Poisson Operator Research Articles

An adaptive lattice Green's function method for external flows with two unbounded and one homogeneous directions

We solve the incompressible Navier-Stokes equations using a lattice Green's function (LGF) approach, including immersed boundaries (IB) and adaptive mesh refinement (AMR), for external flows with one homogeneous direction (e.g. infinite cylinders of arbitrary cross-section). We hybridize a Fourier collocation (pseudo-spectral) method for the homogeneous direction with a specially designed, staggered-grid finite-volume scheme on an AMR grid. The Fourier series is also truncated variably according to the refinement level in the other directions. We derive new algorithms to tabulate the LGF of the screened Poisson operator and viscous integrating factor. After adapting other algorithmic details from the fully inhomogeneous case [1], we validate and demonstrate the new method with transitional and turbulent flows over a circular cylinder at Re=300 and Re=12,000, respectively.

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} asymptotic convergence without the assumption of bi-K-invariance.

ОПТИМІЗАЦІЙНІ ХАРАКТЕРИСТИКИ ОПЕРАТОРА З ДЕЛЬТАПОДІБНИМ ЯДРОМ ДЛЯ КВАЗІГЛАДКИХ ФУНКЦІЙ

The paper presents research results combining the methods of approximation theory and optimal decision theory. Namely, the optimization problem for the biharmonic Poisson integral in the upper half-plane is considered as one of the most optimal solutions to the biharmonic equation in Cartesian coordinates. The approximate properties of the biharmonic Poisson operator in the upper half-plane on the classes of quasi-smooth functions are obtained in the form of an exact equality for the deviation of quasi-smooth functions from the positive operator under consideration. Keywords: biharmonic equation in Cartesian coordinates, quasi-smooth functions, global optimization, biharmonic Poisson integral in the upper half-plane.

Helmholtz decomposition with a scalar Poisson equation in elastic anisotropic media

P- and S-wave separation plays an important role in elastic reverse-time migration. It can reduce the artifacts caused by crosstalk between different modes and improve image quality. In addition, P- and S-wave separation can also be used to better understand and distinguish wave types in complex media. At present, the methods for separating wave modes in anisotropic media mainly include spatial non-stationary filtering, low-rank approximation, and vector Poisson equation. Most of these methods require multiple Fourier transforms or the calculation of large matrices, which require high computational costs for problems with large scale. In this paper, an efficient method is proposed to separate the wave mode for anisotropic media by using a scalar anisotropic Poisson operator in the spatial domain. For 2D problems, the computational complexity required by this method is 1/2 of the methods based on solving a vector Poisson equation. Therefore, compared with existing methods based on pseudo-Helmholtz decomposition operators, this method can significantly reduce the computational cost. Numerical examples also show that the P and S waves decomposed by this method not only have the correct amplitude and phase relative to the input wavefield but also can reduce the computational complexity significantly.

Some applied aspects of the Dirac delta function

The study is devoted to some applied aspects of the Dirac delta function. On the basis of this function, an integral representation was found for the deviation of the functions of the Holder class ${H}^{\alpha }$ ($0<\alpha <1$) from their Poisson integrals in the upper half-plane. In the current research, exact equalities of the upper bounds for the deviations of the functions of the Holder class ${H}^{\alpha }$ from the Poisson operators in the upper half-plane were found by applying the known properties of the Dirac delta function.

