Compact Support Research Articles

Topological reconstruction of compact supports of dependent stationary random variables

Topological reconstruction of compact supports of dependent stationary random variables

Multi-domain wavelet boundary element method for calculating two-dimensional stress intensity factors

In order to improve the accuracy of stress intensity factors (SIFs) calculated by traditional boundary element methods (BEM), the multi-domain wavelet boundary element method (WBEM) is proposed. Firstly, by adjusting the nodes of the B-spline wavelet element on the interval, crack-tip elements are constructed. Since B-spline wavelet on the interval (BSWI) has excellent compact support characteristics and is particularly suitable for describing solution domains with large gradient changes, the constructed crack-tip can reduce the numerical oscillation effect near the crack tip. Secondly, the crack-tip elements are implemented into WBEM. And the combination of WBEM and multi-domain technology can effectively handle interface cracks. Thirdly, the crack problem solving strategy based on multi-domain WBEM can directly evaluate the SIFs of cracks. Finally, several numerical examples involving homogeneous media and bi-material models are given to verify that the proposed method is simple and highly accurate.

An innovative Fibonacci wavelet collocation method for the numerical approximation of Emden-Fowler equations

This article presents a novel approach using the Fibonacci wavelet collocation method (FWCM) for the numerical solution of Emden-Fowler-type equations. The Emden-Fowler equations are a class of nonlinear differential equations that arise in various fields of science and engineering, particularly in astrophysics and fluid dynamics. Due to their nonlinear and singular nature, the conventional approaches to solving these equations encounter difficulties. This method is particularly effective in handling problems with singularities, as it adapts naturally to the local behaviour of the solution. Here, we introduced the Fibonacci wavelet collocation method as a powerful numerical technique for tackling Emden-Fowler-type equations. The Fibonacci wavelet basis functions possess remarkable properties, including compact support, making them well-suited for approximating solutions to differential equations. The main advantage of this approach lies in its ability to reduce the computational complexity associated with solving Emden-Fowler equations, resulting in accurate and efficient solutions. Comparative analyses with other established numerical methods reveal its superior accuracy and convergence rate performance. Further, several examples demonstrate the method's flexibility when dealing with different singularity levels. This paper contributes to numerical analysis by introducing the Fibonacci wavelet method as a robust tool for solving Emden-Fowler-type equations.

A Score-Based Deterministic Diffusion Algorithm with Smooth Scores for General Distributions

Score matching based diffusion has shown to achieve the state of art results in generation modeling. In the original score matching based diffusion algorithm, the forward equation is a differential equation for which the probability density equation evolves according to a linear partial differential equation, the Fokker-Planck equation. A drawback of this approach is that one needs the data distribution to have a Lipschitz logarithmic gradient. This excludes a large class of data distributions that have a compact support. We present a deterministic diffusion process for which the vector fields are always Lipschitz and hence the score does not explode for probability measures with compact support. This deterministic diffusion process can be seen as a regularization of the porous media equation equation, which enables one to guarantee long term convergence of the forward process to the noise distribution. Though the porous media equation is itself not always guaranteed to have a Lipschitz vector field, it can be used to understand the closeness of the output of the algorithm to the data distribution as a function of the the time horizon and score matching error. This analysis enables us to show that the algorithm has better dependence on the score matching error than approaches based on stochastic diffusions. Using numerical experiments we verify our theoretical results on example one and two dimensional data distributions which are compactly supported. Additionally, we validate the approach on a modified MNIST data set for which the distribution is concentrated on a compact set. In each of the experiments, the approach using deterministic diffusion performs better that the diffusion algorithm with stochastic forward process, when considering the FID scores of the generated samples.

Collaborative modeling-based improved moving Kriging approach for low-cycle fatigue life reliability estimation of mechanical structures

To effectively estimate the reliability level of low-cycle fatigue (LCF) life of mechanical structures, a novel method of collaborative modeling-based improved moving Kriging (IMKCM) approach is proposed by fusing improved moving Kriging method and collaborative modeling strategy. In this method, the improved moving Kriging model is developed by the Kriging model, moving least squares method and artificial hummingbird algorithm, in which the Kriging model is selected as the basis function, the artificial hummingbird algorithm is used to define the optimal radius of compact support region and determine the effective modeling samples, and the moving least squares method is adopted to resolve the unknown coefficients; the collaborative modeling strategy is developed from the decomposition and coordination method, in which the mathematical models of the decomposed sub-objectives are synchronously built by the cell array theory, and the mathematical model between the objective and the sub-objectives is established by the improved moving Kriging model. Furthermore, a numerical example (i.e., nested nonlinear function) and an engineering example (i.e., turbine blisk LCF life reliability estimation) are employed to verify the effectiveness of the IMKCM approach. The results show that the proposed IMKCM method holds excellent abilities in modeling features and simulation performance.

