Compact Support Research Articles

Non-Lipschitz heterogeneous reaction with a p-Laplacian operator

<abstract><p>The intention along this work is to provide analytical approaches for a degenerate parabolic equation formulated with a p-Laplacian operator and heterogeneous non-Lipschitz reaction. Firstly, some results are discussed and presented in relation with uniqueness, existence and regularity of solutions. Due to the degenerate diffusivity induced by the p-Laplacian operator (specially when $ \nabla u = 0 $, or close zero), solutions are studied in a weak sense upon definition of an appropriate test function. The p-Laplacian operator is positive for positive solutions. This positivity condition is employed to show the regularity results along propagation. Afterwards, profiles of solutions are explored specially to characterize the propagating front that exhibits the property known as finite propagation speed. Finally, conditions are shown to the loss of compact support and, hence, to the existence of blow up phenomena in finite time.</p></abstract>

Small defects reconstruction in waveguides from multifrequency one-side scattering data

<p style='text-indent:20px;'>Localization and reconstruction of small defects in acoustic or electromagnetic waveguides is of crucial interest in nondestructive evaluation of structures. The aim of this work is to present a new multi-frequency inversion method to reconstruct small defects in a 2D waveguide. Given one-side multi-frequency wave field measurements of propagating modes, we use a Born approximation to provide a <inline-formula><tex-math id="M1">\begin{document}$ \text{L}^2 $\end{document}</tex-math></inline-formula>-stable reconstruction of three types of defects: a local perturbation inside the waveguide, a bending of the waveguide, and a localized defect in the geometry of the waveguide. This method is based on a mode-by-mode spacial Fourier inversion from the available partial data in the Fourier domain. Indeed, in the available data, some high and low spatial frequency information on the defect are missing. We overcome this issue using both a compact support hypothesis and a minimal smoothness hypothesis on the defects. We also provide a suitable numerical method for efficient reconstruction of such defects and we discuss its applications and limits.</p>

Gradient of the single layer potential and quantitative rectifiability for general Radon measures

We identify a set of sufficient local conditions under which a significant portion of a Radon measure μ on Rn+1 with compact support can be covered by an uniformly n-rectifiable set, at the level of a ball B⊂Rn+1 such that μ(B)≈r(B)n. This result involves a flatness condition, formulated in terms of the so-called β1-number of B, and the L2(μ|B)-boundedness, as well as a control on the mean oscillation on the ball, of the operatorTμf(x)=∫∇xE(x,y)f(y)dμ(y). Here E(⋅,⋅) is the fundamental solution for a uniformly elliptic operator in divergence form associated with an (n+1)×(n+1) matrix with Hölder continuous coefficients. This generalizes a work by Girela-Sarrión and Tolsa for the n-Riesz transform. The motivation for our result stems from a two-phase problem for the elliptic harmonic measure.

Topology optimization of irregular flow domain by parametric level set method in unstructured mesh

ABSTRACT In this study, we present a novel method for the topology optimization of the irregular flow domain using a parametric level set method (PLSM). Some improvement was applied on the CS-RBFs (radial basis functions with compact support)-based PLSM to make it suitable for nonuniform mesh, expanding the range field of engineering application of the PLSM. The optimization problem is solved by a gradient-based algorithm with Stokes equations as state constraints, and the objective is set to minimize the power dissipation subject to the volume constraint of flow channels. A PLSM is introduced to avoid the direct solving of the Hamilton–Jacobi partial differential equation, which can have the potential to break through the restriction of relying on structured meshes because no finite difference scheme is required. Then, a self-adaption support radius approach is presented to allow the parametric level set to be evolved on the nonuniformed mesh, which can expand the application of the PLSM to more complicated engineering problems with irregular geometric shapes. A volume integration scheme is applied during the design sensitivity analysis to calculate the shape derivatives, allowing the nucleation of new holes. Numerical examples in two and three dimensions are provided to demonstrate the effectiveness of the proposed method.

The Compact Support Neural Network.