Cluster solutions for the FitzHugh-Nagumo system with Neumann boundary conditions

We study cluster solution of the following FitzHugh-Nagumo system:{ε2Δu+f(u)−v=0in Ω,Δv−γv+βu=0in Ω,∂u∂n=∂v∂n=0on ∂Ω, where Ω is a smooth bounded domain in R2 and ε,γ,β are positive parameters. Under appropriate parameter regimes, given any integer k≥2, this system admits a cluster solution uε with k-boundary spikes that accumulate at Q0, a nondegenerate local maximum point of boundary curvature, whenever the term βεγ is sufficiently large or sufficiently small. Moreover, we show that these k-boundary spikes are separated from each other at a distance of O(log⁡(βεγ)γ) as βεγ→∞ or O((βε)12) as βεγ→0, respectively. The latter case proposes a new concentration feature of the activator-inhibitor type reaction-diffusion system which is barely observed in Gierer-Meinhardt system. The mechanism behind the scene is driven by two competing forces: a short range gravitational force with mass positioned at a local maximum point of the curvature and a long range repelling force between spikes whose major part comes from the Green's function of screened Poisson operator on the half space.

Approximation of continuous functions given on the real axis by three-harmonic Poisson operators

Approximation of continuous functions given on the real axis by three-harmonic Poisson operators

Approximation of continuous functions given on the real axis by three-harmonic Poisson operators

Approximation properties of three-harmonic Poisson operators on the classes of $(\psi,\beta)$-differentiable functions of low smoothness given on the real axis have been studied. Asymptotic equalities have been obtained that provide in some cases a solution to the Kolmogorov--Nikol'skii problem for three-harmonic Poisson operators $P_{3,\sigma}(f;x)$ on the classes $\widehat{C}_{\beta,\infty}^{\psi}$, $\beta\in\mathbb{R}$, in the uniform metric.

Boundedness of the differential transforms for the generalized Poisson operators generated by Laplacian

Boundedness of the differential transforms for the generalized Poisson operators generated by Laplacian

HipBone: A performance-portable graphics processing unit-accelerated C++ version of the NekBone benchmark

We present hipBone, an open-source performance-portable proxy application for the Nek5000 (and NekRS) computational fluid dynamics applications. HipBone is a fully GPU-accelerated C++ implementation of the original NekBone CPU proxy application with several novel algorithmic and implementation improvements which optimize its performance on modern fine-grain parallel GPU accelerators. Our optimizations include a conversion to store the degrees of freedom of the problem in assembled form in order to reduce the amount of data moved during the main iteration and a portable implementation of the main Poisson operator kernel. We demonstrate near-roofline performance of the operator kernel on three different modern GPU accelerators from two different vendors. We present a novel algorithm for splitting the application of the Poisson operator on GPUs which aggressively hides MPI communication required for both halo exchange and assembly. Our implementation of nearest-neighbor MPI communication then leverages several different routing algorithms and GPU-Direct RDMA capabilities, when available, which improves scalability of the benchmark. We demonstrate the performance of hipBone on three different clusters housed at Oak Ridge National Laboratory, namely, the Summit supercomputer and the Frontier early-access clusters, Spock and Crusher. Our tests demonstrate both portability across different clusters and very good scaling efficiency, especially on large problems.

Multimode theory of electron hole transverse instability

We present Vlasov–Poisson three-dimensional linear stability analysis of an initially planar electron hole structure, solving for the distribution function by integration along unperturbed orbits. The non-sinusoidal potential perturbation shape (parallel to$B$) is expanded in eigenfunctions of the adiabatic Poisson operator, generalizing the prior assumption of a rigid shift of the equilibrium. We show that the shiftmode is then modified by a second discrete mode plus an integral over a continuum of wave-like modes. A rigorous treatment shows that the continuum can be approximated effectively by a single mode that satisfies the external wave dispersion relation, thus making the perturbation a weighted sum of three modes. We find numerically the solution for the complex instability frequency, and the corresponding three mode amplitudes determining the perturbation eigenmode. This multimode analysis refines the accuracy of the prior single-mode results, giving slightly higher growth rates at most parameters, as expected from the extra mode shape freedom. Oscillating modes near stability boundaries have larger mode distortions which help explain particle-in-cell simulations that observe instability up to${\sim }20$ % beyond the prior shiftmode thresholds, and narrowing of the perturbation. At high magnetic field, the multimode analysis predicts a reduction of the already small growth rate.

