AbstractWe solve the Milne, constant-source, and albedo problems for isotropic scattering in a two-dimensional “Flatland” half-space via the Wiener–Hopf method. The Flatland H-function is derived a...
In the study of energy-dependent neutron transport in three-dimensional volumes with plane symmetry, @cite_0 presented a general family of solutions to the Milne problem using the method of singular eigenfunctions. When their variable- @math factor @math takes on the specific quantity (in our notation) @math , their energy-dependent 3D solution becomes equivalent to our monoenergetic Flatland solution. Thus, the Flatland @math -function has, if only inadvertently, been presented long ago.
@cite_0
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Abstract Mathematical equivalence is demonstrated between problems for a separable scattering kernel in energy-dependent neutron transport and for the uniform line-blanketing model of nongrey radiative transfer. As an example, the solution of the Milne problem is rederived by the method of singular eigenfunctions, with the application of orthogonality relations. AbstractWe solve the Milne, constant-source, and albedo problems for isotropic scattering in a two-dimensional “Flatland” half-space via the Wiener–Hopf method. The Flatland H-function is derived a... Abstract Two classic problems of radiative transfer and neutron transport are solved for a spatially-uniform semi-infinite medium with isotropic scattering. General analytical equations are derived for (1) angular moments of the outward current and (2) spatial moments of the total flux within the half-space. Such moments, for example, can provide analytically explicit equations for determining the surface albedo of the medium as well as the mean depth and mean square distance of travel within the medium. The analysis is done with the Case-style eigenmode method as expressed in terms of the Chandrasekhar H -function and its moments. We study the flatland (two dimensional) linear transport equation, under an angular2@pperiodicity assumption both on particle density function @j(x,y,@q) and on the differential scattering @s"s(@q). We consider the beam problem, with a forward peaked source on phase-space, and derive P"1 approximation with a diffusion coefficient of 1 2@s"t"r, (versus 1 3@s"t"r of the three dimensional problem), where @s"t"r is the transport cross section. Further assumptions as peaked @s"s(@q) near @q=0 (small angle of scattering), and small angle of flight (@q 0) yield Fokker-Planck and Fermi approximations with the diffusion coefficients @s"t"r (rather than @s"t"r 2 of the three dimensional case). We discretize the problem using four different Galerkin schemes and justify the results through some numerical examples. This paper describes a Benders decomposition-based framework for solving the large scale energy management problem that was posed for the ROADEF 2010 challenge. The problem was taken from the power industry and entailed scheduling the outage dates for a set of nuclear power plants, which need to be regularly taken down for refueling and maintenance, in such a way that the expected cost of meeting the power demand in a number of potential scenarios is minimized. We show that the problem structure naturally lends itself to Benders decomposition; however, not all constraints can be included in the mixed integer programming model. We present a two phase approach that first uses Benders decomposition to solve the linear programming relaxation of a relaxed version of the problem. In the second phase, integer solutions are enumerated and a procedure is applied to make them satisfy constraints not included in the relaxed problem. To cope with the size of the formulations arising in our approach we describe efficient preprocessing techniques to reduce the problem size and show how aggregation can be applied to each of the subproblems. Computational results on the test instances show that the procedure competes well on small instances of the problem, but runs into difficulty on larger ones. Unlike heuristic approaches, however, this methodology can be used to provide lower bounds on solution quality. This paper offers the first variational approach to the problem of dense 3D reconstruction of non-rigid surfaces from a monocular video sequence. We formulate non-rigid structure from motion (nrsfm) as a global variational energy minimization problem to estimate dense low-rank smooth 3D shapes for every frame along with the camera motion matrices, given dense 2D correspondences. Unlike traditional factorization based approaches to nrsfm, which model the low-rank non-rigid shape using a fixed number of basis shapes and corresponding coefficients, we minimize the rank of the matrix of time-varying shapes directly via trace norm minimization. In conjunction with this low-rank constraint, we use an edge preserving total-variation regularization term to obtain spatially smooth shapes for every frame. Thanks to proximal splitting techniques the optimization problem can be decomposed into many point-wise sub-problems and simple linear systems which can be easily solved on GPU hardware. We show results on real sequences of different objects (face, torso, beating heart) where, despite challenges in tracking, illumination changes and occlusions, our method reconstructs highly deforming smooth surfaces densely and accurately directly from video, without the need for any prior models or shape