Bayesian source detection and parameter estimation of a plume model based on sensor network measurements

We consider a network of sensors that measure the intensities of a complex plume composed of multiple absorption–diffusion source components. We address the problem of estimating the plume parameters, including the spatial and temporal source origins and the parameters of the diffusion model for each source, based on a sequence of sensor measurements. The approach not only leads to multiple‐source detection, but also the characterization and prediction of the combined plume in space and time. The parameter estimation is formulated as a Bayesian inference problem, and the solution is obtained using a Markov chain Monte Carlo algorithm. The approach is applied to a simulation study, which shows that an accurate parameter estimation is achievable. Copyright © 2010 John Wiley & Sons, Ltd.     

Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation

An initial–boundary value problem for the two-dimensional heat equation with a source is considered. The source is the sum of two unknown functions of spatial variables multiplied by exponentially decaying functions of time. The inverse problem is stated of determining two unknown functions of spatial variables from additional information on the solution of the initial–boundary value problem, which is a function of time and one of the spatial variables. It is shown that, in the general case, this inverse problem has an infinite set of solutions. It is proved that the solution of the inverse problem is unique in the class of sufficiently smooth compactly supported functions such that the supports of the unknown functions do not intersect. This result is extended to the case of a source involving an arbitrary finite number of unknown functions of spatial variables multiplied by exponentially decaying functions of time.     

Uniqueness for a seismic inverse source problem modeling a subsonic rupture

We consider an inverse source problem for an inhomogeneous wave equation with discrete-in-time sources, modeling a seismic rupture. The inverse source problem, with an arbitrary source term on the right-hand side of the wave equation, is not uniquely solvable. Here we formulate conditions on the source term that allow us to show uniqueness and that provide a reasonable model for the application of interest. We assume that the source term is supported on a finite set of times and that the support in space moves with subsonic velocity. Moreover, we assume that the spatial part of the source is singular on a hypersurface, an application being a seismic rupture along a fault plane. Given data collected over time on a detection surface that encloses the spatial projection of the support of the source, we show how to recover the times and locations of sources microlocally and then reconstruct the smooth part of the source assuming that it is the same at each source location.     

Fujita type critical exponent for a free boundary problem with spatial–temporal source

In this paper, we investigate a nonlinear free boundary problem incorporating with nontrivial spatial and exponential temporal weighted source. To portray the asymptotic behavior of the solution, we first derive some sufficient conditions for finite time blowup. Furthermore, the global vanishing solution is also obtained for a class of small initial data. Finally, a sharp threshold trichotomy result is provided in terms of the size of the initial data to distinguish the blowup solution, the global vanishing solution, and the global transition solution. In particular, our results show that such a problem always possesses a Fujita type critical exponent whenever the spatial source is just equivalent to a trivial constant, or is an extreme one, such as “very negative” one in the sense of measure or integral.     

A spatial filtering specification for the auto-Poisson model

The auto-Poisson model describes georeferenced data consisting of counts exhibiting spatial dependence. Its conventional specification is plagued by being restricted to only situations involving negative spatial autocorrelation, and an intractable normalizing constant. Work summarized here accounts for spatial autocorrelation in the mean response specification by incorporating latent map pattern components. Results are reported for seven empirical datasets available in the literature.     

The spatial behavior of rotating two-component Bose-Einstein condensates

In the understanding of the spatial behavior of interacting components of rotating two-component Bose-Einstein condensates, a central problem is to establish whether coexistence of all the components occurs, or the interspecies interaction leads to extinction, that is, configurations where one or more densities are null. In this paper, we prove that the interspecies interaction leads to extinction in the Thomas-Fermi approximation in dimension three.     

Nonclassical symmetry solutions for reaction–diffusion equations with explicit spatial dependence

Nonclassical symmetry methods are used to study the linear diffusion equation with a nonlinear source term which includes explicit spatial dependence. Mathematical forms for the spatial dependence are found which enable strictly nonclassical symmetries to be admitted when the nonlinearity is cubic. A number of new exact solutions are constructed, and an application of one of these solutions to diploid population genetics is discussed.     

