Asymptotic Arguments Research Articles

Ergodicity for two class stochastic partial differential equations with anisotropic viscosity

We prove the ergodicity on anisotropic space H˜0,1 for two class stochastic PDEs with anisotropic viscosity: the Navier–Stokes equations, the primitive equations. The Maslowski-Seidler theory and an asymptotic coupling argument are invoked to overcome several aspects challenges causing by the fact that systems only have partial dissipation and the strong nonlinearity construction.

A note on the one-dimensional critical points of the Ambrosio–Tortorelli functional

This note addresses the question of convergence of critical points of the Ambrosio–Tortorelli functional in the one-dimensional case under pure Dirichlet boundary conditions. An asymptotic analysis argument shows the convergence to two possible limits points: either a globally affine function or a step function with a single jump at the middle point of the space interval, which are both critical points of the one-dimensional Mumford–Shah functional under a Dirichlet boundary condition. As a byproduct, non minimizing critical points of the Ambrosio–Tortorelli functional satisfying the energy convergence assumption as in (Babadjian, Millot and Rodiac (2022)) are proved to exist.

Drop splashing after impact onto immiscible pools of different viscosities

Droplet impact onto liquid pools is a canonical scenario relevant to numerous natural phenomena and industrial processes. However, despite their ubiquity, multi-fluid systems with the drop and pool consisting of different liquids are far less well understood. Our hypothesis is that the post-impact dynamics greatly depends on the pool-to-droplet viscosity ratioμp/μd, which we explore over a range of six orders of magnitude using a combination of experiments and theoretical approaches (mathematical modelling and direct numerical simulation). Our findings indicate that in this scenario the splashing threshold and the composition of the ejecta sheet are controlled by the viscosity ratio. We uncover that increasing the pool viscosity decreases the splashing threshold for high viscosity pools (μp/μd≳35) when the splash comes from the droplet. By contrast, for low viscosity pools, the splash sheet comes from the pool and increasing the pool viscosity increases the splashing threshold. Surprisingly, there are conditions for which no splashing is observed under the conditions attainable in our laboratory. Furthermore, considering the interface velocity together with asymptotic arguments underlying the generation of the ejecta has allowed us to understand meaningful variations in the pressure during impact and rationalise the observed changes in the splashing threshold.

A fractional nonlocal elastic model for lattice wave analysis

This paper investigates the link between lattice elasticity and nonlocal continuum mechanics from one-dimensional and multi-dimensional wave propagation problems. Eringen (1983) closed the bridge between one-dimensional linear lattice elasticity (called Lagrange lattice) and nonlocal elastic continuum. The Born-Kármán wave dispersive properties of the one-dimensional lattice are accurately fitted by Eringen's nonlocal differential elastic law using asymptotic or phenomenological arguments. A fractional version of Eringen's nonlocal model based on the introduction of a fractional Laplacian for the nonlocal differential operators can improve the accuracy in matching the wave dispersive curve of lattice dynamics. A multi-dimensional generalization of Lagrange lattice for cubic crystals is the lattice of Gazis et al. composed of central and angular interactions. We show in this paper that the differential Eringen's nonlocal elasticity differs from the lattice-based nonlocal elasticity continuum. In particular, Eringen's nonlocal elasticity preserves the isotropic property, whereas the cubic lattice is only isotropic in the low frequency regimes. The fractional generalization of Eringen's model accurately fitted the wave dispersive properties of the cubic crystal in the first Brillouin zone. The fractional generalization of Eringen's model highlights an isotropic response in the low frequency regimes and a cubic symmetry in the high frequency regime.

Large-Degree Asymptotics of Rational Painlevé-IV Solutions by the Isomonodromy Method

The Painlevé-IV equation has two families of rational solutions generated, respectively, by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann–Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painlevé-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painlevé-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement.

Solvability of discrete Helmholtz equations

Abstract We study the unique solvability of the discretized Helmholtz problem with Robin boundary conditions using a conforming Galerkin finite element method. Well-posedness of the discrete equations is typically investigated by applying a compact perturbation argument to the continuous Helmholtz problem so that a `sufficiently rich' discretization results in a `sufficiently small' perturbation of the continuous problem and well-posedness is inherited via Fredholm’s alternative. The qualitative notion `sufficiently rich', however, involves unknown constants and is only of asymptotic nature. Our paper is focussed on a fully discrete approach by mimicking the tools for proving well-posedness of the continuous problem directly on the discrete level. In this way, a computable criterion is derived, which certifies discrete well-posedness without relying on an asymptotic perturbation argument. By using this novel approach we obtain (a) new existence and uniqueness results for the $hp$-FEM for the Helmholtz problem, (b) examples for meshes such that the discretization becomes unstable (Galerkin matrix is singular) and (c) a simple checking Algorithm MOTZ `marching-of-the-zeros', which guarantees in an a posteriori way that a given mesh is certified for a well-posed Helmholtz discretization.

