Singular Approximation Research Articles

Further PDE methods for weak KAM theory

We introduce and make estimates for several new approximations that in appropriate asymptotic limits yield the key PDE for weak KAM theory, namely a Hamilton–Jacobi type equation for a potential u and a coupled transport equation for a measure σ. We revisit as well a singular variational approximation introduced in Evans (Calc Vari Partial Differ Equ 17:159–177, 2003) and demonstrate “approximate integrability” of certain phase space dynamics related to the Hamiltonian flow. Other examples include a pair of strongly coupled PDE suggested by the Lions–Lasry theory (Lasry and Lions in Japan J Math 2:229–260, 2007) of mean field games and a new and extremely singular elliptic equation suggested by sup-norm variational theory.

Pointwise approximation of corner singularities for a singularly perturbed reaction-diffusion equation in an $L$-shaped domain

A singularly perturbed reaction-diffusion equation is posed in a two-dimensional $L$-shaped domain $\Omega$ subject to a continuous Dirchlet boundary condition. Its solutions are in the Hölder space $C^{2/3}(\bar \Omega )$ and typically exhibit boundary layers and corner singularities. The problem is discretized on a tensor-product Shishkin mesh that is further refined in a neighboorhood of the vertex of angle $3\pi /2$. We establish almost second-order convergence of our numerical method in the discrete maximum norm, uniformly in the small diffusion parameter. Numerical results are presented that support our theoretical error estimate.

Theoretical Linear Solutions of Steady and Axisymmetric Stewartson-Layer Flow Developed at an Arbitrarily-Shaped Step by Through-Flow

In our previous experimental and numerical studies, we developed accurate submicron-particle classifiers using an almost rigidly rotating flow. In the present paper, with the final aim of improving the classification performance and devising new better classifiers on the numerical and theoretical bases of fluid and particle motions, we have ingeniouly performed a linear analysis of the steady and axisymmetric flow of attached and detached Stewartson layers developing at a step of a lower wall between rigidly rotating upper and lower walls shaped arbitrarily. We have used a singular perturbation method and boundary layer approximations, and have used, as boundary conditions, the theoretical linear solutions of Ekman layers and interior regions derived in our previous study. Consequently, we have derived theoretical linear solutions of E1/4- and E1/3-Stewartson layers and their Ekman extensions containing arbitrary additive constants that cannot be obtained by the conventional analysis following Van Heijst. The conventional solutions disagree with numerical solutions, but there is every possibility that the additive constants make the present solutions agree with numerical solutions qualitatively.

Elastic moduli of anisotropic clay

Clay minerals are important components in shales, controlling their elastic properties and anisotropy. The elasticity of crystalline clay minerals differs significantly from that of clay in situ because of the ability of clay particles to bind water. In the ma-jority of published works, only isotropic moduli for in situ clays are reported. However, anisotropy is inherent in the clay elas-ticity. We develop an inversion technique for determination of the stiffness tensor of in situ clay from the shale’s stiffness tensor. As an example, we obtain the stiffness tensor of a “water-clay” composite from the data on the water-saturated Greenhorn shale sample, whose clay composition consists of almost equal amounts of illite and smectite and comparable amounts of kaolinite and chlorite. The stiffness tensor of the water-clay composite is found for the Greenhorn shale with step-by-step inversion based upon an effective medium theory. The inversion usesa nonlinear optimization technique with bounds imposed on the estimated parameters. In the inversion, we apply different approaches of the effective medium theory using a published method referred to here as the generalized singular approxi-mation (GSA). The GSA method makes it possible to take into account the microstructure of shales. The resulting elasticity constants of the anisotropic (transversely isotropic) in situ clay composite are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] (in GPa); and the density equals [Formula: see text]. The Thomsen parameters for the clay composite are [Formula: see text], [Formula: see text], and [Formula: see text]. The elasticity constants found for this clay composite can be used in the theoretical analysis of shales that have a similar composition of clay but with different mineral compositions. The inversion technique developed can be used for general shale water-clay composites when the mineral composition and orientation of the clay platelets are known.

