• Kolloquium ModNumDiff SS 2011 [1]

Sparse and Low Rank Recovery
Dienstag, den 25.10.2011, 16.15 Uhr in MA 313
Abstract:

Compressive Sensing (sparse recovery) predicts that sparse vectors can be recovered from what was previously believed to be highly incomplete linear measurements. Efficient algorithms such as convex relaxations and greedy algorithms can be used to perform the reconstruction. Remarkably, all good measurement matrices known so far in this context are based on randomness. Recently, it was observed that similar findings also hold for the recovery of low rank matrices from incomplete information. Again, convex relaxations and randomness are crucial ingredients.

The talk gives an introduction and overview on sparse and low rank recovery with emphasis on results due to the speaker.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 621 - everybody's welcome.

Novel methods to compute the distance to instability of a stable matrix and the H-infinity norm of a linear system
Dienstag, den 25.10.2011, 16.15 Uhr in MA 313
Abstract:

Using the key property that the epsilon-pseudospectrum is determined via perturbations by low rank matrices, we derive dynamical systems leading to the critical perturbations associated with the searched extremal points.

The knowledge of the critical perturbation allows an inexpensive and direct computation of the derivative in the root-finding Newton process associated to the computation of the distance to instability of a stable system.

The method turns out to be fast and appears to be promising in dealing with large-size, sparse problems.

With a similar technique we find an algorithm to compure the H-infinity norm of a linear dynamical system, which is still based on a discrete dynamical system in the manifold of rank-1 matrices.

The talk is based on joint works with Michael Overton and Mert Gurbuzbalaban (Courant Institute, New York University).

Numerical Methods for the Simulation of Two-phase Incompressible Flows
Dienstag, den 08.11.2011, 18.15 Uhr in MA 001 Abstract:

We consider a flow problem with two diffrent immiscible incompressible newtonian phases (fluid-fluid or fluid-gas). A standard model for this consists of the Navier-Stokes equations with a viscosity and density that are discontinuous across the interface and with a localized force at the interface that describes surface tension effects. This fluid dynamics model can be coupled with a model for mass transport between the phases and a model for transport of surfactants on the interface. In the past few years we developed, analyzed and implemented numerical methods for the 3D simulation of such two-phase flow models, cf. [1,2]. In this talk we present certain aspects of our solver in more detail and give illustrations by means of results of numerical simulations of droplet sedimentation.

[1] www.igpm.rwth-aachen.de/DROPS

[2] S. Gross, A. Reusken, Numerical Methods for Two-phase Incomprssible Flows, Springer SCM Vol. 40, 2011.

Computational surface PDEs
Dienstag, den 15.11.2011, 16.15 Uhr in MA 313
Abstract:

I will motivate the computational solution of surface PDES with some examples from biology (cell motility and phase separation on biomembranes) and material science (surface dissolution and the formation of nanoporosity) which couple PDEs on surfaces to the evolution of the surfaces. I will formulate the evolving surface finite element method. I will describe the finite element error analysis which yields optimal order bounds for piece-wise linear elements in the semi-discrete and the fully discrete backward Euler schemes. The error analysis is joint work with G. Dziuk.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

After the talk there will be a joint dinner with the speaker - everybody's welcome.

Multicentric representation of holomorphic functions and von Neumann spectral sets
Dienstag, den 22.11.2011, 16.15 Uhr in MA 313 Abstract:

If A is a contraction in a Hilbert space and f is analytic and bounded in the unit disc, then the norm of f(A) can be bounded by the sup of |f| in the unit disc.
This is due to von Neumann, from 1951. In another language it says that discs are spectral sets.
We show how combining some old ideas (that is, from Lagrange, Jacobi,…) we get a new way to represent holomorphic functions in sets bounded by level curves of polynomials, i.e. lemniscates. This was done in [2].
In this talk we concentrate in demonstrating the power of the approach by generalizing the von Neumann theorem to sets bounded by lemniscates. In particular, we show that
Sets of the form
{z : |p(z)| ≤ ||p(A)|| }
are spectral sets, assuming that the bounding lemniscate does not pass thru any critical point.
There are recent generalizations [M. Crouzeix, B.and F. Delyon] of the von Neumann result to convex sets containing the numerical range. As far as we know our results in [3] are the first ones which apply to nonconvex and even to disconnected sets. In particular, the approach gives both a computational tool and nontrivial estimates for Riesz spectral projections.
To obtain a relevant multicentric representation, one has to first run an algorithm to locate the spectrum and to obtain a global representation for the resolvent. This is explained in [1], but in the talk we only highlite the basic results of that part and concentrate in the generalization of von Neumann theorem.