Plasmonic resonances of slender nanometallic rings

We develop an approximate quasi-static theory describing the low-frequency plasmonic resonances of slender nanometallic rings and configurations thereof. First, we use asymptotic arguments to reduce the plasmonic eigenvalue problem governing the geometric (material- and frequency-independent) modes of a given ring structure to a 1D-periodic integro-differential problem in which the eigenfunctions are represented by azimuthal voltage and polarization-charge profiles associated with each ring. Second, we obtain closed-form solutions to the reduced eigenvalue problem for azimuthally invariant rings (including torus-shaped rings but also allowing for non-circular cross-sectional shapes), as well as coaxial dimers and chains of such rings. For more general geometries, involving azimuthally non-uniform rings and non-coaxial structures, we solve the reduced eigenvalue problem using a semi-analytical scheme based on Fourier expansions of the reduced eigenfunctions. Third, we used the asymptotically approximated modes, in conjunction with the quasi-static spectral theory of plasmonic resonance, to study and interpret the frequency response of a wide range of nanometallic slender-ring structures under plane-wave illumination.

Model-Based and Data-Driven Approaches for Downlink Massive MIMO Channel Estimation

We study downlink channel estimation in a multi-cell Massive multiple-input multiple-output (MIMO) system operating in time-division duplex. The users must know their effective channel gains to decode their received downlink data. Previous works have used the mean value as the estimate, motivated by channel hardening. However, this is associated with a performance loss in non-isotropic scattering environments. We propose two novel estimation methods that can be applied without downlink pilots. The first method is model-based and asymptotic arguments are utilized to identify a connection between the effective channel gain and the average received power during a coherence interval. The second method is data-driven and trains a neural network to identify a mapping between the available information and the effective channel gain. Both methods can be utilized for any channel distribution and precoding. For the model-aided method, we derive all expressions in closed form for the case when maximum ratio or zero-forcing precoding is used. We compare the proposed methods with the state-of-the-art using the normalized mean-squared error and spectral efficiency (SE). The results suggest that the two proposed methods provide better SE than the state-of-the-art when there is a low level of channel hardening, while the performance difference is relatively small with the uncorrelated channel model.

Effective Channel Blind Estimation in Cell-Free Massive MIMO Networks

This letter investigates the performance of a new blind method for effective channel estimation in user-centric cell-free massive multiple-input multiple-output (MIMO) networks. To this end, comparisons with the perfect and statistical channel state information (CSI) methods are performed assuming different precoding choices. The results demonstrate that blind estimation can significantly reduce normalized mean-square error and improve spectral efficiency compared with the statistical CSI method. Additionally, blind estimation works even when the asymptotic arguments that inspire the algorithm do not hold, performs reasonably well in the presence of pilot contamination, and can boost the performance of systems using maximum-ratio (MR) precoding.

Integral parts of reciprocals of the Hurwitz zeta function and the polynomial families that arise in the study thereof

The main purpose of this paper is to investigate the existence of a general evaluatory formula for integral parts of reciprocals of tails of the series defining the Riemann ζ-function for a fixed value of s: a non-unitary positive integer. To that end, we introduce two families of polynomials - one given by a convolution recurrence involving Bernoulli numbers, and the other defined in terms of the previous family. Through an asymptotic argument, we provide the desired formula in terms of said polynomials. Subsequently, using the methodology established, a direct derivation of some particular formulae (corresponding to the cases s=7 and s=9) is made. We then proceed to study the properties of the polynomials mentioned, in particular obtaining a rather surprising symmetry property about the ‘critical line’.