Neural networks are popular and useful in many fields, but they have the problem of giving high confidence responses for examples that are away from the training data. This makes the neural networks very confident in their prediction while making gross mistakes, thus limiting their reliability for safety-critical applications such as autonomous driving and space exploration, etc. This paper introduces a novel neuron generalization that has the standard dot-product-based neuron and the radial basis function (RBF) neuron as two extreme cases of a shape parameter. Using a rectified linear unit (ReLU) as the activation function results in a novel neuron that has compact support, which means its output is zero outside a bounded domain. To address the difficulties in training the proposed neural network, it introduces a novel training method that takes a pretrained standard neural network that is fine-tuned while gradually increasing the shape parameter to the desired value. The theoretical findings of the paper are bound on the gradient of the proposed neuron and proof that a neural network with such neurons has the universal approximation property. This means that the network can approximate any continuous and integrable function with an arbitrary degree of accuracy. The experimental findings on standard benchmark datasets show that the proposed approach has smaller test errors than the state-of-the-art competing methods and outperforms the competing methods in detecting out-of-distribution samples on two out of three datasets.

Functor of Idempotent Probability Measures with Compact Support and Open Mappings

In this paper, we show that the functor of idempotent probability measures with compact support acting in the category of Tikhonov spaces and their continuous mappings is normal. It is found that this functor is monodic. Further, it is proved that the functor of idempotent probability measures with compact support preserves the openness of continuous mappings of Tikhonov spaces.

An Azimuth Signal-Reconstruction Method Based on Two-Step Projection Technology for Spaceborne Azimuth Multi-Channel High-Resolution and Wide-Swath SAR

Azimuth non-uniform signal-reconstruction is a critical step for azimuth multi-channel high-resolution wide-swath (HRWS) synthetic aperture radar (SAR) data processing. However, the received non-uniform signal has noise in the actual azimuth multi-channel SAR (MCSAR) operation, which leads to the serious reduction in the signal-to-noise ratio (SNR) of the results processed by a traditional reconstruction algorithm. Aiming to address the problem of reducing the SNR of the traditional reconstruction algorithm in the reconstruction of non-uniform signal with noise, a novel signal-reconstruction algorithm based on two-step projection technology (TSPT) for the MCSAR system is proposed in this paper. The key part of the TSPT algorithm consists of a two-step projection. The first projection is to project the given signal into the selected intermediate subspace, spanned by the integer conversion of the compact support kernel function. This process generates a set of sparse equations, which can be solved efficiently by using the sparse equation solver. The second key projection is to project the first projection result into the subspace of the known sampled signal. The secondary projection can be achieved with a digital linear translation invariant (LSI) filter and generate a uniformly spaced signal. As a result, compared with the traditional azimuth MCSAR signal-reconstruction algorithm, the proposed algorithm can improve SNR and reduce the azimuth ambiguity-signal-ratio (AASR). The processing results of simulated data and real raw data verify the effectiveness of the proposed algorithm.

Boubaker Wavelets Functions: Properties and Applications

تم في هذا البحث تقديم شرح تفصيلي لدوال متعددة حدود بوبكر المتعامدة مع بعض الخواص ذات الاهمية، كذلك استنتاج تعريف متعددات حدود بوبكر الموجية في الفترة (1, 0] وذلك بالاستفادة من بعض الخواص المهمة لمتعددة حدود بوبكر. تمتلك هذه الدوال الاساسية خاصية العيارية المتعامدة بالإضافة الى ضرورة تواجد المنطلق المرصوص. لهذه الدوال الموجية العديد من المزايا وقد استخدمت في المجال النظري بالإضافة الى المجال العملي وتم استخدامها مع متعددات الحدود المتعامدة لغرض طرح طريقة جديدة للتعامل مع العديد من المسائل في العلوم والهندسة ولذلك تعتبر طريقة استخدام الموجبات ذات اهمية كبيرة عند الاستفادة منها في المجالات ذات العلاقة. بالإضافة الى الاستفادة من موجبات بوبكر للحصول على خاصية جديدة وهي مشتقات دالة بوبكر الموجية. استخدمت موجية بوبكر مع طريقة الترصيف للحصول على حل عددي تقريبي لمعادلات لان ايمدن من النوع الخطي المنفرد. تصف معادلات لان ايمدن العديد من الظواهر المهمة في علم الرياضيات والفيزياء السماوي مثل الانفجارات الحرارية الكونية وتكوين النجوم. وتعتبر احدى حالات مسائل القيمة الابتدائية المنفردة للمعادلات التفاضلية اللاخطية من الرتبة الثانية. تقوم هذه الطريقة المقترحة بتحويل معادلة لان ايمدن الى نظام من المعادلات التفاضلية الخطية والتي يمكن حلها بسهولة باستخدام الحاسبة. بناءً على هذا فقد ظهر تطابق الحل العددي مع الحل التحليلي بالرغم من استخدام عدد قليل من متعددات حدود بوبكر الموجية لغرض ايجاد هذا الحل. كذلك، تم في هذا البحث البرهنة على قيمه قيد الخطأ المستخرج من هذه الطريقة. وتضمن هذا البحث على ثلاث امثلة عددية من نوع معادلات لان ايمدن لتوضيح قابلية استخدام الطريقة المقترحة. تم توضيح النتائج الحقيقة مع النتائج التقريبية في شكل جداول ورسوم هندسية لغرض المقارنة.