Approximate Characteristics of Generalized Poisson Operators on Zygmund Classes

Approximate Characteristics of Generalized Poisson Operators on Zygmund Classes

Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems

In this paper we consider a mean field particle systems whose confinement potentials have many local minima. We establish some explicit and sharp estimates of the spectral gap and logarithmic Sobolev constants uniform in the number of particles. The uniform Poincaré inequality is based on the work of Ledoux (In Séminaire de Probabilités, XXXV (2001) 167–194, Springer) and the uniform logarithmic Sobolev inequality is based on Zegarlinski’s theorem for Gibbs measures, both combined with an explicit estimate of the Lipschitz norm of the Poisson operator for a single particle from (J. Funct. Anal. 257 (2009) 4015–4033). The logarithmic Sobolev inequality then implies the exponential convergence in entropy of the McKean–Vlasov equation with an explicit rate, We need here weaker conditions than the results of (Rev. Mat. Iberoam. 19 (2003) 971–1018) (by means of the displacement convexity approach), (Stochastic Process. Appl. 95 (2001) 109–132; Ann. Appl. Probab. 13 (2003) 540–560) (by Bakry–Emery’s technique) or the recent work (Arch. Ration. Mech. Anal. 208 (2013) 429–445) (by dissipation of the Wasserstein distance).

Quantum-mechanical model of dielectric losses in nanometer layers of solid dielectrics with hydrogen bonds at ultra-low temperatures

Upon based the finite difference methods construct the solutions for Liouville quantum kinetic equation linearized by the external field, in complex with the stationary Schrodinger equation and the Poisson operator equation, for an ensemble of non-interacting hydrogen ions (protons) migrating in the field of a crystal lattice perturbed by a variable polarizing field. The influence of the phonon subsystem is not taken into account. The equilibrium (non-balanced) proton density matrix is calculated using quantum Boltzmann statistics. The temperature spectra of dielectric losses tangent angle for hydrogen bonded crystals (HBC) in a wide temperature range (50–550 K) are calculated. At the theoretical level detected the effects of nano-crystalline states (1–10 nm) during the polarization of HBC in the region of ultra-low temperatures (4–25 K).

Numerical Experimental for an Inverse Source Problem Method

This paper examines extensions of an iterative method for inverse evaluation of the source function for two elliptic systems. The method begins with a starting value for the undetermined source. Next, a background field and equations for the error field are obtained. 2-D domains are considered. This method is suitable for Helmholtz and Poisson operators. In the presence of finite-difference grid resolution, a varying amount of boundary data, and methods of filtering the noise in the boundary data and the noise intensity of the boundary data, the performance, accuracy, and iteration count of the algorithm are investigated. Keywords: Source, Inverse Problems, Poisson, Noise, Ill-Posedness, Well-Posed

Квантовые свойства диэлектрических потерь в нанометровых слоях твердых диэлектриков при сверхнизких температурах