templates. In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ℝ d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ℬ R ≈(ℒ−λ I)−1, where ℒ is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ℝ d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e −i κ‖x‖ ‖x‖, x∈ℝ d , κ∈ℝ+, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ |log e|2 n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2), even in the case κ=O(n). We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy @math in the localized phase. Assume the density of states function is not too flat near @math . Restrict it to some large cube @math . Consider now @math , a small energy interval centered at @math that asymptotically contains infintely many eigenvalues when the volume of the cube @math grows to infinity. We prove that, with probability one in the large volume limit, the eigenvalues of the random Hamiltonian restricted to the cube inside the interval are given by independent identically distributed random variables, up to an error of size an arbitrary power of the volume of the cube. As a consequence, we derive * uniform Poisson behavior of the locally unfolded eigenvalues, * a.s. Poisson behavior of the joint distibutions of the unfolded energies and unfolded localization centers in a large range of scales. * the distribution of the unfolded level spacings, locally and globally, * the distribution of the unfolded localization centers, locally and globally. Abstract A new “2D 1D” equation is proposed to approximate the 3D linear Boltzmann equation. The approximate 2D 1D equation preserves the exact transport physics in the radial directions x and y but employs diffusion physics in the axial direction z . The 2D 1D equation can be systematically discretized, yielding accurate simulation methods for 3D reactor core problems. The resulting 2D 1D solutions are more accurate than 3D diffusion solutions, and are less expensive to calculate than standard 3D transport solutions. In this paper, we (i) show that the simplest 2D 1D equation has certain desirable properties, (ii) systematically discretize this equation, (iii) derive stable iteration schemes for solving the discrete system of equations, and (iv) give numerical results for simple problems that confirm the theoretical predictions of accuracy and iterative stability. In this paper we give a unified proof to the soliton resolution conjecture along a sequence of times, for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations, including the four dimensional radial Yang Mills equation, without using outer energy type inequalities. Such inequalities have played a crucial role in previous works with similar results. Roughly speaking, we prove that along a sequence of times @math (the maximal time of existence), the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space, plus a free radiation term in the global case or a regular part in the finite time blow up case. The main difficulty is that in general (for instance for the radial four dimensional Yang Mills case and the radial six dimensional semilinear wave case) we do not have a favorable outer energy inequality for the associated linear wave equations. Our main new input is the simultaneous use of two virial identities. 1. IntroductionIn this work we address to the problem of the membership to the ideal of r-nuclearoperators for integral operators of the form(1.1) Tf(x) =Z G(x)F(xy)H(y)f(y)dy;where Gand H are bounded functions and F is a smooth function admitting asuitable Taylor expansion. A such kind of operator arises in the theory of laser,particularly in the form of the Fox-Li operator which has attracted the interest ofresearchers in recent years(cf. [BBIN10], [BGI12], [BGHI12]). In those works, theauthors were focused in the investigation of the asymptotic distribution of the sin-gular values and eigenvalues. More recently, the r-nuclearity and its connection withGrothendieck-Lidskii’s formula on compact Lie groups and applications to some dif-ferential operators have been studied in [DR13]. Applications of nuclear operators tothe study of periodic pseudo-di erential operators have been established in [DW13].The nuclearity of operators on spaces of Bochner integrable functions(cf. [DE09])and the corresponding relation with the martingale vector-maximal function for thestudy of traces have been considered in [Del]. Recent progress on the nuclearity andsingular values can also be found in [FMO09] and [MO12].The Fox-Li operator was introduced in [FL61], A. G. Fox and T. Li considered therepeated reection of an electromagnetic wave between two plane-parallel rectangularmirrors. The current density at the surface of each reector satis es the eigenvalueintegral equationTf(x) = f(x):For a good account on the physics of such kind of operators we recommend the chapterXI, section 60 in the book [NE05] by Trefethen and Embree. The Fox-Li operator is The linear Boltzmann equation can be solved with separation of variables in one dimension, i.e., in three-dimensional space with planar symmetry. In this method, solutions are given by superpositions of eigenmodes which are sometimes called singular eigenfunctions. In this paper, we explore the singular-eigenfunction approach in flatland or two-dimensional space. A new numerical algorithm for solving the symmetric eigenvalue problem is presented. The technique deviates fundamentally from the traditional Krylov subspace iteration based techniques (Arnoldi and Lanczos algorithms) or