Parameter identification for nonlocal phase field models for tumor growth via optimal control and asymptotic analysis

In this paper, we introduce the problem of parameter identification for a coupled nonlocal Cahn–Hilliard-reaction-diffusion PDE system stemming from a recently introduced tumor growth model. The inverse problem of identifying relevant parameters is studied here by relying on techniques from optimal control theory of PDE systems. The parameters to be identified play the role of controls, and a suitable cost functional of tracking-type is introduced to account for the discrepancy between some a priori knowledge of the parameters and the controls themselves. The analysis is carried out for several classes of models, each one depending on a specific relaxation (of parabolic or viscous type) performed on the original system. First-order necessary optimality conditions are obtained on the fully relaxed system, in both the two- and three-dimensional cases. Then, the optimal control problem on the non-relaxed models is tackled by means of asymptotic arguments, by showing convergence of the respective adjoint systems and the minimization problems as each one of the relaxing coefficients vanishes. This allows obtaining the desired necessary optimality conditions, hence to solve the parameter identification problem, for the original PDE system in case of physically relevant double-well potentials.

Exact Mobility Edges in 1D Mosaic Lattices Inlaid with Slowly Varying Potentials

AbstractA family of 1D mosaic models inlaid with a slowly varying potential is proposed. Combining the asymptotic heuristic argument with the theory of trace map of transfer matrix, mobility edges (MEs), and pseudo‐MEs (PMEs) in their energy spectra are solved semi‐analytically, where ME separates extended states from weakly localized ones and PME separates weakly localized states from strongly localized ones. The nature of eigenstates in extended, critical, weakly localized, pseudo‐critical, and strongly localized is diagnosed by the local density of states, the Lyapunov exponent, the localization tensor, and fractal dimension. Numerical calculation results are in excellent quantitative agreement with theoretical predictions.

Fbst: An R package for the Full Bayesian Significance Test for testing a sharp null hypothesis against its alternative via the e value

Hypothesis testing is a central statistical method in psychology and the cognitive sciences. However, the problems of null hypothesis significance testing (NHST) and p values have been debated widely, but few attractive alternatives exist. This article introduces the fbst R package, which implements the Full Bayesian Significance Test (FBST) to test a sharp null hypothesis against its alternative via the e value. The statistical theory of the FBST has been introduced more than two decades ago and since then the FBST has shown to be a Bayesian alternative to NHST and p values with both theoretical and practical highly appealing properties. The algorithm provided in the fbst package is applicable to any Bayesian model as long as the posterior distribution can be obtained at least numerically. The core function of the package provides the Bayesian evidence against the null hypothesis, the e value. Additionally, p values based on asymptotic arguments can be computed and rich visualizations for communication and interpretation of the results can be produced. Three examples of frequently used statistical procedures in the cognitive sciences are given in this paper, which demonstrate how to apply the FBST in practice using the fbst package. Based on the success of the FBST in statistical science, the fbst package should be of interest to a broad range of researchers and hopefully will encourage researchers to consider the FBST as a possible alternative when conducting hypothesis tests of a sharp null hypothesis.

Renormalizability of Alternative Theories of Gravity: Differences between Power Counting and Entropy Argument

It is well known that General Relativity cannot be considered under the standard of a perturbatively renormalizable quantum field theory, but asymptotic safety is taken into account as a possibility for the formulation of gravity as a non-perturbative renormalizable theory. Recently, the entropy argument has however stepped into the discussion claiming for a “no-go” to the asymptotic safety argument. In this paper, we present simple counter-examples, considering alternative theories of gravity, to the entropy argument as further indications, among others, on the possible flows in the assumptions on which the latter is based. We considered different theories, namely curvature-based extensions of General Relativity as f(R), f(G), extensions of teleparallel gravity as f(T), and Horava–Lifshitz gravity, working out the explicit spherically symmetric solutions in order to make a comparison between power counting and the entropy argument. Even in these cases, inconsistencies were found.

Optimal Sampling Pattern for the Free Final Time Linear Quadratic Regulator

The optimal sampling problem is the selection of the optimal sampling instants together with the optimal control actions such that a given cost function is minimized. In this article, we solve the optimal sampling problem for the free final time linear quadratic regulator. Each optimal sampling instant is computed as the minimization of a maximum eigenvalue problem that is formulated at each stage by previously applying dynamic programming. The solution provides the optimal sampling instants, control actions, and the optimal final time in a recursive and constructive way. Furthermore, the solution is optimal for any arbitrary number of control moves <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$N \geq 1$</tex></formula> , as it is not based on asymptotic arguments. Two application examples show the feasibility of the approach.

