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Scalable Frames
(2012)
Tight frames can be characterized as those frames which possess optimal numerical stability properties. In this paper, we consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors; a process which can also be regarded as perfect preconditioning of a frame by a diagonal operator. A frame is called scalable, if such a diagonal operator exists. We derive various characterizations of scalable frames, thereby including the infinite-dimensional situation. Finally, we provide a geometric interpretation of scalability in terms of conical surfaces.
A mathematical model for instationary magnetization
processes is considered, where the underlying spatial domain
includes electrically conducting and nonconducting regions. The
model accounts for the magnetic induction law that couples the given
electrical voltage with the induced electrical current in the
induction coil. By a theorem of Showalter on degenerate parabolic
equations, theorems on existence, uniqueness, and regularity of the
solution to the associated Maxwell integrodifferential system are
proved.
Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka [Adv. Math. Econ. 6, 69–83, 2004] and Lyons–Victoir [Proc. R. Soc. Lond. Ser. A 460, 169–198, 2004], involve the solution to numerous auxiliary ordinary differential equations. With focus on the Ninomiya-Victoir algorithm [Appl. Math. Fin. 15, 107–121, 2008], which corresponds to a concrete level 5 cubature method, we study some parametric diffusion models motivated from financial applications, and exhibit structural conditions under which all involved ODEs can be solved explicitly and efficiently. We then enlarge the class of models for which this technique applies, by introducing a (model-dependent) variation of the Ninomiya-Victoir method. Our method remains easy to implement; numerical examples illustrate the savings in computation time.
Density expansions for hypoelliptic diffusions (X1^,...,X^d) are revisited. In particular, we are interested in density expansions of the projection (X^1_T,...,X^l_T) at time $T>0$, with $l \le d$. Global conditions are found which replace the well-known ”not-in-cutlocus” condition known from heat-kernel asymptotics; cf. G. Ben Arous (88). Our small noise expansion allows for a ”second order” exponential factor. Applications include tail and implied volatility asymptotics in some correlated stochastic volatility models; in particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009).
A robust implementation of a Dupire type local volatility model is an important issue for every option trading floor. In the present note we provide new analytic insights into the asymptotic behavior of local volatility in the wings. We present a general approximation formula and specialize it to the Heston model, showing that local variance is linear in the wings. This further justifies the choice of certain local volatility parametrizations.
Flows over time generalize classical ``static'' network flows by introducing a temporal dimension. They can thus be used to model non-instantaneous travel times for flow and variation of flow values over time, both of which are crucial characteristics in many real-world routing problems. There exist two different models of flows over time with respect to flow conservation: one where flow might be stored temporarily at intermediate nodes and a stricter model where flow entering an intermediate node must instantaneously progress to the next arc. While the first model is in general easier to handle, the second model is often more realistic since in applications like, e.\,g., road traffic, storage of flow at intermediate nodes is undesired or even prohibited. The main contribution of this paper is a fully polynomial time approximation scheme (FPTAS) for (min-cost) multi-commodity flows over time without intermediate storage. This improves upon the best previously known $(2+\varepsilon)$-approximation algorithm presented 10 years ago by Fleischer and Skutella (IPCO~2002).
Some mathematical problems related to the 2nd order optimal shape of a crystallization interface
(2012)
We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.
We characterize the Smith form of skew-symmetric matrix polynomials
over an arbitrary field $\F$,
showing that all elementary divisors occur with even multiplicity.
Restricting the class of equivalence transformations to unimodular congruences,
a Smith-like skew-symmetric canonical form
for skew-symmetric matrix polynomials is also obtained.
These results are used to analyze the eigenvalue and elementary divisor structure
of matrices expressible as products of two skew-symmetric matrices,
as well as the existence of structured linearizations
for skew-symmetric matrix polynomials.
By contrast with other classes of structured matrix polynomials
(e.g., alternating or palindromic polynomials),
every regular skew-symmetric matrix polynomial
is shown to have a structured strong linearization.
While there are singular skew-symmetric polynomials of even degree
for which a structured linearization is impossible,
for each odd degree we develop a skew-symmetric companion form
that uniformly provides a structured linearization
for every regular and singular skew-symmetric polynomial
of that degree.
Finally, the results are applied to the construction of minimal
symmetric factorizations of skew-symmetric rational matrices.
We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem and a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with respect to the variation of the underlying
probability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularity conditions, we derive Aubin's property of the solution set mapping with respect to the change of probability measure. The established results are
applied to stability analysis of stationary points of classical one stage and two stage stochastic minimization problems, two stage stochastic mathematical programs with equilibrium constraints and stochastic programs with second order dominance constraints.