other Davidson-Jacobi techniques, and takes its inspiration from the contour integration and density matrix representation in quantum mechanics. It will be shown that this new algorithm - named FEAST - exhibits high efficiency, robustness, accuracy and scalability on parallel architectures. Examples from electronic structure calculations of Carbon nanotubes (CNT) are presented, and numerical performances and capabilities are discussed. This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher–Rao metric and is averaged with the transportation term. This gives rise to a convex variational problem defining the new metric. Our first contribution is a proof of the existence of geodesics (i.e., solutions to this variational problem). We then show that (generalized) optimal transport and Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. Lastly, we propose a numerical scheme making use of first-order proximal splitting methods and we show an application of this new distance to image interpolation. We propose a method for solving the vibrational Schrodinger equation with which one can compute hundreds of energy levels of seven-atom molecules using at most a few gigabytes of memory. It uses nested contractions in conjunction with the reduced-rank block power method (RRBPM) described in J. Chem. Phys. 2014, 140, 174111. Successive basis contractions are organized into a tree, the nodes of which are associated with eigenfunctions of reduced-dimension Hamiltonians. The RRBPM is used recursively to compute eigenfunctions of nodes in bases of products of reduced-dimension eigenfunctions of nodes with fewer coordinates. The corresponding vectors are tensors in what is called CP-format. The final wave functions are therefore represented in a hierarchical CP-format. Computational efficiency and accuracy are significantly improved by representing the Hamiltonian in the same hierarchical format as the wave function. We demonstrate that with this hierarchical RRBPM it is possible to compute energy levels of a 6... We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units. We present an energy-stable low-storage high order discontinuous Galerkin method for the wave equation.We describe algorithms for efficient implementation on Graphics Processing Units (GPUs).We derive order-dependent global and local timestep restrictions for DG on hybrid meshes.We verify high order convergence and computational efficiency with numerical experiments. This paper is part of the radial asymptotic stability analysis of the ground state soliton for either the cubic nonlinear Schrodinger or Klein–Gordon equations in three dimensions. We demonstrate by a rigorous method that the linearized scalar operators which arise in this setting, traditionally denoted by L±, satisfy the gap property, at least over the radial functions. This means that the interval (0, 1] does not contain any eigenvalues of L± and that the threshold 1 is neither an eigenvalue nor a resonance. The gap property is required in order to prove scattering to the ground states for solutions starting on the centre-stable manifold associated with these states. This paper therefore provides the final instalment in the proof of this scattering property for the cubic Klein–Gordon equation in the radial case, see the recent theory of Nakanishi and the third author. The method developed here is quite general, and applicable to other spectral problems which arise in the theory of nonlinear equations. The development of energy selective, photon counting X-ray detectors allows for a wide range of new possibilities in the area of computed tomographic image formation. Under the assumption of perfect energy resolution, here we propose a tensor-based iterative algorithm that simultaneously reconstructs the X-ray attenuation distribution for each energy. We use a multilinear image model rather than a more standard stacked vector representation in order to develop novel tensor-based regularizers. In particular, we model the multispectral unknown as a three-way tensor where the first two dimensions are space and the third dimension is energy. This approach allows for the design of tensor nuclear norm regularizers, which like its 2D counterpart, is a convex function of the multispectral unknown. The solution to the resulting convex optimization problem is obtained using an alternating direction method of multipliers approach. Simulation results show that the generalized tensor nuclear norm can be used as a standalone regularization technique for the energy selective (spectral) computed tomography problem and when combined with total variation regularization it enhances the regularization capabilities especially at low energy images where the effects of noise are most prominent. We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree T, and study the existence of the free energy density ϕ, the limit of the log-partition function divided by the number of vertices n as n tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity ϕ subject to uniqueness of a relevant Gibbs measure for the factor model on T. By way of example we compute ϕ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on ϕ. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on T. In the special case that T has a Galton–Watson law, this formula coincides with the nonrigorous “Bethe prediction” obtained by statistical physicists using the “replica” or “cavity” methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.