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New mathematical results are given for the radiative transfer equations alone and coupled with the temperature equation of a fluid: existence, uniqueness, a maximum principle, and a convergent monotone iterative scheme. Numerical tests for Earth's atmosphere and the heating of a pool by the Sun are included. [ABSTRACT FROM AUTHOR]

This paper develops a finite approximation approach to find the solution F(x) of integral equations in the form F(x) = ξ(x) + R R κ(x, u)dF(u) ∀x ∈ R, where the unknown distributional solution F(x) defines the measure associated with the integration, and F(x) may not be continuous. To the best of our knowledge, the equation solutions in this framework have never been studied before. However, such equations arise frequently when modeling stochastic systems. We construct a Banach space of (right-continuous) distribution functions and reformulate the problem into an operator equation. We provide general necessary and sufficient conditions that allow us to show convergence of the approximation approach developed in this paper. We then provide two specific choices of approximation sequences and show that the properties of these sequences are sufficient to generate approximate equation solutions that converge to the true solution assuming solution uniqueness and some additional mild regularity conditions. Our analysis is performed under the supremum norm, allowing wider applicability of our results. Worst-case error bounds are also available from solving a linear program. We demonstrate the viability and computational performance of our approach by constructing three examples. The solution of the first example can be constructed manually but demonstrates the correctness and convergence of our approach. The second application example involves stationary distribution equations of a stochastic model and demonstrates the dramatic improvement our approach provides over the use of computer simulation. The third example solves a problem involving an everywhere nondifferentiable function for which no closed-form solution is available. [ABSTRACT FROM AUTHOR]

The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of two-dimensional scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system coupling (1) a standard finite element discretization localized around the scatterer and (2) integral equations whose unknowns are traces of the solution on the boundaries of a finite number of overlapping half-planes contained in the domain. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness and equivalence of this HSM formulation to the original scattering problem have been established for complex wavenumbers only. In the present paper we show, in the case of a homogeneous background, that the HSM formulation is equivalent to the original scattering problem also for real wavenumbers, and so is well-posed, provided the traces satisfy radiation conditions at infinity analogous to the standard Sommerfeld radiation condition. As a key component of our argument we show that if the trace on the boundary of a half-plane satisfies our new radiation condition, then the corresponding solution to the half-plane Dirichlet problem satisfies the Sommerfeld radiation condition in a slightly smaller half-plane. We expect that this last result will be of independent interest, in particular in studies of rough surface scattering. [ABSTRACT FROM AUTHOR]

A two-dimensional diffusion process is considered. The distribution of the first exit point of such a process from an arbitrary domain of its values is determined, as a function of the initial point of the process, by an elliptic second-order differential equation, and corresponds to the solution of the Dirichlet problem for this equation (the case of nonconstant coefficients). We examine the distribution density of the first exit point of the process from the small circular neighborhood of its initial point and study its relation to the Dirichlet problem. In terms of this asymptotics, we prove two theorems, which provide sufficient conditions and necessary conditions for the distribution of the first exit point, as a function of the initial point of the process, to satisfy a certain second-order elliptic differential equation corresponding to the standard Wiener process with drift and break. The removable second-order terms of the expansion in powers of the radius of the small circular neighborhood of the initial point of the process are identified. In terms of the removable terms, these two theorems are combined as a single theorem giving a necessary and sufficient condition for correspondence to this Wiener process. [ABSTRACT FROM AUTHOR]

The half-space matching (HSM) method has recently been developed as a new method for the solution of two-dimensional scattering problems with complex backgrounds, providing an alternative to perfectly matched layers or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system of integral equations in which the unknowns are restrictions of the solution to the boundaries of a finite number of overlapping half-planes contained in the domain: this integral equation system is coupled to a standard finite element discretization localized around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness and equivalence to the original scattering problem have been established only for complex wavenumbers. In the present paper, by combining the HSM framework with a complex-scaling technique, we provide a new formulation for real wavenumbers which is provably well-posed and has the attraction for computation that the complex-scaled solutions of the integral equation system decay exponentially at infinity. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results. [ABSTRACT FROM AUTHOR]