References

[1] O. Nevanlinna, Computing the Spectrum and Representing the Resolvent, Numer. Funct. Anal. Optimiz. Volume 30, Issue 9 and 10 (2009) 1025 - 1047

[2] O. Nevanlinna, Multicentric Holomorphic Calculus;
to appear in Computational Methods & Function Theory
preprint:
math.tkk.fi/reports/a596.pdf [2]

[3] O. Nevanlinna, Multicentric representation and von Neumann spectral sets, submitted,
preprint: math.tkk.fi/reports/st_2011_11.pdf [3]

Shearlet frames and optimally sparse approximations
Dienstag, den 13.12.2011, 16.15 Uhr in MA 313
Abstract:

Shearlet theory has become a central tool in analyzing and representing 2D and 3D data with anisotropic features. Shearlet systems are systems of functions generated by one single generator with parabolic scaling, shearing, and translation operators applied to it, in much the same way wavelet systems are dyadic scalings and translations of a single function, but including a directional parameter. The success of shearlets owes to an extensive list of desirable properties: Shearlet systems can be generated by one function, they provide precise resolution of wavefront sets, they allow compactly supported analyzing elements, they are associated with fast decomposition algorithms, and they provide a unified treatment of the continuum and the digital world.

This talk gives an introduction to shearlet theory with focus on separable and compactly supported shearlets in 2D and 3D. We will consider constructions of band-limited and compactly supported shearlet frames in those dimensions. Finally, we will show that compactly supported shearlet frames satisfying weak decay, smoothness, and directional moment conditions provide optimally sparse approximations of a generalized model of cartoon-like images comprising of piecewise C² functions that are smooth apart from piecewise C² discontinuity edges.

This talk is based on joint with G. Kutyniok and W.-Q Lim (TU Berlin).

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 621 - everybody's welcome.

Quadratic Eigenvalue problems: Canonical forms and inverse problems
Dienstag, den 10.01.2012, 16.15 Uhr in MA 313
Abstract:

Recent results on the existence of real canonical forms for matrix polynomials will be presented and applied to the design of systems with prescribed spectral properties.

(This is collaborative work with Ion Zaballa, and Uwe Prells.)

An adaptive shearlet method for first order transport equations
Dienstag, den 31.01.2012, 16.15 Uhr in MA 313
Abstract:

Adaptive wavelet methods, pioneered by Cohen, Dahmen, and DeVore, provide optimal solvers for elliptic problems. However, solutions of hyperbolic conservation laws or more generally of transport dominated equations are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds. Thus, wavelets can not be employed as trial spaces to derive optimal solvers for this class of PDEs, since wavelets fail to povide optimally sparse approximations of the respective solutions. Recently, shearlets were introduced as a means to optimal sparsely encode anisotropic singularities of multivariate data while allowing compactly supported analyzing elements. This raises the question whether they might serve as a optimal discretization concept for transport dominated equations.

An important conceptual foundation of our approach to tackling this question is a stable adaptive Petrov-Galerkin formulation that has recently been developed by Dahmen, Huang, Schwab, Welper, which we will first introduce. We will then discuss how this can be further developed to accommodate shearlet frames in the adaptive refinement process. We indicate how to establish theoretical guarantees for an asymptotically optimal approximation behavior. Finally, we will present numerical results which show that our adaptive scheme generates effective approximations of the solutions of first order transport equations. This is joint work with Dahmen, Kutyniok, Schwab and Welper.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Frames on the road to perfection
Dienstag, den 7.02.2012, 16.15 Uhr in MA 313
Abstract:

Redundant, stable expansions with frames have become central to many applications of mathematics in today’s technology. Frames have gained a broad relevance in problems ranging from remote sensing to wireless transmissions, in analog-digital conversion such as audio and video encoding, in packet-based network communications, noise-insensitive quantum computing and recently also in compressed sensing. Optimal frames for certain purposes have been characterized and systematically constructed, for example equiangular tight frames by combinatorial techniques, or frames with smallest support by “spectral tetris” algorithms. However, after many years of efforts the known optimal examples reside mostly in small dimensions or their construction relies on specific group-representation properties.

This talk presents an alternative to either of the conventional, structured design methods; we let frames evolve under flows which drive them towards optimality, instead of constructing them directly. The general objectives are to find appropriate frame dynamics, suitable initializations, and to obtain deterministic control of frame performance measures for certain applications. Known results for the construction of equal-norm Parseval frames are explained as well as recent work with Helen Elwood on the construction of equiangular tight frames.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Operator splitting methods for fourth-order nonlinear PDEs in imaging
Dienstag, den 27.03.2012, 16.15 Uhr in MA 313
Abstract:

In this talk I will discuss operator splitting techniques such as Strang splitting, convexity splitting and Split Bregman for the solution of nonlinear fourth-order PDEs that arise as models for image enhancement and restoration. In order to tackle the expensive computation of solutions - due to the high differential order and the nonlinear nature of the equation - splitting methods are aimed to break the problem down to a series of simpler and/or less expensive problems to solve. The talk is furnished with applications in image inpainting.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.