Prior Sample Size Extensions for Assessing Prior Impact and Prior-Likelihood Discordance

AbstractThis paper outlines a framework for quantifying the prior’s contribution to posterior inference in the presence of prior-likelihood discordance, a broader concept than the usual notion of prior-likelihood conflict. We achieve this dual purpose by extending the classic notion of prior sample size, M, in three directions: (I) estimating M beyond conjugate families; (II) formulating M as a relative notion that is as a function of the likelihood sample size k, M(k), which also leads naturally to a graphical diagnosis; and (III) permitting negative M, as a measure of prior-likelihood conflict, that is, harmful discordance. Our asymptotic regime permits the prior sample size to grow with the likelihood data size, hence making asymptotic arguments meaningful for investigating the impact of the prior relative to that of likelihood. It leads to a simple asymptotic formula for quantifying the impact of a proper prior that only involves computing a centrality and a spread measure of the prior and the posterior. We use simulated and real data to illustrate the potential of the proposed framework, including quantifying how weak is a ‘weakly informative’ prior adopted in a study of lupus nephritis. Whereas we take a pragmatic perspective in assessing the impact of a prior on a given inference problem under a specific evaluative metric, we also touch upon conceptual and theoretical issues such as using improper priors and permitting priors with asymptotically non-vanishing influence.

A reformulation of the stochastic load combination problem

This paper revisits a classical problem of stochastic load combination, in which the maximum load generated by combinations of various types of stochastic processes are analysed. This paper takes a fresh look at the combination of a renewal pulse with a homogeneous Poisson shock process, and proves that the problem can be transformed into an equivalent renewal process. This approach leads to an accurate solution for the distribution of maximum combined load in the form of an integral equation, which can be solved by a trapezoidal integration scheme. The proposed solution is accurate for all small and large load levels and time intervals, since this approach does not rely on any asymptotic arguments. This paper also analyses various assumptions and relative accuracies of popular approximations of the load combination problem reported in literature. The proposed formulation advances the understanding of the load combination problem and provides an accurate and original solution for an important problem of structural reliability analysis.

Inverse Source on Conformal Conic Geometries

The inverse source problem has a number of applications in antenna analysis and synthesis. The properties of the radiation operator, connecting the source current to the far zone field, depend on the source geometry and can be analyzed by its singular value decomposition. Here, first, we present useful upper bounds about the number of degrees of freedom (NDF) for some 2-D source geometries (i.e., for elliptical and parabolic arc sources) and examine the role of two different representation variables. These results were obtained from asymptotic arguments and allow to define the maximum number of independent sources and patterns that can be radiated by each geometry. They are verified to fit the numerically computed ones, too. Next, we examine the point source reconstructions by considering the point spread function. An approximate closed form evaluation reveals that the arc length representation variable leads to a space invariant behavior. The role of the source electrical length in determining the NDF is pointed out, too. Finally, the radiation properties of different source geometries are compared by means of a synthetic index and examples of radiation pattern synthesis and array diagnostics confirm the need to investigate the role of the source geometry.

Near-Field Transverse Resolution in Planar Source Reconstructions

This article deals with the classical question of estimating the achievable resolution in terms of the configuration parameters in inverse source problems. In particular, the study focuses on the case of a planar surface magnetic current which is to be reconstructed from near-field observed over a bounded rectangular aperture parallel to the source domain. Resolution formulas are well known for far-field or Fresnel-zone configurations, and also for near-field cases, when data are full-view or a measurement aperture is an unbounded plane. For the case of bounded near-field observations, the resolution estimations mainly rely on some asymptotic arguments in order to mimic far-field reasoning. Here, the plan is to improve these results and to work out a resolution estimation that precisely captures the spatially varying behavior entailed by the near-field and aspect-limited configuration. To this end, the pertinent radiation operator is inverted by an adjoint inversion scheme (a back-propagation-like method) and the corresponding point-spread function is analytically estimated. Numerical examples show that the derived resolution estimation clearly points out the role of the geometrical parameters of the configuration and it is more accurate than other literature results.

Optimal sampling pattern for free final time linear quadratic regulator: the scalar case

The optimal sampling problem is the selection of the optimal sampling instants together with the optimal control actions such that a given cost function is minimised. In this article, we solve the optimal sampling problem for free final time linear quadratic regulator with scalar dynamical system. The solution provides the optimal sampling instants, control actions and the optimal final time in a recursive and constructive way for any arbitrary number of samples , as it is not based on asymptotic arguments. An application example shows the feasibility of the approach.