101.820 VO (2std Vorlesung, Sommersemester 2019)
AKNUM:Nichtlokale Operatoren - Analysis und Numerik
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Date and location

The lecture will be slightly blocked. Rather than doing 1.5 hours a week, I will give twice a week a 1 hour lecture. This compensates for weeks without lectures.

Aim of the lecture

The aim of this lecture is to understand operator equations with non-local operators analytically and numerically.

Subject of course

What are non-local operators?

A non-local operator A (mapping between two function spaces) is characterized by the property that the evaluation of Af(x) does depend on the whole f and not only on values of f in a neighborhood of x. As various classical tools in analysis and numerics are not applicable to this class of operators, we introduce appropriate techniques for such problems.

Examples of non-local operators

Differential operators such as the Laplacian Δ are local operators. However, integral operators such as the Fourier transform are non-local. In this lecture, we study integral operators of the form
                     
and integral equations Au = f analytically and numerically. More specific, we deal with (boundary) integral operators that arise from the representation formula of the Laplace equation as well as fractional differential operators
                     
for 0 < s < 1. Such operators appear naturally in more refined models of anomalous diffusion in physics, biology or finance. Mathematically, there are various definition of the fractional Laplacian, which are not equivalent on bounded domains. In this lecture, we will consider two different definitions - the spectral and integral fractional Laplacian - and discuss numerical methods for these two definitions.

Lectures

#DateTopicLecturer
110.10.2019IntroductionFaustmann
215.10.2019Representation formulaFaustmann
317.10.2019Function spacesFaustmann
22.10.2019No lecture
24.10.2019No lecture
429.10.2019Conormal derivativeFaustmann
531.10.2019The Newton potentialFaustmann
65.11.2019The single layer potentialFaustmann
77.11.2019The Calderon systemFaustmann
812.11.2019Integral representationsFaustmann
914.11.2019Integral representationsFaustmann
19.11.2019No lecture
1021.11.2019Jump conditions, ellipticityFaustmann
1126.11.2019Ellipticity, Galerkin methodFaustmann
1228.11.2019A-priori estimatesFaustmann
133.12.2019Introduction to fractional operatorsFaustmann
5.12.2019No lecture due to distinguished PDE lecture
1410.12.2019The Fourier definitionFaustmann
1512.12.2019The Caffarelli Silvestre extensionFaustmann
1617.12.2019FL on bounded domains, localizationFaustmann
19.12.2019-6.1.2020Christmas Holidays
177.1.2020Localization, regularityFaustmann
189.1.2020FEM for integral FLFaustmann
1914.1.2020Weak formulation, regularity spectral FLFaustmann
2016.1.2020FEM for spectral FLFaustmann
21.1.2020no lecture - PhD defense of colleague
2123.1.2020Dunford-Taylor approachFaustmann

Downloads

In the following you can find the lecture notes for the first part of the lecture.

The current iteration of the lecture notes also includes - for sake of completeness - proofs I did not do in the lecture.

I have not done the proofreading yet. Therefore, I would appreciate if you would tell me any mistakes you find!

22.01.2020Lecture Notes[pdf]

Literature

W. McLean: Strongly elliptic systems and boundary integral equations Link

S. Sauter, C. Schwab: Boundary Element Methods Link

A. Bonito, J.P. Borthagaray, R.H. Nochetto, E. Otarola, A.J. Salgado: Numerical methods for fractional diffusion Link

Required Knowledge

In principle, there is no specific previous knowledge required. However, basic knowledge in functional analysis, numerics and partial differential equations is helpful.

Examination modalities

Oral examination, make an appointment by e-mail.