Marco Avellaneda

Adaptive greedy approximations

The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NP-hard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the function’s structures. A matching pursuit provides a means of quickly computing compact, adaptive function approximations.Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and by constructing a stochastic differential equation model.We derive a notion of the coherence of a signal with respect to a dictionary from our characterization of the approximation errors of a pursuit. The dictionary elements slected during the initial iterations of a pursuit correspond to a function’s coherent structures. The tail of the expansion, on the other hand, corresponds to a noise which is characterized by the invariant measure of the pursuit map.When using a suitable dictionary, the expansion of a function into its coherent structures yields a compact approximation. We demonstrate a denoising algorithm based on coherent function expansions.

Magnetoelectric Effect in Piezoelectric/Magnetostrictive Multilayer (2-2) Composites

We consider the magnetoelectric effect arising in a multilayer composite consisting of bonded layers of a piezoelectric ceramic and a magnetostrictive material operating in the linear regime. Magnetoelectric coupling arises from the mechanical contacts between layers and the elec tric/mechanical and magnetic/mechanical coupling in each phase. Calculations of the magnetoelec tric coefficient α* as well as of the figures of merit kme and kme,cl are presented. These calculations fully take into account the electric, magnetic and mechanical mismatch in the material properties, as well as the volume-fractions of both materials. We derive the optimal volume fraction fcrit 1 of piezoelectric material needed to maximize the figure of merit k fme or kme,cl for the composite and give criteria for optimizing the magnetoelectric effect in terms of the choice of the individual constit uents. We computed the figures of merit of a variety of piezoelectric/magnetostrictive combinations. The combinations CoFe2O4/PZT5H (kme = 14%), Terfenol D/PZT5A (kme = 22%) and Terfenol D/PZT4 (kme = 30%) show reasonable energy transfer to be useful as magnetoelectric transducers.

High-frequency trading in a limit order book

The role of a dealer in securities markets is to provide liquidity on the exchange by quoting bid and ask prices at which he is willing to buy and sell a specific quantity of assets. Traditionally,...

Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porous media

A rigorous expression is derived that relates exactly the static fluid permeability k for flow through porous media to the electrical formation factor F (inverse of the dimensionless effective conductivity) and an effective length parameter L, i.e., k=L2/8F. This length parameter involves a certain average of the eigenvalues of the Stokes operator and reflects information about electrical and momentum transport. From the exact relation for k, a rigorous upper bound follows in terms of the principal viscous relation time Θ1 (proportional to the inverse of the smallest eigenvalue): k≤νΘ1/F, where ν is the kinematic viscosity. It is also demonstrated that νΘ1≤DT1, where T1 is the diffusion relaxation time for the analogous scalar diffusion problem and D is the diffusion coefficient. Therefore, one also has the alternative bound k≤DT1/F. The latter expression relates the fluid permeability on the one hand to purely diffusional parameters on the other. Finally, using the exact relation for the permeability, a ...

Mathematical models with exact renormalization for turbulent transport

AbstractThe advection-diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and/or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media. A variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts. In this paper, a simple model problem for the advection-diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor. Explicit formulas for the anomalous time scaling in various regimes as well as the Greens function for the large-scale, long-time, ensemble average are developed here. Formulas for the renormalized higher order statistics are also developed. The simple form of the model problem is deceptive; the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied. These phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter

Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model

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On the effective conductivity of polycrystals and a three‐dimensional phase‐interchange inequality

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