Analytic Three-Dimensional Model of a Double-Shielded Giant Magnetoresistive Perpendicular Head Using a Singular Function Expansion

By matching a singular function approximation to the Fourier solution at the air-bearing surface (ABS), we have derived a three-dimensional (3-D) analytic model of a shielded giant magnetoresistive (GMR) head, with side shields, for perpendicular replay. We present an explicit expression for the potential in the ABS and give values for the parameters in that expression for a range of practical head dimensions. Using only a few terms of this singular potential model, we show that the vertical field is accurate to within 1% of the sensor potential in the region of the medium for the majority of head dimensions suitable for magnetic recording.

Higher correlations of divisor sums related to primes II: variations of the error term in the prime number theorem

We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for DirichletL-functions, we obtain an Ω±-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ω-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes.

Priority queueing systems: from probability generating functions to tail probabilities

Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. To obtain these probabilities in discrete-time queueing systems, we have to invert probability generating functions, since most important distributions in discrete-time queueing systems can be determined in the form of probability generating functions. In this paper, we calculate the tail probabilities of two particular random variables in discrete-time priority queueing systems, by means of the dominant singularity approximation. We show that obtaining these tail probabilities can be a complex task, and that the obtained tail probabilities are not necessarily exponential (as in most ‘traditional’ queueing systems). Further, we show the impact and significance of the various system parameters on the type of tail behavior. Finally, we compare our approximation results with simulations.

Convergence of a singular Euler-Poisson approximation of the incompressible Navier-Stokes equations

In this note, we rigorously justify a singular approximation of the incompressible Navier-Stokes equations. Our approximation combines two classical approximations of the incompressible Euler equations: a standard relaxation approximation, but with a diffusive scaling, and the Euler-Poisson equations in the quasineutral regime.

A note on singularity approximation

A simple method is considered for approximating the locations of singularities and the associated jumps of a piecewise constant function. The locations of jump discontinuities of a function are recovered approximately, one by one, by means of ratios of so called higher order Fourier–Jacobi coefficients of the function. It is shown that the location of singularity of a piecewise constant function with one discontinuity is recovered exactly and the locations of singularities of a piecewise constant function with multiple discontinuities are recovered with exponential accuracy. The method is applicable to piecewise smooth functions as well, however the accuracy of the approximation sharply declines. In addition, the stability and complexity of the method is discussed and some numerical examples are presented.

Galaxy Structures and External Perturbations in Gravitational Lenses

Abstract In modeling strong gravitational lens systems, one often adopts simple models, such as singular isothermal elliptical density plus lowest-order external perturbation. However, such simple models may mislead us if the real mass distribution is more complicated than that in the assumed models. In particular, assumptions on mass models are crucial in studying flux ratio anomalies that have been suggested as evidence for a cold dark-matter substructure. We reinvestigated four quadruple lens systems using power-law Fourier models, which have advantages of clear physical meanings and the applicability of a linear method, as well as a simple singular isothermal elliptical density model. We also investigated the effect of external perturbations, including a singular isothermal sphere, lowest-order expansion, and next-order expansion. We have found that the $\cos 3\theta$ terms of the primary galaxy and/or of external perturbation significantly reduce the $\chi^2$ in PG 1115$+$080 and B1422$+$231. In particular, we could reproduce the flux ratios of B1422$+$231 with next-order external perturbation assuming 5% flux uncertainties, suggesting that external perturbation cannot be described by a simple singular isothermal sphere approximation. On the other hand, we could not fit B0712$+$472 and B2045$+$265 very well even with our models, although the $\chi^2$ were reduced compared with the case of using simple models. Our results clearly demonstrate that both the primary lens galaxy and the external perturbation are often more complicated than we usually assume.

Iterative computation of the smallest singular value and the corresponding singular vectors of a matrix

Influenced by some techniques used for computing singular points of nonlinear equations, a generalized inverse iteration method is proposed for approximating the smallest singular value σ n and the associated left and right singular vectors u, v of a matrix A∈ R n×n . In the practically relevant case σ n >0 the method is mathematically equivalent to inverse iteration with AA T. However, unlike classic inverse iteration, the new method works with matrices B k∈ R (n+1)×(n+1) obtained by bordering A in such a way that the B k have uniformly bounded condition numbers. This allows using iterative Krylov-type solvers for large problems. If σ n−1 > σ n the singular vector approximations convergence linearly with factor κ= σ n / σ n−1 <1. Moreover, a certain generalized Rayleigh quotient σ ( k) obtained as a byproduct has a relative error ( σ ( k) − σ n )/ σ n which goes to zero R-linearly with factor κ 2. Some numerical examples confirm the theoretical results and show that the algorithm works reliable also for almost singular matrices and when using Krylov solvers.