We propose to compute a sparse approximate inverse Cholesky factor L of a dense covariance matrix \Theta by minimizing the Kullback--Leibler divergence between the Gaussian distributions \scrN (0, \Theta) and \scrN (0,L \top L 1), subject to a sparsity constraint. Surprisingly, this problem has a closed-form solution that can be computed efficiently, recovering the popular Vecchia approximation in spatial statistics. Based on recent results on the approximate sparsity of inverse Cholesky factors of \Theta obtained from pairwise evaluation of Green's functions of elliptic boundary-value problems at points \{xi\} 1\leq i\leq N \subset Rd, we propose an elimination ordering and sparsity pattern that allows us to compute \epsilon -approximate inverse Cholesky factors of such \Theta in computational complexity \scrO (N log(N/\epsilon)d) in space and \scrO (N log(N/\epsilon)2d) in time. To the best of our knowledge, this is the best asymptotic complexity for this class of problems. Furthermore, our method is embarrassingly parallel, automatically exploits low-dimensional structure in the data, and can perform Gaussian-process regression in linear (in N) space complexity. Motivated by its optimality properties, we propose applying our method to the joint covariance of training and prediction points in Gaussian-process regression, greatly improving stability and computational cost. Finally, we show how to apply our method to the important setting of Gaussian processes with additive noise, compromising neither accuracy nor computational complexity. [ABSTRACT FROM AUTHOR]

This paper is concerned with direct and inverse scattering by a locally perturbed infinite plane (called a locally rough surface in this paper) on which a Neumann boundary condition is imposed. A novel integral equation formulation is proposed for the direct scattering problem which is defined on a bounded curve (consisting of a bounded part of the infinite plane containing the local perturbation and the lower part of a circle) with two corners and some closed smooth artificial curve. It is a nontrivial extension of our previous work on direct and inverse scattering by a locally rough surface from the Dirichlet boundary condition to the Neumann boundary condition [SIAM J. Appl. Math., 73 (2013), pp. 1811{1829]. For the Dirichlet boundary condition, the integral equation obtained is uniquely solvable in the space of bounded continuous functions on the bounded curve, and it can be solved eficiently by using the Nystrom method with a graded mesh. However, the Neumann condition case leads to an integral equation which is solvable in the space of squarely integrable functions on the bounded curve rather than in the space of bounded continuous functions, making the integral equation very dificult to solve numerically with the classic and eficient Nystrom method. In this paper, we make use of the recursively compressed inverse preconditioning method developed by Helsing to solve the integral equation which is eficient and capable of dealing with large wave numbers. For the inverse problem, it is proved that the locally rough surface is uniquely determined from a knowledge of the far-field pattern corresponding to incident plane waves. Moreover, based on the novel integral equation formulation, a Newton iteration method is developed to reconstruct the locally rough surface from a knowledge of multiple frequency far-field data. Numerical examples are also provided to illustrate that the reconstruction algorithm is stable and accurate even for the case of multiple-scale profiles. [ABSTRACT FROM AUTHOR]

We develop a mathematical theory for the excitation of surface plasmon resonance on an infinitely thick metallic slab with a nano-gap defect. Using layer potential techniques, we establish the well-posedness of the underlying scattering problem. We further obtain the asymptotic expansion of the scattering solution in order to characterize the leading-order term of the surface plasmonic waves, and derive sharp estimates for both the plasmonic and nonplasmonic parts of the solution. The explicit dependence of the surface plasmon resonance on the size of the nano-gap, and the real and imaginary parts of the metal dielectric constant, are given. [ABSTRACT FROM AUTHOR]

Every student of numerical linear algebra is familiar with block matrices and vectors. The same ideas can be applied to the continuous analogues of operators, functions, and functionals. It is shown here how the explicit consideration of block structures at the continuous level can be a useful tool. In particular, block operator diagrams lead to templates for spectral discretization of differential and integral equation boundary-value problems in one space dimension by the rectangular differentiation, identity, and integration matrices introduced recently by Driscoll and Hale. The templates are so simple that we are able to present them as executable MATLAB codes just a few lines long, developing ideas through a sequence of 12 increasingly advanced examples. The notion of the rectangular shape of a linear operator is made mathematically precise by the theory of Fredholm operators and their indices, and the block operator formulations apply to nonlinear problems too. We propose the convention of representing nonlinear blocks as shaded. At each step of a Newton iteration for a nonlinear problem, the structure is linearized and the blocks become unshaded, representing Fréchet derivative operators, square or rectangular. [ABSTRACT FROM AUTHOR]

