Classical Approximation Research Articles

Novel Girsanov correction based Milstein schemes for analysis of nonlinear multi-dimensional stochastic dynamical systems

This work proposes Girsanov corrected explicit, semi-implicit, and implicit Milstein approximations for the solution of nonlinear stochastic differential equations. The solution trajectories provided by the Milstein schemes are corrected by employing the change of measures, aimed at removing the error associated with the diffusion process incurred due to the transformation between two probability measures. The change of measures invoked in the Milstein schemes ensures that the solution from the mapping is measurable with respect to the filtration generated by the error process. The proposed scheme incorporates the error between the approximated mapping and the exact representation as an innovation that is accounted in the Milstein trajectories as an additive term. Numerical demonstrations using parametrically and non-parametrically excited stochastic oscillators, a practical problem involving ring type gyroscope, and a 7-degree-of-freedom nonlinear structural system subjected to both stationary and non-stationary excitation demonstrate the improvement in the solution accuracy for the proposed schemes with much coarser time steps when compared with the classical Milstein approximation with finer time steps.

Asymptotic distributions for likelihood ratio tests for the equality of covariance matrices

Consider k independent random samples from p-dimensional multivariate normal distributions. We are interested in the limiting distribution of the log-likelihood ratio test statistics for testing for the equality of k covariance matrices. It is well known from classical multivariate statistics that the limit is a chi-square distribution when k and p are fixed integers. Jiang and Qi (Scand J Stat 42:988–1009, 2015) and Jiang and Yang (Ann Stat 41(4):2029–2074, 2013) have obtained the central limit theorem for the log-likelihood ratio test statistics when the dimensionality p goes to infinity with the sample sizes. In this paper, we derive the central limit theorem when either p or k goes to infinity. We also propose adjusted test statistics which can be well approximated by chi-squared distributions regardless of values for p and k. Furthermore, we present numerical simulation results to evaluate the performance of our adjusted test statistics and the log-likelihood ratio statistics based on classical chi-square approximation and the normal approximation.

Approximation Speed of Quantized Versus Unquantized ReLU Neural Networks and Beyond

We deal with two complementary questions about approximation properties of ReLU networks. First, we study how the uniform quantization of ReLU networks with real-valued weights impacts their approximation properties.We establish an upper-bound on the minimal number of bits per coordinate needed for uniformly quantized ReLU networks to keep the same polynomial asymptotic approximation speeds as unquantized ones. We also characterize the error of nearest-neighbour uniform quantization of ReLU networks. This is achieved using a new lower-bound on the Lipschitz constant of the map that associates the parameters of ReLU networks to their realization, and an upper-bound generalizing classical results. Second, we investigate when ReLU networks can be expected, or not, to have better approximation properties than other classical approximation families. Indeed, several approximation families share the following common limitation: their polynomial asymptotic approximation speed of any set is bounded from above by the encoding speed of this set.We introduce a new abstract property of approximation families, called ∞-encodability, which implies this upper-bound. Many classical approximation families, defined with dictionaries or ReLU networks, are shown to be ∞-encodable. This unifies and generalizes several situations where this upper-bound is known.

Response Matrix/Discrete Ordinates Solution of the 1D Fokker-Planck Equation

The Fokker-Planck equation (FPE) is one of the quintessential equations of particle transport theory. Representing small angle scattering characteristics of electron and photon transport by differential scattering indeed is a mathematical/numerical challenge. Here, we address the challenge with the method of response matrix applied to the Sn approximation to arrive at a nearly six-place-precision benchmark. Our approach aligns with the response matrix solution of the radiative transfer equation for anisotropic scattering published previously. We conclude with the comparison of the response matrix benchmark to a classical finite difference approximation.

A Novel “Finite Element-Meshfree” Triangular Element Based on Partition of Unity for Acoustic Propagation Problems

The accuracy of the conventional finite element (FE) approximation for the analysis of acoustic propagation is always characterized by an intractable numerical dispersion error. With the aim of enhancing the performance of the FE approximation for acoustics, a coupled FE-Meshfree numerical method based on triangular elements is proposed in this work. In the proposed new triangular element, the required local numerical approximation is built using point interpolation mesh-free techniques with polynomial-radial basis functions, and the original linear shape functions from the classical FE approximation are employed to satisfy the condition of partition of unity. Consequently, this coupled FE-Meshfree numerical method possesses simultaneously the strengths of the conventional FE approximation and the meshfree numerical techniques. From a number of representative numerical experiments of acoustic propagation, it is shown that in acoustic analysis, better numerical performance can be achieved by suppressing the numerical dispersion error by the proposed FE-Meshfree approximation in comparison with the FE approximation. More importantly, it also shows better numerical features in terms of convergence rate and computational efficiency than the original FE approach; hence, it is a very good alternative numerical approach to the existing methods in computational acoustics fields.