Nearly singular approximations of CPV integrals with end- and corner-singularities for the numerical solution of hypersingular boundary integral equations

A local numerical approach to cope with the singular and hypersingular boundary integral equations (BIEs) in non-regularized forms is presented in the paper for 2D elastostatics. The approach is based on the fact that the singular boundary integrals can be represented approximately by the mean values of two nearly singular boundary integrals and on the techniques of distance transformations developed primarily in previous work of the authors. The nearly singular approximations in the present work, including the normal and the tangential distance transformations, are designed for the numerical evaluation of boundary integrals with end-singularities at junctures between two elements, especially at corner points where sufficient continuity requirements are met. The approach is very general, which makes it possible to solve the hypersingular BIE numerically in a non-regularized form by using conforming C 0 quadratic boundary elements and standard Gaussian quadratures without paying special attention to the corner treatment. With the proposed approach, an infinite tension plate with an elliptical hole and a pressurized thick cylinder were analyzed by using both the formulations of conventional displacement and traction boundary element methods, showing encouragingly the efficiency and the reliability of the proposed approach. The behaviors of boundary integrals with end- and corner-singular kernels were observed and compared by the additional numerical tests. It is considered that the weakly singularities remain but should have been cancelled with each other if used in pairs by the corresponding terms in the neighboring elements, where the corner information is included naturally in the approximations.

A note on quasi-maximum likelihood solutions to an inverse problem for Poisson processes

Recent results on unfolding Poisson process intensity function are improved. Singular matrix approximation of the folding operator is allowed. For Euclidean spaces, the assumptions are expressed in terms of the decay rate of the singular values of the original folding operator rather than those of the approximating matrix. L 2-convergence rates and approximation effects are discussed.

Underresolved Simulations of Heat Baths

Some recent numerical and theoretical studies indicate that it is possible to accurately simulate the macroscopic motion of a particle in a heat bath, comprising coupled oscillators, without accurately resolving the fast frequencies in the heat bath itself. Here we study this issue further by performing numerical experiments on a wide variety of mechanical heat bath models, all generalizations of the Ford–Kac oscillator model. The results indicate that the nature of the particle-bath damping in the macroscopic limit crucially affects the ability of underresolved simulations to correctly predict macroscopic behaviour. In particular, problems for which the damping is local in time pose more severe problems for approximation. The root cause is that local damping typically arises from the degeneration of a memory kernel to a delta singularity in the macroscopic limit. The approximation of such singularities is a more delicate issue than the approximation of smoother memory kernels.

The crystallographic texture and properties of metal materials, received by the PDME-method

The researches of polycrystalline materials received by the PDME-method as ribbons are carried out. The crystallographic textures of aluminum materials (as example) are investigated in nine layers on thickness with help of the X-ray diffraction and then ODFs of them are reconstructed. The general singularity approximation of random fields is used for the evaluation of effective elastic modulus and parameters of anisotropy of these layers. On the other hand for each layers it is possible to calculate velocities of melt cooling. It allows to connect values of the velocity of cooling with the structure and properties of investigated layers. The connection between velocities of cooling of a melt and elastic properties of rapidly quenched materials and crystallographic textures of materials is established as the theoretical model based on initial experimental data. The results are presented by the figures.

The p and hp finite element method for problems on thin domains

The p and hp versions of the finite element method allow the user to change the polynomial degree to increase accuracy. We survey these methods and show how this flexibility can be exploited to counter four difficulties that occur in the approximation of problems over thin domains, such as plates, beams and shells. These difficulties are: (1) control of modeling error, (2) approximation of corner singularities, (3) resolution of boundary layers, and (4) control of locking. Our guidelines enable the efficient resolution of these difficulties when a p/ hp code is available.

Optimization of Secondary Metabolite Production Using Singular Approximation and Minimum Principle

Optimal control profiles as calculated with two control algorithms, singular approximation and minimum principle, are compared in this article. Switching points were determined using the singular approximation by mathematical calculation. The optimal growth rate was calculated using minimum principle. With an increased number of switching points, the calculated optimal control profiles approached the theoretical optimal control profile as calculated using the minimum principle. With three switching times, the product concentration approached 96% of the theoretical optimal control profile. From these results, optimal control can be achieved with more than a three-switching-point approximation.