Tight wavelet frames on the space of M-positive vectors

Wavelets on the sets of [Formula: see text]-positive vectors in the Euclidean space are studied. These sets are multidimensional analogs of the half-line in the Walsh analysis. Following the ideas of the Walsh analysis, the space of [Formula: see text]-positive vectors is equipped with a coordinate-wise addition. Harmonic analysis on this space is also similar to the Walsh harmonic analysis, and the Fourier transform is such that there exists a class of so-called test functions (with a compact support of the function itself and of its Fourier transform). Tight wavelet frames consisting of the test functions are studied. A complete description of the masks generating such frames is given, and an algorithmic method for constructing them is developed. These frames may be very useful for applications to signal processing because some examples of such systems on the half-line were already investigated in this aspect, and it appeared that they have an advantage over classical wavelet systems when used for processing fractal signals and images.

Reflectionless Schrodinger operators and Marchenko parametrization

Let $T_q=-d^2/dx^2 +q$ be a Schr\"odinger operator in the space $L_2(\mathbb{R})$. A potential $q$ is called reflectionless if the operator $T_q$ is reflectionless. Let $\mathcal{Q}$ be the set of all reflectionless potentials of the Schr\"odinger operator, and let $\mathcal{M}$ be the set of nonnegative Borel measures on $\mathbb{R}$ with compact support. As shown by Marchenko, each potential $q\in\mathcal{Q}$ can be associated with a unique measure $\mu\in\mathcal{M}$. As a result, we get the bijection $\Theta\colon \mathcal{Q}\to \mathcal{M}$. In this paper, we show that one can define topologies on $\mathcal{Q}$ and $\mathcal{M}$, under which the mapping $\Theta$ is a homeomorphism.

Quasi-stationary distribution of a single species model under demographic stochasticity and Allee effects

In this paper, we utilize an absorbed diffusion process to model the dynamics of a single species under the influence of demographic stochasticity and component Allee effects. The trajectories of stochastic solutions exhibits multi-scale dynamics distinct from those of the corresponding mean-field model. The primary focus is on analyzing transient dynamics before extinction, which is described by the quasi-stationary distribution. The paper provides results on the existence, uniqueness, and exponential convergence to the quasi-stationary distribution for initial distributions with compact support. Due to the singularity of the noise at 0, we employ the idea of changing variables to transfer the model into one-dimensional Kolmogorov diffusions with a drift possibly exploding at 0, to prove the main results.

Cohomology and geometry of Deligne–Lusztig varieties for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}

We give a description of the cohomology groups of the structure sheaf on smooth compactifications X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document} of Deligne–Lusztig varieties X(w) for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}, for all elements w in the Weyl group. As a consequence, we obtain the modpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{mod}\\ p^m$$\\end{document} and integral p-adic étale cohomology of X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document}. Moreover, using our result for X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document} and a spectral sequence associated to a stratification of X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document}, we deduce the modpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{mod}\\ p^m$$\\end{document} and integral p-adic étale cohomology with compact support of X(w). In our proof of the main theorem, in addition to considering the Demazure–Hansen smooth compactifications of X(w), we show that a similar class of constructions provide smooth compactifications of X(w) in the case of GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}. Furthermore, we show in the appendix that the Zariski closure of X(w), for any connected reductive group G and any w, has pseudo-rational singularities.

Compactons in Higher-Order Nesterenko's-Type Equations

Abstract A model describing propagation of waves in a prestressed granular media is considered. The model, having the form of evolutionary partial differential equation (PDE), is obtained from the system of ordinary differential equations (ODEs) describing dynamics of a chain of prestressed granules by means of formal asymptotic expansion. It is shown in our previous papers that in the lowest asymptotic approximation, in which both nonlinear effects and the presence of media structure are taken into account, the model equation possesses traveling wave (TW) solutions with compact support (compactons) manifesting soliton properties. In this paper, we study a higher-order evolutionary PDE obtained by taking into account previously discarded terms of the asymptotic expansion, as well as another PDE (called analogue), differing from the original one in the values of parameters and having compacton solutions expressed in analytical form. Numerical and analytical studies of both the higher-order model and its analogue allow to conclude that both models have compacton solutions exhibiting some properties of “true” solitons. This, in turn, testifies the stability of the previously used model with respect to the inclusion of the discarded terms of the asymptotic expansion.