Spherical Distributions Used in Evolutionary Algorithms

Performance of evolutionary algorithms in real space is evaluated by local measures such as success probability and expected progress. In high-dimensional landscapes, most algorithms rely on the normal multi-variate, easy to assemble from independent, identically distributed components. This paper analyzes a different distribution, also spherical, yet with dependent components and compact support: uniform in the sphere. Under a simple setting of the parameters, two algorithms are compared on a quadratic fitness function. The success probability and the expected progress of the algorithm with uniform distribution are proved to dominate their normal mutation counterparts by order n!!.

Application of dynamic programming approach to computation of atomic functions

The special class of atomic functions is considered. The atomic function is a solution with compact support of linear differential functional equation with constant coefficients and linear transformations of the argument. The functions considered are used in discrete atomic compression (DAC) of digital images. The algorithm DAC is lossy and provides better compression than JPEG, which is de facto a standard for compression of digital photos, with the same quality of the result. Application of high precision values of atomic functions can improve the efficiency of DAC, as well as provide the development of new technologies for data processing and analysis. This paper aims to develop a low complexity algorithm for computing precise values of the atomic functions considered. Precise values of atomic functions at the point of dense grids are the subject matter of this paper. Formulas of V. O. Rvachev and their generalizations are used. Direct application of them to the computation of atomic functions on dense grids leads to multiple calculations of a great number of similar expressions that should be reduced. In this research, the reduction required is provided. The goal is to develop an algorithm based on V. O. Rvachev’s formulas and their generalizations. The following tasks are solved: to convert these formulas to reduce the number of arithmetic operations and to develop a verification procedure that can be used to check results. In the current research, methods of atomic function theory and dynamic programming algorithms development principles are applied. A numerical scheme for computation of atomic functions at the points of the grid with the step, which is less than each predetermined positive real number, is obtained and a dynamic algorithm based on it is developed. Also, a verification procedure, which is based on the properties of atomic functions, is introduced. The following results are obtained: 1) the algorithm developed provides faster computation than direct application of the corresponding formulas; 2) the algorithm proposed provides precise computation of atomic functions values; 3) procedure of verification has linear complexity in the number of values to be checked. Moreover, the algorithms proposed are implemented using Python programming language and a set of tables of atomic functions values are obtained. Conclusions: results of this research are expected to improve existing data processing technologies based on atomic functions, especially the algorithm DAC, and accelerate the development of new ones.

Fluctuations of a swarm of Brownian bees.

The "Brownian bees" model describes an ensemble of N independent branching Brownian particles. When a particle branches into two particles, the particle farthest from the origin is eliminated so as to keep the number of particles constant. In the limit of N→∞, the spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation. At long times the particle density approaches a spherically symmetric steady-state solution with a compact support. Here, we study fluctuations of the "swarm of bees" due to the random character of the branching Brownian motion in the limit of large but finite N. We consider a one-dimensional setting and focus on two fluctuating quantities: the swarm center of mass X(t) and the swarm radius ℓ(t). Linearizing a pertinent Langevin equation around the deterministic steady-state solution, we calculate the two-time covariances of X(t) and ℓ(t). The variance of X(t) directly follows from the covariance of X(t), and it scales as 1/N as to be expected from the law of large numbers. The variance of ℓ(t) behaves differently: It exhibits an anomalous scaling (1/N)lnN. This anomaly appears because all spatial scales, including a narrow region near the edges of the swarm where only a few particles are present, give a significant contribution to the variance. We argue that the variance of ℓ(t) can be obtained from the covariance of ℓ(t) by introducing a cutoff at the microscopic time 1/N where the continuum Langevin description breaks down. Our theoretical predictions are in good agreement with Monte Carlo simulations of the microscopic model. Generalizations to higher dimensions are briefly discussed.