Minimal decompositions and iterative methods

Summary. We study how singular values and approximation numbers by algebraic operators of a bounded linear operator A on a separable Hilbert space H can be related to the speed of convergence of iterative methods. In infinite dimensional H these two quantities lead us to consider those A that can be split as $A=N+K$ with N normal andK compact. Then the spectrum of N, via a classical polynomial approximation problem, bounds below the speed of convergence of GMRES. In finite dimensional H we combine these two approximation numbers to approximation numbers by normal operators of a matrix A that lead us to consider splittings ofA as $A=N+F$ with N normal andF of smallest possible rank. We show, analogously to infinite dimensions, that the eigenvalues of N can be used, via a new polynomial approximation problem, in assessing the speed of convergence of GMRES. For upper bounds we obtain estimates that can be combined with the results of [28].

Bifurcation of internal solitary wave modes from the essential spectrum

The bifurcation of internal modes from the phonon band in models supporting solitary wave solutions is currently one of the exciting phenomena in the field. We will present a number of analytical and semianalytical techniques for the detection, study, and understanding of these modes. We will see how they appear, without threshold, due to the discretization of the continuum equations. This perturbation is viewed in terms of a singular continuum approximation and analyzed by both perturbation theory and the Evans's function method. It is shown that these methods give equivalent results. Moreover, they are corroborated by mixed analytical-numerical computations based on the recently developed discrete Evans's function method. The extent to which these predictions survive to strong discretizations is discussed. The results will be presented in the context of both the sine-Gordon and the ${\ensuremath{\varphi}}^{4}$ models.

A model for the post-collapse equilibrium of cosmological structure: truncated isothermal spheres from top-hat density perturbations

The post-collapse structure of objects that form by gravitational condensation out of the expanding cosmological background universe is a key element in the theory of galaxy formation. Towards this end, we have reconsidered the outcome of the non-linear growth of a uniform, spherical density perturbation in an unperturbed background universe — the cosmological ‘top-hat’ problem. We adopt the usual assumption that the collapse to infinite density at a finite time predicted by the top-hat solution is interrupted by a rapid virialization caused by the growth of small-scale inhomogeneities in the initial perturbation. We replace the standard description of the post-collapse object as a uniform sphere in virial equilibrium by a more self-consistent one as a truncated, non-singular, isothermal sphere in virial and hydrostatic equilibrium, including for the first time a proper treatment of the finite-pressure boundary condition on the sphere. The results differ significantly from both the uniform sphere and the singular isothermal sphere approximations for the post-collapse objects. The virial temperature that results is more than twice the previously used ‘standard value’ of the post-collapse uniform sphere approximation, but 1.4 times smaller than that of the singular, truncated isothermal sphere approximation. The truncation radius is 0.554 times the radius of the top-hat at maximum expansion, and the ratio of the truncation radius to the core radius is 29.4, yielding a central density that is 514 times greater than at the surface and 1.8×104 times greater than that of the unperturbed background density at the epoch of infinite collapse predicted by the top-hat solution. For the top-hat fractional overdensity ΔL predicted by extrapolating the linear solution into the non-linear regime, the standard top-hat model assumes that virialization is instantaneous at ΔLΔc=1.686 i.e. the epoch at which the non-linear top-hat reaches infinite density. The surface of the collapsing sphere meets that of the post-collapse equilibrium sphere slightly earlier, however, when ΔL=1.52. These results will have a significant effect on a wide range of applications of the Press-Schechter and other semi-analytical models to cosmology. We discuss the density profiles obtained here in relation to the density profiles for a range of cosmic structures, from dwarf galaxies to galaxy clusters, indicated by observation and by N-body simulation of cosmological structure formation, including the recent suggestion of a universal density profile for haloes in the cold dark matter (CDM) model. The non-singular isothermal sphere solution presented here predicts the virial temperature and integrated mass distribution of the X-ray clusters formed in the CDM model as found by detailed, 3D, numerical gas and N-body dynamical simulations remarkably well. This solution allows us to derive analytically the numerically calibrated mass-temperature and radius-temperature scaling laws for X-ray clusters, which were derived empirically by Evrard, Metzler & Navarro from simulation results for the CDM model.