On the Dirichlet problem for beltrami equations with sources in simply connected domains

In this paper, we present our recent results on the solvability of the Dirichlet problem Reω(z) → φ (ζ) as z → ζ, z∈ D, ζ∈ ∂D, with continuous boundary data φ: ∂D ???? R for degenerate Beltrami equations ωz =µ(z)ω2 + σ(z), |µ(z) ˂ 1 a.e., with sources σ: D → C that belong to the class Lp (D), p ˃ 2, and have compact supports in D. In the case of locally uniform ellipticity of the equations, we formulate, in arbitrary simply connected domains D of the complex plane C a series of eff ective integral criteria of the type of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz on singularities of the equations at the boundary for existence of locally Hölder continuous solutions in the class W1.2loc (D) of the Dirichlet problem with their representation through the so-called generalized analytic functions with sources.

On the Existence of Global Solutions for the Vlasov–Poisson System in a Half-space and Plasma Confinement

We consider the Vlasov–Poisson system with external magnetic field in a half-space with the Neumann boundary condition for the electric potential and specular reflection on a boundary. For arbitrary compactly supported initial density distribution functions, we obtain sufficient conditions for external magnetic field, which provide global existence of density distribution functions with compact supports lying at some distance from a boundary.

Drying in nanoporous media with Kelvin effect: Capillary imbibition against evaporation by smoothed particle hydrodynamics method

Understanding drying processes in nanoporous media is of great importance in many technological and industrial situations. To better understand how gas moves through clayey rocks, of interest for underground disposal of radioactive wastes, we propose using pore-scale direct numerical simulations. In this study, we use the Smoothed Particle Hydrodynamics method, which has proved to be an effective approach for simulating complex fluid dynamics within porous media at the nanoscale. Our simulations consider capillary-dominated two-phase flow with evaporation and condensation at liquid–gas interfaces, coupled to the diffusion of water vapor in the gas phase, as well as the Kelvin effect, which is a specific feature of nanopores. Our evaporation-condensation model is validated against analytical solutions. The size of the compact support of kernel function and the particle density required to obtain accurate and stable results of capillary pressure are investigated. Drying regimes, capillary-driven and evaporated-driven, are explored. A specific effort is made to highlight the influence of the Kelvin effect on desaturation and the creation of preferential paths for gas flow as well as its impact on drying rate. The role of condensation due to local vapor concentration conditions is also emphasized.

Multivariate compactly supported C∞ functions by subdivision

This paper discusses the generation of multivariate C∞ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called Up-function, by a non-stationary scheme based on masks of spline subdivision schemes of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based on masks of three-directional box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes than the class we study. With our method for achieving small compact supports, we obtain in the univariate case, Up-like functions with supports [0,1+ϵ] in comparison to the support [0,2] of the Up-function. Examples of univariate and bivariate Up-like functions are given. As in the univariate case, the construction of Up-like functions can motivate the generation of C∞ compactly supported wavelets of small support in any dimension.

Semigroup theory and analysis of solutions for a higher order non‐Lipschitz problem with advection in ℝN$$ {\mathrm{\mathbb{R}}}^N $$

In studying heterogeneous diffusion phenomena beyond the classical order 2 Laplacian, higher order operators emerge as powerful tools. When such operators are applied, the resultant profiles often showcase an oscillatory nature that may be noticeable near the zero stationary solution. These oscillations can pose challenges, especially when they negate the compactness of the support for initial distributions that are both smooth and possess compact support. The presented analysis in this work seeks to delve into the nature of solutions within open finite domains, with an ambition to scale up the findings to the realm of through the use of an extension operator. The foundation for establishing the existence and uniqueness of these solutions leans on the semigroup representation theory. Asymptotic solutions have been carved out via the Hamilton–Jacobi transport equation. To cement the theoretical foundation, numerical simulations have been incorporated shedding light on the accuracy of our initial assumptions within the asymptotic framework.

