An efficient high-order Nystr\"om scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

This text proposes a fast, rapidly convergent Nystr\"{o}m method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of $O(N \log N)$ for an $N$-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper

Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity principle (DRM), this paper presents an inheretnly meshless, exponential convergence, integration-free, boundary-only collocation techniques for numerical solution of general partial differential equation systems. The basic ideas behind this methodology are very mathematically simple and generally effective. The RBFs are used in this study to approximate the inhomogeneous terms of system equations in terms of the DRM, while non-singular general solution leads to a boundary-only RBF formulation. The present method is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of non-singular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does no need to introduce the artificial boundary and results in the symmetric system equations under certain conditions. It is also found that the BKM can solve nonlinear partial differential equations one-step without iteration if only boundary knots are used. The efficiency and utility of this new technique are validated through some typical numerical examples. Some promising developments of the BKM are also discussed.Comment: 36 pages, 2 figures, Welcome to contact me on this paper: Email: [email protected] or [email protected]

First-passage-time statistics of a Brownian particle driven by an arbitrary unidimensional potential with a superimposed exponential time-dependent drift

In one-dimensional systems, the dynamics of a Brownian particle are governed by the force derived from a potential as well as by diffusion properties. In this work, we obtain the first-passage-time statistics of a Brownian particle driven by an arbitrary potential with an exponential temporally decaying superimposed field up to a prescribed threshold. The general system analyzed here describes the sub-threshold signal integration of integrate-and-fire neuron models, of any kind, supplemented by an adaptation-like current, whereas the first-passage-time corresponds to the declaration of a spike. Following our previous studies, we base our analysis on the backward Fokker Planck equation and study the survival probability and the first-passage-time density function in the space of the initial condition. By proposing a series solution we obtain a system of recurrence equations, which given the specific structure of the exponential time-dependent drift, easily admit a simpler Laplace representation. Naturally, the present general derivation agrees with the explicit solution we found previously for the Wiener process in (2012 JPhysA 45 185001). However, to demonstrate the generality of the approach, we further explicitly evaluate the first-passage-time statistics of the underlying Ornstein Uhlenbeck process. To test the validity of the series solution, we extensively compare theoretical expressions with the data obtained from numerical simulations in different regimes. As shown, agreement is precise whenever the series is truncated at an appropriate order. Beyond the fact that both the Wiener and Ornstein Uhlenbeck processes have a direct interpretation in the context of neuronal models, given their ubiquity in different fields, our present results will be of interest in other settings where an additive state-independent temporal relaxation process is being developed as the particle diffuses.Comment: 22 pages (20 pages in the journal version), 3 figures, published in J. Phys.

On the fundamental resonant mode of inhomogeneous soil deposits

The problem of estimating seismic ground deformation is central to state-of-practice procedures of designing and maintaining infrastructure in earthquake-prone areas. Particularly, the problem of estimating the displacement field in a soft shallow layer overlying rigid bedrock induced by simple shear wave excitation has been favored by engineers due to its simplicity combined with inherent relevance for practical scenarios. We here derive analytical estimates for both the fundamental frequency and the amplitude of the first resonant mode of such systems by applying an intuitive argument based on resonance of single-degree-of-freedom systems. Our estimates do not presuppose a continuous velocity distribution, and can be used for fast assessment of site response in seismic hazard assessment and engineering design. On the basis of the said estimates of fundamental frequency and amplitude, we next propose a novel definition of “equivalent homogeneous shear modulus” of the inhomogeneous deposit, and we show that the response of the fundamental mode is controlled primarily by the properties of the layers contiguous to the bedrock. We finally discuss the validity of our argument, and evaluate the accuracy of our results by comparison with analytical and numerical solutions