We present a fast direct solver for two-dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of Lippmann{Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad-tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require O(N3/2) work, where N denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both low and high frequency regimes. [ABSTRACT FROM AUTHOR]

We investigate existence and exact multiplicity of phase-locked solutions of an integro-differential equation derived from a Kuramoto system of coupled oscillators. Under general assumptions on the form of frequency distribution, we derive new, easily verified criteria which guarantee that either (i) exactly one solution exists, or (ii) exactly two solutions coexist over an entire interval of values of the key parameter γ. We illustrate our results with an example in which each of these possibilities occurs. Problems for future research are suggested. [ABSTRACT FROM AUTHOR]

The "drum problem"--finding the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary condition--has many applications, yet remains challenging for general domains when high accuracy or high frequency is needed. Boundary integral equations are appealing for large-scale problems, yet certain difficulties have limited their use. We introduce the following two ideas to remedy this: (1) We solve the resulting nonlinear eigenvalue problem using Boyd's method for analytic root-finding applied to the Fredholm determinant, and we show that this is many times faster than the usual iterative minimization of a singular value. (2) We fix the problem of spurious exterior resonances via a combined field representation. This also provides the first robust boundary integral eigenvalue method for non-simply connected domains. We implement the new method in two dimensions using spectrally accurate Nyström product quadrature. We prove exponential convergence of the determinant at roots for domains with analytic boundary. We demonstrate 13-digit accuracy, and improved efficiency, in a variety of domain shapes including a nonconvex cavity shape with strong exterior resonances. [ABSTRACT FROM AUTHOR]

This paper is concerned with the direct and inverse acoustic or electromagnetic scattering problems by a locally perturbed, perfectly reflecting, infinite plane (which is called a locally rough surface in this paper). We propose a novel integral equation formulation for the direct scattering problem which is defined on a bounded curve (consisting of a bounded part of the infinite plane containing the local perturbation and the lower part of a circle) with two corners. This novel integral equation can be solved efficiently by using the Nyström method with a graded mesh introduced previously by Kress and is capable of dealing with large wave number cases. For the inverse problem, we propose a Newton iteration method to reconstruct the local perturbation of the plane from multiple-frequency far-field data, based on the novel integral equation formulation. Numerical examples are carried out to demonstrate that our reconstruction method is stable and accurate even for the case of multiple-scale profiles. [ABSTRACT FROM AUTHOR]

In this work, we consider the problem of boundary stabilization for a quasilinear 2 x 2 system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves H² exponential stability of the closedloop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type 4 x 4 system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them. [ABSTRACT FROM AUTHOR]

In this paper, we present a boundary integral method for computing the quasi-steady evolution of an epitaxial island. The problem consists of all adatom diffusion equation (with desorption) on terrace and a kinetic boundary condition at the step (island boundary). The normal velocity for step motion is determined by a two-sided flux. The integral formulation of the problem involves both double- and single-layer potentials due to the kinetic boundary condition. Numerical tests on a growing/shrinking circular or a slightly perturbed circular island are ill excellent agreement with the linear analysis, demonstrating that the method is stable, efficient, and spectrally accurate in space. Nonlinear simulations for the growth of perturbed circular islands show that sharp tips and fiat edges will form during growth instead of the usual dense branching morphology seen throughout physical and biological systems driven out of equilibrium. In particular, Bales-Zangwill instability is manifested ill the form of wave-like fronts (meandering instability) around the tip regions. The numerical techniques presented here can be applied generally to a class of free/moving boundary problems ill physical and biological science. [ABSTRACT FROM AUTHOR]

We present an analytic proof of the Pecherskii-Rogozin identity and the Wiener-Hopf factorization. The proof is rather general and requires only one mild restriction on the tail of the Lévy measure. The starting point of the proof of the Pecherskii-Rogozin identity is a two-dimensional integral equation satisfied by the joint distribution of the first passage time and the overshoot. This equation is reduced to a one-dimensional Wiener-Hopf integral equation, which is then solved using classical techniques from the theory of the Riemann boundary value problems. The Wiener-Hopf factorization is then derived as a corollary of the Pecherskii-Rogozin identity. [ABSTRACT FROM AUTHOR]

