Continuous Kernel Research Articles

The Dirichlet-to-Neumann operator on $C(\partial \Omega)$

Let $\Omega \subset {\bf R}^d$ be an open bounded set with Lipschitz boundary $\Gamma$. Let $D_V$ be the Dirichlet-to-Neumann operator with respect to a purely second-order symmetric divergence form operator with real Lipschitz continuous coefficients and a positive potential $V$. We show that the semigroup generated by $-D_V$ leaves $C(\Gamma)$ invariant and that the restriction of this semigroup to $C(\Gamma)$ is a $C_0$-semigroup. We investigate positivity and spectral properties of this semigroup. We also present results where $V$ is allowed to be negative. Of independent interest is a new criterium for semigroups to have a continuous kernel.

On generalized principal eigenvalues of nonlocal operators witha drift

This article is concerned with the following spectral problem: to find a positive function φ∈C1(Ω) and λ∈R such that q(x)φ′(x)+∫ΩJ(x,y)φ(y)dy+a(x)φ(x)+λφ(x)=0forx∈Ω, where Ω⊂R is a non-empty domain (open interval), possibly unbounded, J is a positive continuous kernel, and a and q are continuous coefficients. Such a spectral problem naturally arises in the study of nonlocal population dynamics models defined in a space–time varying environment encoding the influence of a climate change through a spatial shift of the coefficient. In such models, working directly in a moving frame that matches the spatial shift leads to consider a problem where the dispersal of the population is modeled by a nonlocal operator with a drift term. Assuming that the drift q is a positive function, for rather general assumptions on J and a, we prove the existence of a principal eigenpair (λp,φp) and derive some of its main properties. In particular, we prove that λp(Ω)=limR→+∞λp(ΩR), where ΩR=Ω∩(−R,R) and λp(ΩR) corresponds to the principal eigenvalue of the truncation operator defined in ΩR. The proofs especially rely on the derivation of a new Harnack type inequality for positive solutions of such problems.

A binary-tree element subdivision method for evaluation of nearly singular domain integrals with continuous or discontinuous kernel

An adaptive and efficient volume element subdivision method using binary tree for evaluation of nearly singular domain integrals with continuous or discontinuous kernel in three-dimensional (3-D) boundary element method (BEM) has been presented. In the Conventional Subdivision Method (CSM) for evaluation of nearly singular integrals, the patches are obtained by simply connecting the source point with each vertex of the element. Thus, the accuracy of the integral obtained with CSM is easily affected by the shape of the element and the location of the source point. In contrast, the proposed Binary-Tree Subdivision Method (BTSM) is more convenient to implement and can guarantee successful patch generation under any circumstances for accurate evaluation of nearly singular domain integrals with continuous or discontinuous kernel. Numerical results for volume elements of arbitrary type with various relative locations of the source point demonstrate robustness, accuracy and efficiency of the proposed method. • The BTSM is proposed for evaluating nearly singular integrals with various kernel. • The BTSM is applicable to different kinds of volume element with arbitrary shape. • The BTSM is more convenient to implement by the single binary-tree data structure. • The BTSM can guarantee the convergence of element subdivision in any situation. • The BTSM possesses better accuracy and efficiency than the conventional method.

Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance

In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including n n -dimensional cube [ 0 , 1 ] n [0, 1]^n are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on [ 0 , 1 ] n [0, 1]^n , density of the medium is homogeneous and represented by the Lebesgue measure. Our study includes densities which are singular to the homogeneous one. We establish a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on [ 0 , 1 ] 2 [0, 1]^2 and self-similar measures. We are going to show the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, we obtain diagonal lower and upper estimates of the heat kernel as time tends to 0 0 . In particular, to express the principal part of the lower diagonal heat kernel estimate, we introduce “protodistance”associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.

Scale effect and higher-order boundary conditions for generalized lattices, with direct and indirect interactions

The effects of higher-order boundary conditions in the dynamics behaviour of some higher-order lattices are studied from exact and asymptotic solutions. Higher-order lattices considered herein are generalized axial lattices with direct and indirect symmetrical elastic interactions. More specifically, this two-neighbour interaction lattice is composed of different springs connected to adjacent nodes and to next to adjacent nodes, with possible different stiffness values for each interaction. The boundary nodes at each extremity of this generalized lattice are assumed to be fixed. The natural frequencies of such a fixed-fixed generalized lattice with both symmetrical and truncated higher-order boundary conditions are analytically calculated, from the resolution of a fourth-order boundary difference value problem. The physical meaning of both higher-order boundary conditions is discussed. Whereas the so-called symmetrical higher-order boundary condition is associated with a boundary spring twice the internal one, the truncated higher-order boundary condition preserves the stiffness value of the boundary spring to the internal one. For both higher-order boundary conditions, the vibration modes and dimensionless frequencies are exactly calculated. In both cases, the dimensionless frequency of the general lattice is shown to be lower than the asymptotic continuous one. However, an asymptotic analysis shows that the scaling law for such generalized lattice is strongly sensitive to each higher-order boundary condition. A power law of order 1 or order 2 is obtained for the scaling laws associated with each higher-order boundary condition. As generalized lattices can be also understood as the physical discrete support of some distributed nonlocal elastic models with continuous kernels, it is expected that the strong scale dependence observed in this paper also concerns nonlocal elastic problems.

