Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

The author’s presentation of multilevel Monte Carlo path simulation at the MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo methods. This paper reviews the progress since then, emphasising the simplicity, flexibility and generality of the multilevel Monte Carlo approach. It also offers a few original ideas and suggests areas for future research.

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Multilevel Monte Carlo methods

The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic PDEs Abstractly, it can be viewed as determining a “good” n-dime

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Convergence Rates for Greedy Algorithms in Reduced Basis Methods

SUMMARY
In this paper, we propose a method for the numerical solution of linear systems of equations in low rank tensor format. Such systems may arise from the discretisation of PDEs in high dimensions, but our method is not limited to this type of application. We present an iterative scheme, which is based on the projection of the residual to a low dimensional subspace. The subspace is spanned by vectors in low rank tensor format which—similarly to Krylov subspace methods—stem from the subsequent (approximate) application of the given matrix to the residual. All calculations are performed in hierarchical Tucker format, which allows for applications in high dimensions. The mode size dependency is treated by a multilevel method. We present numerical examples that include high-dimensional convection–diffusion equations.Copyright © 2012 John Wiley & Sons, Ltd.

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A projection method to solve linear systems in tensor format

Cartoon-like images, i.e., C^2 functions which are smooth apart from a C^2 discontinuity curve, have by now become a standard model for measuring sparse (non-linear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit (almost) optimally sparse approximation within this model. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications. 
In this paper, we now present the first complete proof of (almost) optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of shearlet frames -- not necessarily required to be tight -- with shearlet generators having compact support and satisfying some weak moment conditions.

Compactly Supported Shearlets are Optimally Sparse

https://www.mathematik.uni-marburg.de/~dereich/Preprints/quad_levy.pdf

A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations

Digitization is transforming our world economically, culturally, and psychologically. The influx of new forms of communication, networking, and business opportunities, as well as new types of distraction, self-observation, and control into our societies represents an epochal challenge. Following Bernard Stiegler's concept of pharmacology, Felix Heidenreich and Florian Weber-Stein propose to view these new forms as digital pharmaka. Properly dosed, they can enable new self-relationships and forms of sociality; in the case of overdose, however, there is a risk of intoxication. In this essay, Felix Heidenreich, Florian Weber-Stein, and, in a detailed interview, Bernard Stiegler analyze this complex change in our world and develop new skills to use digital pharmaka.

Felix Heidenreich

Papers

A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations

The Politics of Digital Pharmacology