Linear inviscid damping for monotone shear flows.
Abstract
exhibit damping close to linear shear flows.
The mechanism behind this "inviscid
In this talk I give a proof of linear stability,
It is well known that several solutions to the Skorokhod problem
optimize certain ``cost''- or ``payoff''-functionals. We use the
theory of Monge-Kantorovich transport to study the corresponding
optimization problem. We formulate a dual problem and establish
duality based on the duality theory of optimal transport. Notably
the primal as well as the dual problem have a natural interpretation
in terms of model-independent no arbitrage theory.
In optimal transport the notion of c-monotonicity is used to
characterize the geometry of optimal transport plans. We derive a
similar optimality principle that provides a geometric
characterization of optimal stopping times. We then use this
principle to derive several known solutions to the Skorokhod
embedding problem and also new ones.
This is joint work with Mathias Beiglböck and Alex Cox.
An old conjecture by A. Zygmund proposes
a Lebesgue Differentiation theorem along a
Lipschitz vector field in the plane. E. Stein
formulated a corresponding conjecture about
the Hilbert transform along the vector field.
If the vector field is constant along
vertical lines, the Hilbert transform along
the vector field is closely related to Carleson's
operator. We discuss some progress in the area
by and with Michael Bateman and by my student
Shaoming Guo.
We present recent numerical techniques for the treatment of integral formulations of Helmholtz boundary value problems in the case of high frequencies. The combination of $H^2$-matrices with further developments of the adaptive cross approximation allows to solve such problems with logarithmic-linear complexity independent of the frequency. An advantage of this new approach over existing techniques such as fast multipole methods is its stability over the whole range of frequencies, whereas other methods are efficient either for low or high frequencies.
A factorability structure on a group G is a specification of normal forms of group elements as words over a fixed generating set. There is a chain complex computing the (co)homology of G. In contrast to the well-known bar resolution, there are much less generators in each dimension of the chain complex. Although it is often difficult to understand the differential, there are examples where the differential is particularly simple, allowing computations by hand. This leads to the cohomology ring of hv-groups, which I define at the end of the talk in terms of so called "horizontal" and "vertical" generators.
A factorability structure on a group G is a specification of normal forms
of group elements as words over a fixed generating set. There is a chain
complex computing the (co)homology of G. In contrast to the well-known bar
resolution, there are much less generators in each dimension of the chain
complex. Although it is often difficult to understand the differential,
there are examples where the differential is particularly simple, allowing
computations by hand. This leads to the cohomology ring of hv-groups,
which I define at the end of the talk in terms of so called "horizontal"
and "vertical" generators.
When managing risk, frequently only imperfect hedging instruments are at hand.
We show how to optimally cross-hedge risk when the spread between the hedging
instrument and the risk is stationary. At the short end, the optimal hedge ratio
is close to the cross-correlation of the log returns, whereas at the long end, it is
optimal to fully hedge the position. For linear risk positions we derive explicit
formulas for the hedge error, and for non-linear positions we show how to obtain
numerically effcient estimates. Finally, we demonstrate that even in cases with no
clear-cut decision concerning the stationarity of the spread it is better to allow for
mean reversion of the spread rather than to neglect it.
The talk is based on joint work with Georgi Dimitroff, Gregor Heyne and Christian Pigorsch.
Partial differential equations with a nonlinear pointwise constraint defined through a manifold occur in a variety of applications: The magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rod-like molecules that constitute a liquid crystal is often modeled by a vector field that attains its values in the real projective plane thus respecting the head-to-tail symmetry of the molecules. Other applications arise in geometric
modeling, quantum mechanics, and general relativity. Simple examples reveal that it is impossible to satisfy pointwise constraints exactly by lowest order finite elements. For two model problems we discuss the practical realization of the constraint, the efficient solution of the resulting nonlinear systems of equations, and weak accumulation of approximations at exact solutions.
Sequential Monte Carlo Samplers are a class of stochastic algorithms for
Monte Carlo integral estimation w.r.t. probability distributions, which combine
elements of Markov chain Monte Carlo methods and importance sampling/resampling
schemes. We develop a stability analysis by functional inequalities for a
nonlinear flow of probability measures describing the limit behaviour of the
methods as the number of particles tends to infinity. Stability results are
derived both under global and local assumptions on the generator of the
underlying Metropolis dynamics. This allows us to prove that the combined
methods sometimes have good asymptotic stability properties in multimodal setups
where traditional MCMC methods mix extremely slowly. For example, this holds for
the mean field Ising model at all temperatures.