Solving parabolic Wick-stochastic boundary value problems using a finite element method

A class of linear parabolic stochastic boundary value problems of Wick-type is studied. The equations are understood in a weak sense on a suitable stochastic distribution space, and existence and uniqueness results are provided. The paper continues to discuss a numerical method for this type of problem, based on a Galerkin type of approximation. Estimates showing linear convergence in time and space are derived, and rate of convergence results for the stochastic dimension are reported.     

Robust Parameter Estimation for Stochastic Differential Equations

We consider the estimation of parameters in stochastic differential equations (SDEs). The problem is treated in the setting of nonlinear filtering theory with a degenerate diffusion matrix. A robust stochastic Feynman–Kac representation for solutions of SDEs of Zakai-type is derived. It is verified that these solutions are conditional densities for the conditional measures defined by degenerate filtering problems. We show that the corresponding estimator for the parameters is robust in the following sense: It depends continuously on both the measurement path and on the intensity of the measurement noise. An algorithm based on a Monte-Carlo approach is given for the practical application of the estimator, and numerical results are reported. Mathematics Subject Classifications (2000) Primary: 62M05, 62M20; secondary: 62F15.     

Biodistribution of gold nanoparticles in mouse lung following intratracheal instillation

Background  

The fate of gold nanoparticles, 2, 40 and 100 nm, administered intratracheally to adult female mice was examined. The nanoparticles were traced by autometallography (AMG) at both ultrastructural and light microscopic levels. Also, the gold content was quantified by inductively coupled plasma mass spectrometry (ICP-MS) and neutron activation analysis (NAA). The liver is the major site of deposition of circulating gold nanoparticles. Therefore the degree of translocation was determined by the hepatic deposition of gold. Mice were instilled with 5 intratracheal doses of gold nanoparticles distributed over a period of 3 weeks and were killed 24 h after the last dose. One group of mice were given a single intratracheal dose and were killed after 1 h.     

On the Completions of the Spaces of Metrics on an Open Manifold

We construct a complete metric d₀ on the space of continuous complete Riemannian metrics on a smooth manifold X of dimension n. Using that metric, we are able to show, that the space ^b,m?(X) defined in [1] is complete when suppplied with the uniform structure defined in the same paper.     

Atiyah—Patodi—Singer type index theorems for manifolds with splitting of $ \eta $-invariants

We construct self-adjoint extensions of Dirac operators on manifolds with corners of codimension 2, which generalize the Atiyah—Patodi—Singer boundary conditions. The boundary conditions are related to geometric constructions, which convert problems on manifolds with corners into problems on manifolds with boundary and wedge singularities. In the case, where the Dirac bundle is a super-bundle, we prove two general index theorems, which differ by the splitting formula for -invariants. Further we work out the de Rham, signature and twisted spin complex in closer detail. Finally we give a new proof of the splitting formula for the -invariant. Submitted: October 1999, Revised version: March 2001.     

Equilibrium strategy and population-size effects in lowest unique bid auctions

In lowest unique bid auctions, N players bid for an item. The winner is whoever places the lowest bid, provided that it is also unique. We use a grand canonical approach to derive an analytical expression for the equilibrium distribution of strategies. We then study the properties of the solution as a function of the mean number of players, and compare them with a large data set of internet auctions. The theory agrees with the data with striking accuracy for small population-size N, while for larger N a qualitatively different distribution is observed. We interpret this result as the emergence of two different regimes, one in which adaptation is feasible and one in which it is not. Our results question the actual possibility of a large population to adapt and find the optimal strategy when participating in a collective game.     

Equivalence of Sobolev Spaces

We repair the proof of equivalence of certain L²-Sobolev spaces on manifolds with bounded curvature of all orders from [4]. The results are extended to generalized compatible Dirac operators, fractional order Sobolev spaces and weighted Sobolev spaces. A certain way of doing coordinate free computations is presented.     

An equivalent roughness model for seabed backscattering at very high frequencies using a band-matrix approach

This work concerns modeling of very high frequency (>100 kHz) sonar images obtained from a sandy seabed. The seabed is divided into a discrete number of ID height profiles. For each height profile the backscattered pressure is computed by an integral equation method for interface scattering between two homogeneous media as formulated by Chan [IEEE Trans. Antennas Propag. 46, 142-149 (1998)]. However, the seabed is inhomogeneous, and volume scattering is a major contributor to backscattering. The SAX99 experiments revealed that the density in the unconsolidated sediment within the first 5 mm exhibits a high spatial variation. For that reason, additional roughness is introduced: For each surface point a stochastic realization of the density along the vertical is generated, and the sediment depth at which the density has its maximum value will constitute the new height field value. The matrix of the full integral equation is reduced to a band matrix as the interaction between the point sources on the seabed is neglected from a certain range; this allows computations on long height profiles with lengths up to approximately 25 m (at 300 kHz). The equivalent roughness approach, combined with the band-matrix approach, agrees with SAX99 data at 300 kHz.     

Mixed finite element approximation for the stochastic pressure equation of Wick type

We propose a mixed finite element method for the numericalsolution of the stochastic pressure equation of Wick type. Inthis formulation, the pressure and the velocity are the mostrelevant unknowns. We give existence and uniqueness resultsfor the continuous problem and its approximation. Optimal errorestimates are derived and algorithmic aspects are discussed.Finally, the results of numerical experiments confirm the practicalefficiency of the mixed method.