Kolmogorov Widths on the Sphere via Eigenvalue Estimates for Hölderian Integral Operators

Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere $$S^m$$ in the Euclidean space $${\mathbb {R}}^{m+1}$$ are known to depend upon the Mercer’s expansion of the compact and self-adjoint $$L^2(S^m)$$ -operator associated to the kernel. The estimation of the Kolmogorov nth width of the unit ball of the reproducing kernel Hilbert space in $$L^2(S^m)$$ and the identification of the so-called optimal subspace usually suffice. These Kolmogorov widths can be computed through the eigenvalues of the integral operator associated to the kernel. This paper provides sharp upper bounds for the Kolmogorov widths in the case in which the kernel satisfies an abstract Holder condition. In particular, we follow the opposite direction usually considered in the literature, that is, we estimate the widths from decay rates for the sequence of eigenvalues of the integral operator.

Weak convergence of linear and quadratic forms and related statements on L-approximability

In this paper we obtain central limit theorems for quadratic forms of non-causal short memory linear processes with independent identically distributed innovations. Nabeya and Tanaka (1988) suggested the format, which links the asymptotic distribution to integral operators. In their approach, integral operators had to have continuous symmetric kernels. Mynbaev (2001) employed the theory of approximations to get rid of the continuity requirement. Here we go one step further by lifting the kernel symmetry condition. Also, we establish L p -approximability of the special sequences which arise in the theory of regressions with slowly varying regressors.

Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network

We consider a spatially-extended model for a network of interacting FitzHugh-Nagumo neurons without noise, and rigorously establish its mean-field limit towards a nonlocal kinetic equation as the number of neurons goes to infinity. Our approach is based on deterministic methods, and namely on the stability of the solutions of the kinetic equation with respect to their initial data. The main difficulty lies in the adaptation in a deterministic framework of arguments previously introduced for the mean-field limit of stochastic systems of interacting particles with a certain class of locally Lipschitz continuous interaction kernels. This result establishes a rigorous link between the microscopic and mesoscopic scales of observation of the network, which can be further used as an intermediary step to derive macroscopic models. We also propose a numerical scheme for the discretization of the solutions of the kinetic model, based on a particle method, in order to study the dynamics of its solutions, and to compare it with the microscopic model.

Numerical approach for stochastic differential equations of fragmentation; application to avalanches

We build and develop a unifying method for the construction of a continuous time branching–fragmentation processes on the space of all fragmentation sizes, induced either by continuous fragmentation kernels or by discontinuous ones. This construction leads to a stochastic model for the fragmentation phase of an avalanche. We introduce an approximation scheme for the process which solves the corresponding stochastic differential equations of fragmentation. One of the main achievements of the paper is to compute the distributions of the branching processes approximating the forthcoming branching–fragmentation process. Finally, we present numerical results that confirm the validity of the fractal property which was emphasized by our model for an avalanche.

Efficient global optimization of constrained mixed variable problems

Due to the increasing demand for high performance and cost reduction within the framework of complex system design, numerical optimization of computationally costly problems is an increasingly popular topic in most engineering fields. In this paper, several variants of the Efficient Global Optimization algorithm for costly constrained problems depending simultaneously on continuous decision variables as well as on quantitative and/or qualitative discrete design parameters are proposed. The adaptation that is considered is based on a redefinition of the Gaussian Process kernel as a product between the standard continuous kernel and a second kernel representing the covariance between the discrete variable values. Several parameterizations of this discrete kernel, with their respective strengths and weaknesses, are discussed in this paper. The novel algorithms are tested on a number of analytical test-cases and an aerospace related design problem, and it is shown that they require fewer function evaluations in order to converge towards the neighborhoods of the problem optima when compared to more commonly used optimization algorithms.