Application of Wavelet Methods in Computational Physics

AbstractThe quantitative study of many physical problems ultimately boils down to solving various partial differential equations (PDEs). Wavelet analysis, known as the “mathematical microscope”, has been hailed for its excellent Multiresolution Analysis (MRA) capabilities and its basis functions that possess various desirable mathematical qualities such as orthogonality, compact support, low‐pass filtering, and interpolation. These properties make wavelets a powerful tool for efficiently solving these PDEs. Over the past 30 years, numerical methods such as wavelet Galerkin methods, wavelet collocation methods, wavelet finite element methods, and wavelet integral collocation methods have been proposed and successfully applied in the quantitative analysis of various physical problems. This article will start from the fundamental theory of wavelet MRA and provide a brief summary of the advantages and limitations of various numerical methods based on wavelet bases. The objective of this article is to assist researchers in choosing the appropriate numerical methodologies for their particular physical issues. Furthermore, it will explore prospective advancements in wavelet‐based techniques, offering valuable insights for researchers committed to enhancing wavelet numerical methods in the field of computational physics.

A sharp MLS penalty immersed finite element method for fluid-structure interaction of highly deformable slender body in turbulent flow

This paper presents a new computational approach to simulate challenging fluid-structure interactions (FSI) between fluids and slender deformable structures. Key innovations address limitations of standard immersed boundary methods, including spurious forces, stability at low density ratios, and accuracy at high Reynolds numbers. The method couples a sharp interface immersed boundary technique with detached eddy simulation turbulence modelling to enable precise FSI for high Reynolds number flows. A strong coupling partitioned algorithm stabilized by Aitken relaxation significantly enhances stability for low density ratios down to 1. A moving least square compact support domain approximation reduces spurious oscillations from moving geometries while providing second-order accuracy. Adaptive mesh refinement imposes jump conditions on slim deformable bodies and minimizes grid leakage. The proposed method is evaluated on three conventional FSI benchmarks and four experimental cases, confirming its robustness, accuracy, and stability in low and significantly high Reynolds numbers. A more complex fluids engineering case is considered last to test the solution under challenging conditions with fast moving solid boundaries and fast flowing fluid. A thin deformable membrane is forced through a submersible centrifugal pump under standard operating conditions. The solution is shown to produce a stable solution with good collision handling ability.

Asymptotic growth and decay of two-dimensional symmetric plasmas

We study the large time behavior of classical solutions to the two-dimensional Vlasov-Poisson (VP) and relativistic Vlasov-Poisson (RVP) systems launched by radially-symmetric initial data with compact support. In particular, we prove that particle positions and momenta grow unbounded as $ t \to \infty $ and obtain sharp rates on the maximal values of these quantities on the support of the distribution function for each system. Furthermore, we establish nearly sharp rates of decay for the associated electric field, as well as upper and lower bounds on the decay rate of the charge density in the large time limit. We prove that, unlike (VP) in higher dimensions, smooth solutions do not scatter to their free-streaming profiles as $ t \to \infty $ because nonlinear, long-range field interactions dominate the behavior of characteristics due to the exchange of energy from the potential to the kinetic term. In this way, the system may 'forget' any previous configuration of particles.

Functor of Idempotent Probability Measures with Compact Support and Open Mappings

Functor of Idempotent Probability Measures with Compact Support and Open Mappings

Exploration of Learned Lifting-Based Transform Structures for Fully Scalable and Accessible Wavelet-Like Image Compression.

This paper provides a comprehensive study on features and performance of different ways to incorporate neural networks into lifting-based wavelet-like transforms, within the context of fully scalable and accessible image compression. Specifically, we explore different arrangements of lifting steps, as well as various network architectures for learned lifting operators. Moreover, we examine the impact of the number of learned lifting steps, the number of channels, the number of layers and the support of kernels in each learned lifting operator. To facilitate the study, we investigate two generic training methodologies that are simultaneously appropriate to a wide variety of lifting structures considered. Experimental results ultimately suggest that retaining fixed lifting steps from the base wavelet transform is highly beneficial. Moreover, we demonstrate that employing more learned lifting steps and more layers in each learned lifting operator do not contribute strongly to the compression performance. However, benefits can be obtained by utilizing more channels in each learned lifting operator. Ultimately, the learned wavelet-like transform proposed in this paper achieves over 25% bit-rate savings compared to JPEG 2000 with compact spatial support.