Numerical Algorithms for Estimating Probability Density Function Based on the Maximum Entropy Principle and Fup Basis Functions.

Estimation of the probability density function from the statistical power moments presents a challenging nonlinear numerical problem posed by unbalanced nonlinearities, numerical instability and a lack of convergence, especially for larger numbers of moments. Despite many numerical improvements over the past two decades, the classical moment problem of maximum entropy (MaxEnt) is still a very demanding numerical and statistical task. Among others, it was presented how Fup basis functions with compact support can significantly improve the convergence properties of the mentioned nonlinear algorithm, but still, there is a lot of obstacles to an efficient pdf solution in different applied examples. Therefore, besides the mentioned classical nonlinear Algorithm 1, in this paper, we present a linear approximation of the MaxEnt moment problem as Algorithm 2 using exponential Fup basis functions. Algorithm 2 solves the linear problem, satisfying only the proposed moments, using an optimal exponential tension parameter that maximizes Shannon entropy. Algorithm 2 is very efficient for larger numbers of moments and especially for skewed pdfs. Since both Algorithms have pros and cons, a hybrid strategy is proposed to combine their best approximation properties.

Corrigendum to: Reconstruction of an impenetrable obstacle in anisotropic inhomogeneous background

The purpose of this note is to correct the statements of Lemma 2.2 and Lemma 3.15 in Kow & Wang (2021) and revise the assumption of the obstacle |$D$| in our approach. We also correct some notations and misprints. Lemma 2.2 of Kow & Wang (2021) was misstated for |$x\in \mathbb{R}^d$|⁠. It is only valid for |$x\in \mathbb{R}^d\setminus{\textrm{supp}}(A-I)$|⁠. Recall that |${\textrm{supp}}(A-I)$| is a compact set. In other words, for each |$x \in \mathbb{R}^{d}\setminus{\textrm{supp}}\,(A-I)$|⁠, we have where |$\varPhi _{A,n}^{\infty }(\hat{z},x)$| is the far-field pattern of |$\varPhi _{A,n}(z,x)$| satisfying Here the total field |$u_{{\textrm{ref}}}^{{\textrm{to}}}=u_{{\textrm{ref}}}^{{\textrm{sc}}}+u_{{\textrm{ref}}}^{{\textrm{inc}}}$| satisfies with |$u_{{\textrm{ref}}}^{{\textrm{inc}}}(x,-\hat{z})=e^{-ikx\cdot \hat{z}}$|⁠. We will not repeat the proof here, but only remind the readers of the formula where |$\varPsi (z,x):=\varPhi _{A,n}(z,x)-\varPhi _{I,1}(z,x)$| and the fact that the right hand side of (2) belongs to |$L^2(\mathbb{R}^d)$| with compact support when |$x\in \mathbb{R}^d\setminus{\textrm{supp}}(A-I)$|⁠. In view of (1), we revise the definition of the modified Herglotz wave...

A power series analysis of bound and resonance states of one-dimensional Schrödinger operators with finite point interactions

In this paper we consider one-dimensional Schrödinger operatorsSqu(x)=(−d2dx2+qr(x)+qs(x))u(x),x∈R,where qr∈L∞(R) is a regular potential with compact support, and qs∈D′(R) is a singular potentialqs(x)=∑j=1N(αjδ(x−xj)+βjδ′(x−xj)),αj,βj∈Cthat involves a finite number of point interactions. The eigenenergies associated to the bound states and the complex energies associated to the resonance states of operator Sq are given by the zeros of certain characteristic functions η± that share the same structure up to an algebraic sign. The functions η± are obtained explicitly in the form of power series of the spectral parameter, and the computation of the coefficients of the series is given by a recursive integration procedure. The results here presented are general enough to consider arbitrary regular potentials qr∈L∞(R) with compact support, even complex-valued, and point interactions with complex strengths αj,βj (j=1,…,N). Moreover, our approach leads to an efficient numerical treatment of both the bound and resonance states.