A numerically robust method of moments with number density function reconstruction and its application to soot formation, growth and oxidation

Several method of moments (MOM) models were developed recently to describe the evolution of soot particle populations. However, especially soot oxidation is still very challenging in MOM, as pointwise information of the underlying soot particle number density function (NDF) is usually unknown and thus, the prediction of smallest particles that oxidize completely is not easy. The recently proposed Extended Quadrature Method of Moments (EQMOM) (Yuan, Laurent, & Fox, 2012) has the potential to resolve this issue providing a continuous NDF reconstruction based on the set of transported moments. While its general suitability for soot was already demonstrated in the literature, the EQMOM moment inversion procedure reveals several numerical difficulties.In this work, we propose an EQMOM modification called split-based EQMOM that builds upon the general idea of EQMOM to represent the NDF by a weighted sum of superimposed, non-negative, continuous kernel density functions (KDFs), avoiding, however, the numerical problems of its original version. The model is based on a split of the entire NDF into a sum of overlaying density functions, whose evolution is governed by individual population balance equations (PBEs). These PBEs are derived and implemented in a novel Monte Carlo (MC) framework which is applied to simulate two fuel-rich burner-stabilized laminar premixed flames with different sooting behaviours. Results are compared to a classical MC model to demonstrate the suitability. Next, the MC data is used to assess the suitability of lognormal, gamma and inverse Gaussian distributions to approximate the NDF shape by employing the Wasserstein metric as criterion. This information is finally used to formulate an improved EQMOM soot model, which is then evaluated for both fuel-rich and oxidizing conditions. For the latter, the two-stage burner experiments of Echavarria, Jaramillo, Sarofim, & Lighty (2011) is considered, where particle oxidation is the dominant process. It is demonstrated that the proposed MOM model allows an accurate and numerically robust description of soot formation, growth and also oxidation.

Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind

In this work we obtain approximate solutions for Fredholm integral equations of the second kind by means of Petrov–Galerkin method, choosing “regular pairs” of subspaces, Xn,Yn, which are simply characterized by the positive definitiveness of a correlation matrix. This choice guarantees the solvability and numerical stability of the approximation scheme in an easy way, and the selection of orthogonal basis for the subspaces make the calculations quite simple. Afterwards, we explore an interesting phenomenon called “superconvergence”, observed in the 1970s by Sloan: once the approximations un∈Xn to the solution of the operator equation u-Ku=g are obtained, the convergence can be notably improved by means of an iteration of the method, un*=g+Kun. We illustrate both procedures of approximation by means of two numerical examples: one for a continuous kernel, and the other for a weakly singular one.

On Solutions of the Fuji-Suzuki-Tsuda System

We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlev\'e VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for $c=N-1$.

Oscillating heat kernels on ultrametric spaces

Let (X,d) be a proper ultrametric space. Given a measure m on X and a function B \mapsto C(B) defined on the collection of all non-singleton balls B of X , we consider the associated hierarchical Laplacian L=L_{C}{} . The operator L acts in \mathcal{L}^{2}(X,m), is essentially self-adjoint and has a pure point spectrum. It admits a continuous heat kernel \mathfrak{p}(t,x,y) with respect to m . We consider the case when X has a transitive group of isometries under which the operator L is invariant and study the asymptotic behaviour of the function t\mapsto \mathfrak{p}(t,x,x)=\mathfrak{p}(t) . It is completely monotone, but does not vary regularly. When X=\mathbb{Q}_{p}{} , the ring of p -adic numbers, and L=\mathcal{D}^{\alpha} , the operator of fractional derivative of order \alpha , we show that \mathfrak{p}(t)=t^{-1/\alpha}\mathcal{A} (\log_{p}t) , where \mathcal{A}(\tau) is a continuous non-constant \alpha -periodic function. We also study asymptotic behaviour of min \mathcal{A} and max \mathcal{A} as the space parameter p tends to \infty . When X=S_{\infty}{} , the infinite symmetric group, and L is a hierarchical Laplacian with metric structure analogous to \mathcal{D}^{\alpha}, we show that, contrary to the previous case, the completely monotone function \mathfrak{p}(t) oscillates between two functions \psi(t) and \Psi(t) such that \psi(t)/\Psi(t)\to 0 as t \to \infty .

Strictly Positive Definite Functions on Compact Two-Point Homogeneous Spaces: the Product Alternative

For two continuous and isotropic positive definite kernels on the same compact two-point homogeneous space, we determine necessary and sufficient conditions in order that their product be strictly positive definite. We also provide a similar characterization for kernels on the space-time setting $G \times S^d$, where $G$ is a locally compact group and $S^d$ is the unit sphere in $\mathbb{R}^{d+1}$, keeping isotropy of the kernels with respect to the $S^d$ component. Among other things, these results provide new procedures for the construction of valid models for interpolation and approximation on compact two-point homogeneous spaces.

On the Operator Inequalities u ≤ Ku + f and u ≥ Ku + f with Applications

In this article, the operator inequalities and with linear positive operator K are analyzed in a real Banach space. The obtained results are applied to linear Fredholm and Volterra integral inequalities of the second kind with general continuous nonnegative kernel k(x, s). These results are all new. For a special case of a Volterra inequality, Gronwall’s lemma is deduced, however, in a manner that is different from the known one. The general results may also be applied to Fredholm-type and Volterra-type matrix inequalities.