A class of generalized analytic functions arising from the Oseen equations in the axially symmetric case has been identified. For this class of functions, the generalized Cauchy integral formula has been obtained, and a series representation for the region exterior to a sphere has been constructed. The velocity field of the axially symmetric Oseen flow has been represented in terms of two generalized analytic functions, and it has been shown that for an exterior Oseen flow problem, those functions are uniquely determined, provided that they both vanish at infinity. Also the pressure and vorticity have been determined as the real and imaginary parts of the two functions representing the velocity field, and the drag exerted on a solid body of revolution in the axially symmetric Oseen flow has been expressed in terms of one of the involved generalized analytic functions. The problem of the axially symmetric Oseen flow past a solid body of revolution has been reduced to an integral equation based on the generalized Cauchy integral formula. The integral equation has been shown to have computational advantage over an integral equation based on the Oseenlets. The developed framework of generalized analytic functions has been illustrated in solving the problem of the Oseen flow past a solid sphere and solid bispheroids. For different Reynolds numbers, a minimum drag spheroid of fixed volume has been found. [ABSTRACT FROM AUTHOR]

An integral equation constrained optimization approach to finding minimum-drag shapes for solid bodies of revolution in Stokes flows subject to constraints on body's volume and shape has been developed. An axially symmetric Stokes flow problem has been formulated in the form of an integral equation (state equation), and finding an optimal shape has been reduced to an integral equation constrained optimization problem. The total variation of the Lagrangian functional of the problem has been reduced to the variation only with respect to the shape by the adjoint equation-based method, and the steepest descent direction for the coefficients in the function series representing the shape has been found analytically. It has been shown that solutions to the state and adjoint integral equations are related by a simple algebraic formula, which eliminates the need for solving the adjoint equation. The approach has been illustrated in finding the minimum-drag shape for a solid unit volume particle and in finding the optimal shape for the fixed-volume nose of a solid torpedo-shaped body encountering minimum drag along its axis of revolution. In the later problem, the dependence of the nose's optimal shape on the torpedo's length has been investigated. [ABSTRACT FROM AUTHOR]

We consider an optimal stopping problem for the Russian option in the Bachelier model with a finite time horizon. We obtain an integral equation, which yields us a border between stopping and continuation sets. Also the asymptotic behavior of this border at 0 and infinity is found. [ABSTRACT FROM AUTHOR]

The framework of generalized analytic functions arising from the related potentials (so-called k-harmonically analytic functions) has been developed in application to three-dimensional (3D) axially symmetric Stokes flow problems. Cauchy's integral formula for the class of k-harmonically analytic functions has been obtained, and series representations for k-harmonically analytic functions for the regions exterior to sphere and prolate and oblate spheroids have been derived. As the central result in the developed framework, a solution form representing the velocity field and pressure for 3D axially symmetric Stokes flows has been constructed in terms of two 0-harmonically analytic functions. It has also been shown that it uniquely determines an external velocity field vanishing at infinity. With the obtained solution form, the problem of 3D Stokes flows due to the axially symmetric translation of a solid body of revolution has been reduced to a boundary-value problem for two 0-harmonically analytic functions, and the resisting force exerted on the body has been expressed in terms of a 0-harmonically analytic function entering the solution form. For regions in which Laplace's equation admits separation of variables, the boundary-value problem can be solved analytically via representations of 0-harmonically analytic functions in corresponding curvilinear coordinates. This approach has been demonstrated for the axially symmetric translation of solid sphere and solid prolate and oblate spheroids. As the second approach, the boundary-value problem has been reduced to an integral equation based on Cauchy's integral formula for k-harmonically analytic functions. As an illustration, the integral equation has been solved for the axially symmetric translation of solid bispheroids and the solid torus of elliptical cross-section for various values of a geometrical parameter. [ABSTRACT FROM AUTHOR]