Using Fractional Bernoulli Wavelets for Solving Fractional Diffusion Wave Equations with Initial and Boundary Conditions

In this paper, fractional-order Bernoulli wavelets based on the Bernoulli polynomials are constructed and applied to evaluate the numerical solution of the general form of Caputo fractional order diffusion wave equations. The operational matrices of ordinary and fractional derivatives for Bernoulli wavelets are set via fractional Riemann–Liouville integral operator. Then, these wavelets and their operational matrices are utilized to reduce the nonlinear fractional problem to a set of algebraic equations. For solving the obtained system of equations, Galerkin and collocation spectral methods are employed. To demonstrate the validity and applicability of the presented method, we offer five significant examples, including generalized Cattaneo diffusion wave and Klein–Gordon equations. The implementation of algorithms exposes high accuracy of the presented numerical method. The advantage of having compact support and orthogonality of these family of wavelets trigger having sparse operational matrices, which reduces the computational time and CPU requirements.

A machine learning approach of finding the optimal anisotropic SPH kernel

It is presented a machine learning approach to find the optimal anisotropic SPH kernel, whose compact support consists of an ellipsoid that matches with the convex hull of the self-regulating k-nearest neighbors of the smoothing particle (query).

Optimization in the construction of cardinal and symmetric wavelets on the line

We present an optimization approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimization problem. The transform warrants parametrization of the quadrature mirror filter in terms of the possible integer sample values of the scaling function and the associated wavelet. The parameters are predicated to satisfy constraints derived from the conditions of regularity, compact support and orthonormality. This approach allows for the construction of nearly cardinal scaling functions when an objective function that measures deviation from cardinality is minimized. A similar objective function based on a measure of symmetry is also proposed to facilitate the construction of nearly symmetric scaling functions on the line.

A parametric level set method for the optimization of composite structures with curvilinear fibers

This paper presents a parametric level set method for the optimization of composite structures with curvilinear fibers. We use iso-contours of a scalar function, also called the level set function, to represent the fibers, and the level set function is constructed by a set of radial basis functions with compact support. The orientation of fiber at any arbitrary point is defined by the orientation of the tangent vector of iso-contour. In order to prevent gaps and overlaps between neighboring fiber tows from arising during the manufacturing of composite structures, a constraint is formulated that the norm of gradient vector of the level set function is equal to 1 almost everywhere. Furthermore, these gradient constraints are aggregated into a single one by using the p-norm method to improve the efficiency of optimization. The compliance minimization problem is considered, and the coefficients of radial basis functions are taken as the design variables. The results of optimizations proved that the proposed method can provide an optimized composite structure with equally spaced curvilinear fibers.

On well-posedness and singularity formation for the Euler–Riesz system

In this paper, we investigate the initial value problem for the Euler–Riesz system, where the interaction forcing is given by ∇(−Δ)sρ for some −1<s<0, with s=−1 corresponding to the classical Euler–Poisson system. We develop a functional framework to establish local-in-time existence and uniqueness of classical solutions for the Euler–Riesz system. In this framework, the fluid density could decay fast at infinity, and the Euler–Poisson system can be covered as a special case. Moreover, we prove local well-posedness for the pressureless Euler–Riesz system when the potential is repulsive, by observing hyperbolic nature of the system. Finally, we present sufficient conditions on the finite-time blowup of classical solutions for the isentropic/isothermal Euler–Riesz system with either attractive or repulsive interaction forces. The proof, which is based on estimates of several physical quantities, establishes finite-time blowup for a large class of initial data; in particular, it is not required that the density is of compact support.

Discontinuous Galerkin methods through the lens of variational multiscale analysis

In this article, we present a theoretical framework for integrating discontinuous Galerkin methods in the variational multiscale paradigm. Our starting point is a projector-based multiscale decomposition of a generic variational formulation that uses broken Sobolev spaces and Lagrange multipliers to accommodate the non-conforming nature at the boundaries of discontinuous Galerkin elements. We show that existing discontinuous Galerkin formulations, including their penalty terms, follow immediately from a specific choice of multiscale projector. We proceed by defining the “fine-scale closure function”, which captures the closure relation between the remaining fine-scale term in the discontinuous Galerkin formulation and the coarse-scale solution via a single integral expression for each basis function in the coarse-scale test space. We show that the projectors that correspond to discontinuous Galerkin methods lead to fine-scale closure functions with more compact support and smaller amplitudes compared to the fine-scale closure function of the classical (conforming) finite element method. This observation provides a new perspective on the natural stability of discontinuous Galerkin methods for hyperbolic problems, and may open the door to rigorously designed variational multiscale based fine-scale models that are suitable for DG methods.