Spatial autoregression and related spatio-temporal models

We propose a spatial autoregressive random field of order p on the spatial domain for p?2 in this paper, whose univariate margins are the continuous-time autoregression of order p on the real line, and introduce a class of semiparametric spatio-temporal covariance models stationary in space with the spatial autoregressive margin.     

Characterization of the Reynolds-stress and dissipation-rate decay and anisotropy from DNS of grid-generated turbulence

Grid–generated turbulence is a classical but still controversial topic, one open issue being the spatial decay rate of turbulent energy. We study the influence of grid geometry on the Reynolds–stress and dissipation–rate tensors, including the range and exponent of their self–similar spatial decay. DNS using a validated lattice Boltzmann code at mean–flow Reynolds numbers up to 1400 are performed, comparing square grids with blockage ratios from 0.05 to 0.49. A clear picture of spatial distribution and self–similarity emerges for the statistics of interest: Axisymmetry is excellently confirmed. A consistent power law decay is found in the self–similar decay region beyond 10 grid stride lengths downstream. Its exponent of –5/3 can be obtained, for weak turbulence, from a spatial flux balance reminiscent of the constant transport through the inertial range of isotropic turbulence. In the near–grid region, on the other hand, differences in Reynolds stress components are pronounced while those between dissipation tensor components are only recognizable very close to the grid, where a strong dependence on grid porosity is found. A normalization with respect to porosity is proposed that collapses the data from all runs. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the analyticity and the almost periodicity of the solution to the Euler equations with non-decaying initial velocity

The Cauchy problem of the Euler equations in the whole space is considered with non-decaying initial velocity in the frame work of . It is proved that if the initial velocity is real analytic then the solution is also real analytic in spatial variables. Furthermore, a new estimate for the size of the radius of convergence of Taylor's expansion is established. The key of the proof is to derive the suitable estimates for the higher order derivatives of the bilinear terms. It is also shown the propagation of the almost periodicity in spatial variables.     

Convex source support in three dimensions

This work extends the algorithm for computing the convex source support in the framework of the Poisson equation to a bounded three-dimensional domain. The convex source support is, in essence, the smallest (nonempty) convex set that supports a source that produces the measured (nontrivial) data on the boundary of the object. In particular, it belongs to the convex hull of the support of any source that is compatible with the measurements. The original algorithm for reconstructing the convex source support is inherently two-dimensional as it utilizes M?bius transformations. However, replacing the M?bius transformations by inversions with respect to suitable spheres and introducing the corresponding Kelvin transforms, the basic ideas of the algorithm carry over to three spatial dimensions. The performance of the resulting numerical algorithm is analyzed both for the inverse source problem and for electrical impedance tomography with a single pair of boundary current and potential as the measurement data.     

Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media

Nonlinear dynamical stochastic models are ubiquitous in different areas. Their statistical properties are often of great interest, but are also very challenging to compute. Many excitable media models belong to such types of complex systems with large state dimensions and the associated covariance matrices have localized structures. In this article, a mathematical framework to understand the spatial localization for a large class of stochastically coupled nonlinear systems in high dimensions is developed. Rigorous \linebreak mathematical analysis shows that the local effect from the diffusion results in an exponential decay of the components in the covariance matrix as a function of the distance while the global effect due to the mean field interaction synchronizes different components and contributes to a global covariance. The analysis is based on a comparison with an appropriate linear surrogate model, of which the covariance propagation can be computed explicitly. Two important applications of these theoretical results are discussed. They are the spatial averaging strategy for efficiently sampling the covariance matrix and the localization technique in data assimilation. Test examples of a linear model and a stochastically coupled FitzHugh-Nagumo model for excitable media are adopted to validate the theoretical results. The latter is also used for a systematical study of the spatial averaging strategy in efficiently sampling the covariance matrix in different dynamical regimes.     

Symmetry analysis of the two-dimensional diffusion equation with a source term

A symmetry analysis is performed on a (2+1)-dimensional linear diffusion equation with a nonlinear source term involving the dependent variable and its spatial derivatives. In the first part of the paper, we use the classical method to classify source terms where the original equation admits a nontrivial symmetry. In the second part of the paper, we use the nonclassical method and show that we simply recover the classical symmetries.     