Using the methods of quasi-classical kinetic theory, continuum electrodynamics, and non-relativistic quantum theory, we construct and study the quantum kinetic equation of proton relaxation, which, together with the Poisson operator equation describes the mechanism of diffusion tunneling transport of hydrogen ions (protons) in the potential field of a crystal lattice perturbed by a polarizing field (quantum diffusion polarization) in crystals with hydrogen bonds. Using the apparatus of the density matrix (statistical matrix), by complete quantum-mechanical averaging of the polarization operator, studies are carried out of the experimental value of the polarization of the dielectric, as a function of the parameters of the external electric field (amplitude, frequency of electromotive force) and temperature. When calculating the equilibrium density matrix for an ensemble of basic relaxers (hydrogen ions), the proton-proton and proton-phonon interactions are not taken into account, and the Hamilton operator for the phonon subsystem is assumed to be a numerical constant for a given crystal under given experimental conditions (calculated by computer method as a parameter for comparing the theory with the experiment). The influence of the phonon subsystem on the kinetics of the relaxation process is reduced to a weak spatially homogeneous force field acting on protons moving in the field of the main forces of hydrogen bonds. The Hamilton of the proton subsystem is constructed for the model of an ideal proton gas in equilibrium with the ionic subsystem of the crystal lattice, and the equilibrium statistical operator of the proton subsystem is written using the Boltzmann quantum statistics. Theoretically, the size effects are found to be manifested in shifts of the low-temperature (50–100 K) maxima of the dielectric loss angle tangent towards ultra-low temperatures (4–25 K) with a decrease in the amplitudes of the maxima by 3-4 orders of magnitude, with a reduction in the thickness of the crystal layer from 1–10 microns to 1–10 nm. The effect of anomalous displacements of low-temperature maxima, which is explained by the abnormally high quantum transparency of the potential barrier for protons (0.8-0.9) in thin films of a crystal with hydrogen bonds (1-10 nm), causes, near the temperatures of the shifted maxima of dielectric losses (4–25 K), a quasi-ferroelectric state, which is also characterized by abnormally high values of the real component of the complete dielectric permittivity (2.5–3.5millions).

A scheduling policy to save 10% of communication time in parallel fast Fourier transform

SummaryThe fast Fourier transform (FFT) has applications in almost every frequency related study, for example, in image and signal processing, and radio astronomy. It is also used as a Poisson operator inversion kernel in partial differential equations in fluid flows, in density functional theory, many‐body theory, and others. The three‐dimensional FFT has large time complexity . Hence, parallelization is used to compute such FFTs. Popular libraries perform slab division or pencil decomposition of data. None of the existing libraries achieve perfect inverse scaling of time with cores because FFT requires all‐to‐all communication and clusters hitherto do not have physical all‐to‐all connections. Dragonfly, one of the popular topologies for the interconnect, supports hierarchical connections among the components. We show that if we align the all‐to‐all communication of FFT with the physical connections of Dragonfly topology we will achieve a better scaling and reduce communication time.

Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted Lq-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt A_{infty }-class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the Ap-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.

Angles between Haagerup–Schultz projections and spectrality of operators

We investigate angles between Haagerup–Schultz projections of operators belonging to finite von Neumann algebras, in connection with a property analogous to Dunford's notion of spectrality of operators. In particular, we show that an operator is similar to the sum of a normal and an s.o.t.-quasinilpotent operator that commute if and only if the angles between its Haagerup–Schultz projections are uniformly bounded away from zero (and we call this the uniformly non-zero angles property). Moreover, we show that spectrality is equivalent to this uniformly non-zero angles property plus decomposability. Finally, using this characterization, we construct an easy example of an operator which is decomposable but not spectral, and we show that Voiculescu's circular operator is not spectral (nor is any of the circular free Poisson operators).

Quantum-mechanical model of thermally stimulated depolarization in layered dielectrics at low temperatures

Built the quantum-mechanical scheme for investigating the spectrum of thermally stimulated currents of depolarization (TCDP), which allows to study the processes of relaxation of hetero-and homo-charge in hydrogen bonded crystals (HBC) with complex crystalline structure (layered silicates, crystal-hydrates), taking into account the distribution of relaxers (protons) over the energy levels of the quasi-discrete spectrum in potential image of the crystal lattice, and allows us to calculate the parameters of low-temperature relaxers by the Klinger method in the quadratic approximation over the external electric field. However, the calculation of the excess proton concentration operator was carried out in the quasi-classical approximation on the basis of the Fokker-Planck operator equation, which is solved together with the Poisson operator equation. By method of density matrix in the quadratic approximation in the external field calculated thermally stimulated depolarization currents on the basis of which the investigated size effects in the nanometer size layers associated with the shift of the maximum current of thermo-depolarization to low temperatures while reducing the thickness of the crystal layer.