interplay of their geometric and analytic propertie...
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Hilbert space methods for elliptic equations
We started by introducing Sobolev spaces in dimension one only, or more precisely for functions defined on an open interval in ℝ ...
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Spectral decomposition and evolution equations
The spectral decomposition of the Laplacian with suitable boundary conditions gives us precise information about the solvability of the equation ...
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Book
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Book
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Article
Open AccessLinear/Ridge expansions: enhancing linear approximations by ridge functions
We consider approximations formed by the sum of a linear combination of given functions enhanced by ridge functions—a Linear/Ridge expansion. For an explicitly or implicitly given objective function, we reform...
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Multiskalenprobleme in der Numerik–Lehre – Ein Erfahrungsbericht
Der Bologna–Prozess hat u. a. bewirkt, dass Lehrveranstaltungen von Studierenden mit vermehrt heterogenen Eingangskompetenzen besucht werden. Bei gleichen angestrebten Ausgangskompetenzen stellt dies die Lehre...
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Chapter and Conference Paper
A Variational Formulation for LTI-Systems and Model Reduction
We consider a variational formulation of Linear Time-Invariant (LTI)-systems and derive a model reduction in dimension and time inspired by space-time variational reduced basis (RB) methods for parabolic probl...
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Chapter and Conference Paper
Matrix Oriented Reduction of Space-Time Petrov-Galerkin Variational Problems
Variational formulations of time-dependent PDEs in space and time yield (d + 1)-dimensional problems to be solved numerically. This increases the number of unknowns as well as the storage amount. On the other han...
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Article
HT-AWGM: a hierarchical Tucker–adaptive wavelet Galerkin method for high-dimensional elliptic problems
This paper is concerned with the construction, analysis, and realization of a numerical method to approximate the solution of high-dimensional elliptic partial differential equations. We propose a new combinat...
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Open AccessIntraday renewable electricity trading: advanced modeling and numerical optimal control
As an extension of (Progress in industrial mathematics at ECMI 2018, pp. 469–475, 2019), this paper is concerned with a new mathematical model for intraday electricity trading involving both renewable and convent...