Image Domain Gridding: a fast method for convolutional resampling of visibilities

In radio astronomy obtaining a high dynamic range in synthesis imaging of wide fields requires a correction for time and direction-dependent effects. Applying direction-dependent correction can be done by either partitioning the image in facets and applying a direction-independent correction per facet, or by including the correction in the gridding kernel (AW-projection). An advantage of AW-projection over faceting is that the effectively applied beam is a sinc interpolation of the sampled beam, where the correction applied in the faceting approach is a discontinuous piece wise constant beam. However, AW-projection quickly becomes prohibitively expensive when the corrections vary over short time scales. This occurs, for example, when ionospheric effects are included in the correction. The cost of the frequent recomputation of the oversampled convolution kernels then dominates the total cost of gridding. Image domain gridding is a new approach that avoids the costly step of computing oversampled convolution kernels. Instead low-resolution images are made directly for small groups of visibilities which are then transformed and added to the large uv grid. The computations have a simple, highly parallel structure that maps very well onto massively parallel hardware such as graphical processing units (GPUs). Despite being more expensive in pure computation count, the throughput is comparable to classical W-projection. The accuracy is close to classical gridding with a continuous convolution kernel. Compared to gridding methods that use a sampled convolution function, the new method is more accurate. Hence, the new method is at least as fast and accurate as classical W-projection, while allowing for the correction for quickly varying direction-dependent effects.

Blind in a commutative world: Simple illustrations with functions and chaotic attractors

The paper is devoted to investigate three different points including the importance, usefulness of the Bode diagram in calculus including classical and fractional on one hand. On the other hand to answer and disprove the statements made about fractional derivatives with continuous kernels. And finally to show researchers what we see and we do not see in a commutative world. To achieve this, we considered first the Caputo–Fabrizio derivative and used its Laplace transform to obtain a transfer function. We represented the Bode, Nichols, and the Nyquist diagrams of the corresponding transfer function. We in order to assess the effect of exponential decay filter used in Caputo–Fabrizio derivative, compare the transfer function associate to the Laplace transform of the classical derivative and that of Caputo–Fabrizio, we obtained surprisingly a great revelation, the Caputo–Fabrizio kernel provide better information than first derivative according to the diagram. In this case, we concluded that, it was not appropriate to study the Bode diagram of transfer function of Caputo–Fabrizio derivative rather, it is mathematically and practically correct to see the effect of the kernel on the first derivative as it is well-established mathematical operators. The Caputo–Fabrizio kernel Bode diagram shows that, the kernel is low past filter which is very good in signal point of view. We consider the Mittag–Leffler kernel and its corresponding Laplace transform and find out that due to the fractional order, the corresponding transfer function does not exist therefore the Bode diagram cannot be presented as there is no so far a mathematical formula that help to find transfer function of such nature. It is therefore an opened problem, how can we construct exactly a transfer function with the following term (iw)α(iw)α+b for instance? We proved that fractional derivative with continous kernel are best to model real world problems, as they do not inforce a non-singular model to become singular due to the singularity of the kernel. We show that, by considering initial time to be slightly above the origin then the Riemann–Liouville and Caputo-power derivatives are fractional derivatives with continuous kernel. We considered some interesting chaotic models and presented their numerical solutions in different ways to show what we see or do not see if a commutative world. To end, we presented the terms to be followed to provide a new fractional derivative.

Weighted Conjugate Gradient-Type Methods for Solving Quadrature Discretization of Fredholm Integral Equations of the First Kind

A variant of conjugate gradient-type methods, called weighted conjugate gradient (WCG), is given to solve quadrature discretization of various first-kind Fredholm integral equations with continuous kernels. The WCG-type methods use a new inner product instead of the Euclidean one arising from discretization of $$L^2$$ -inner product by the quadrature formula. On this basis, the proposed algorithms generate a sequence of vectors which are approximations of solution at the quadrature points. Numerical experiments on a few model problems are used to illustrate the performance of the new methods compared to the CG-type methods.

Density regression using repulsive distributions

ABSTRACTFlexible regression is a traditional motivation for the development of non-parametric Bayesian models. A popular approach for this involves a joint model for responses and covariates, from which the desired result arises by conditioning on the covariates. Many such models involve the convolution of a continuous kernel with some discrete random probability measure defined as an infinite mixture of i.i.d. atoms. Following this strategy, we propose a flexible model that involves the concept of repulsion between atoms. We show that this results in a more parsimonious representation of the regression than the i.i.d. counterpart. The key aspect is that repulsion discourages mixture components that are near each other, thus favouring parsimony. We show that the conditional model retains the repulsive features, thus facilitating interpretation of the resulting flexible regression, and with little or no sacrifice of model fit compared to the infinite mixture case. We show the utility of the methodology by way of a small simulation study and an application to a well-known data set.