A skeletal muscle is composed of motor units, each consisting of a motoneuron and the muscle fibers it innervates. The input to the motor units is formed of electrical signals coming from higher motor centers and propagated to the motoneurons along a network of nerve fibers. Because of its complexity, this network still escapes actual direct observations. The present model describes the steady state activation of a muscle, i.e., of its motor units. It incorporates the network as an unknown quantity and, given the latter, predicts the input-force relation (activation curve) of the muscle. Conversely, given a suitable activation curve, our model enables the recovery of the network. This step is performed by using experimental data about the activation curve, and the whole activation process of a muscle can then be theoretically investigated. In this way, this approach provides a link between the macroscopic (activation curve) and microscopic (network) levels. From a mathematical viewpoint, solving the preceding inverse problem is equivalent to solving an integral equation of a new type. [ABSTRACT FROM AUTHOR]

Changes to the relative separation of molecules or other interacting species on account of diffusion accompany their associative or dissociative reaction. The molecules are symbolized, for two distinct types, A,B, by the relations A +B AB, and, if [A], [B], and [AB] denote the corresponding densities, the equation d/dt [AB] = k+[A][B] specifies an associative process with forward rate constant k+. An approximate version of the preceding takes the form of a linear differential equation, which can be employed to obtain significant estimates for both k+ and the flux function d[AB]/dt. Such estimates are presented in different circumstances, including the localization of A,B on a common planar surface or their distribution in space, and also when the domain of A is a half space whereas that of B is a bounding planar surface. It proves advantageous to reformulate the last, a mixed boundary value problem, in terms of a linear integral equation. Biological applications are discussed, including the mechanism for the observed phosphorylation of proteins in resting cells and the incipience of phototransduction in rod photoreceptors. [ABSTRACT FROM AUTHOR]

In this paper we analyze a Galerkin procedure, based on a combination of finite and spectral elements, for approximating a time-harmonic acoustic wave scattered by a bounded inhomogeneity. The finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition, which consists in a linear integral equation. This integral equation is discretized by a spectral Galerkin approximation method. We provide error estimates for the Galerkin method, propose fully discrete schemes based on elementary quadrature formulas, and show that the perturbation due to this numerical integration gives rise to a quasi-optimal rate of convergence. We also suggest a method for implementing the algorithm using the preconditioned GMRES method and provide some numerical results. [ABSTRACT FROM AUTHOR]

Considering a diffusion process in a d-dimensional space, we study the distribution of the first exit point of a process from a small neighborhood of its initial state. A weak asymptotic expansion in terms of the small scale parameter is obtained for the density of the distribution. In the case of spherical neighborhoods, simple formulas are deduced for the first three coefficients of the expansion, reflecting the probabilistic sense of the coefficients of an elliptic partial differential equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the implementation and investigation of integral equation based solvers as preconditioners for finite difference discretizations of Poisson equations in geo- metrically complex domains. The target discretizations are those associated with ‘cut-out’ grids. We discuss such grids and also describe a software structure which enables their rapid construction. Computational results are presented. [ABSTRACT FROM AUTHOR]

We consider a two-dimensional problem of scattering of a time-harmonic electromagnetic plane wave by an infinite inhomogeneous conducting or dielectric layer at the interface between semi-infinite homogeneous dielectric half-spaces. The magnetic permeability is assumed to be a fixed positive constant. The material properties of the media are characterized completely by an index of refraction, which is a bounded measurable function in the layer and takes positive constant values above and below the layer, corresponding to the homogeneous dielectric media. In this paper, we examine only the transverse magnetic (TM) polarization case. A radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for difraction gratings. With the help of the radiation condition the problem is reformulated as an equivalent mixed system of boundary and domain integral equations, consisting of second-kind integral equations over the layer and interfaces within the layer. Assumptions on the variation of the index of refraction in the layer are then imposed which prove to be suficient, together with the radiation condition, to prove uniqueness of solution and nonexistence of guided wave modes. Recent, general results on the solvability of systems of second kind integral equations on unbounded domains establish existence of solution and continuous dependence in a weighted norm of the solution on the given data. The results obtained apply to the case of scattering by a rough interface between two dielectric media and to many other practical configurations. [ABSTRACT FROM AUTHOR]