Adaptive multilevel methods in space and time for parabolic problems-the periodic case

The aim of this paper is to display numerical results that show the interest of some multilevel methods for problems of parabolic type. These schemes are based on multilevel spatial splittings and the use of different time steps for the various spatial components. The spatial discretization we investigate is of spectral Fourier type, so the approximate solution naturally splits into the sum of a low frequency component and a high frequency one. The time discretization is of implicit/explicit Euler type for each spatial component. Based on a posteriori estimates, we introduce adaptive one-level and multilevel algorithms. Two problems are considered: the heat equation and a nonlinear problem. Numerical experiments are conducted for both problems using the one-level and the multilevel algorithms. The multilevel method is up to 70% faster than the one-level method.

     

Determining an unknown source in the heat equation by a wavelet dual least squares method

We consider the problem of determining an unknown source, which depends only on the spatial variable, in a heat equation. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. For a reconstruction of the unknown source from measured data the dual least squares method generated by a family of Meyer wavelet subspaces is applied. An explicit relation between the truncation level of the wavelet expansion and the data error bound is found, under which the convergence result with the error estimate is obtained.     

Solution of the advection-dispersion equation with free exit boundary

I investigated the exit boundary condition for the advection-dispersion equation and found that in numerical solutions of this equation, using Galekin finite elements, a free exit boundary condition requiring no a priori information is possible, provided the advective component in the numerical equations is of sufficient magnitude relative to the dispersive component. Since the relationship between these two components is controlled by the spatial discretization through the grid Peclet number, the free exit boundary condition can in fact be applied whenever there is a non-zero advective component. The numerical solution in a finite domain with free exit boundary, using a correctly proportioned spatial discrezation, behaves like an infinite-domain solution.     

A circular embedding of a graph in Euclidean 3-space

A spatial embedding of a graph G is an embedding of G into the 3-dimensional Euclidean space . J.H. Conway and C.McA. Gordon proved that every spatial embedding of the complete graph on 7 vertices contains a nontrivial knot. A linear spatial embedding of a graph is an embedding which maps each edge to a single straight line segment. In this paper, we construct a linear spatial embedding of the complete graph on 2n−1 (or 2n) vertices which contains the torus knot T(2n−5,2) (n4). A circular spatial embedding of a graph is an embedding which maps each edge to a round arc. We define the circular number of a knot as the minimal number of round arcs in among such embeddings of the knot. We show that a knot has circular number 3 if and only if the knot is a trefoil knot, and the figure-eight knot has circular number 4.     

Rotating multicomponent Bose–Einstein condensates

In the understanding of the spatial behavior of interacting components of multicomponent Bose–Einstein condensates (BECs), a central problem is to establish whether coexistence of all the components occurs, or the interspecies interaction leads to extinction, that is, configurations where one or more densities are null. In this paper, for the rotating k-mixture BEC, we prove that the interspecies interaction leads to extinction in the Thomas–Fermi approximation.     

Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws

The stability of traveling wave solutions of scalar viscous conservation laws is investigated by decomposing perturbations into three components: two far-field components and one near-field component. The linear operators associated to the far-field components are the constant coefficient operators determined by the asymptotic spatial limits of the original operator. Scaling variables can be applied to study the evolution of these components, allowing for the construction of invariant manifolds and the determination of their temporal decay rate. The large time evolution of the near-field component is shown to be governed by that of the far-field components, thus giving it the same temporal decay rate. We also give a discussion of the relationship between this geometric approach and previous results, which demonstrate that the decay rate of perturbations can be increased by requiring that initial data lie in appropriate algebraically weighted spaces.     

Adaptive iterative thresholding algorithms for magnetoencephalography (MEG)

We provide fast and accurate adaptive algorithms for the spatial resolution of current densities in MEG. We assume that vector components of the current densities possess a sparse expansion with respect to preassigned wavelets. Additionally, different components may also exhibit common sparsity patterns. We model MEG as an inverse problem with joint sparsity constraints, promoting the coupling of non-vanishing components. We show how to compute solutions of the MEG linear inverse problem by iterative thresholded Landweber schemes. The resulting adaptive scheme is fast, robust, and significantly outperforms the classical Tikhonov regularization in resolving sparse current densities. Numerical examples are included.