Finite temperature considerations in the structure of quadratic GUP-modified white dwarfs

In quantum gravity phenomenology, the effect of the generalized uncertainty principle (GUP) on white dwarf structure has been given much attention in recent literature. However, these studies assume a zero temperature equation of state (EoS), excluding young white dwarfs whose initial temperatures are substantially high. To that cause, this paper calculates the Chandrasekhar EoS and resulting mass-radius relations of finite temperature white dwarfs modified by the quadratic GUP, an approach that extends Heisenberg’s uncertainty principle by a quadratic term in momenta. The EoS was first approximated by treating the quadratic GUP parameter as perturbative, causing the EoS to exhibit expected thermal deviations at low pressures, and conflicting behaviors at high pressures, depending on the order of approximation. We then proceeded with a full numerical simulation of the modified EoS, and showed that in general, finite temperatures cause the EoS at low pressures to soften, while the quadratic GUP stiffens the EoS at high pressures. This modified EoS was then applied to the Tolman–Oppenheimer–Volkoff equations and its classical approximation to obtain the modified mass-radius relations for general relativistic and Newtonian white dwarfs. The relations for both cases were found to exhibit the expected thermal deviations at small masses, where low-mass white dwarfs are shifted to the high-mass regime at large radii, while high-mass white dwarfs acquire larger masses, beyond the Chandrasekhar limit. Additionally, we find that for sufficiently large values of the GUP parameter and temperature, we obtain mass-radius relations that are completely removed from the ideal case, as high-mass deviations due to GUP and low-mass deviations due to temperature are no longer mutually exclusive.

Interaction potential between coplanar uniformly charged disk and ring

We consider a system made up of a uniformly charged disk and a uniformly charged ring that are coplanar with each other. The disk and ring have arbitrary radii and contain arbitrary amounts of net charge that is spread uniformly over area and length, respectively. This study is seen as the first necessary step towards a classical model treatment of a two-dimensional system of electrons in a weak perpendicular magnetic field. In this scenario, the uniformly charged disk represents the neutralizing background in a jellium approximation while a uniformly charged ring is viewed as a classical approximation to the cyclotron motion of the electron in presence of a uniform perpendicular magnetic field. In this work, we obtain an exact integral expression for the interaction potential energy of the system as a function of the separation distance between the centers of the two bodies and discuss various emerging features of this model.

The Fuzzy Width Theory in the Finite-Dimensional Space and Sobolev Space

This paper aims to fuzzify the width problem of classical approximation theory. New concepts of fuzzy Kolmogorov n-width and fuzzy linear n-width are introduced on the basis of α-fuzzy distance which is induced by the fuzzy norm. Furthermore, the relationship between the classical widths in linear normed space and the fuzzy widths in fuzzy linear normed space is discussed. Finally, the exact asymptotic orders of the fuzzy Kolmogorov n-width and fuzzy linear n-width corresponding to a given fuzzy norm in finite-dimensional space and Sobolev space are estimated.

The universal approximation theorem for complex-valued neural networks

We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function σ:C→C in which each neuron performs the operation CN→C,z↦σ(b+wTz) with weights w∈CN and a bias b∈C, and with σ applied componentwise. We completely characterize those activation functions σ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of Cd arbitrarily well. Unlike the classical case of real networks, the set of “good activation functions”—which give rise to networks with the universal approximation property—differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as σ is neither a polynomial, a holomorphic function, or an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of σ is not a polyharmonic function.

Extraction and application of super-smooth cubic B-splines over triangulations

The space of C1 cubic Clough–Tocher splines is a classical finite element approximation space over triangulations for solving partial differential equations. However, for such a space there is no B-spline basis available, which is a preferred choice in computer aided geometric design and isogeometric analysis. A B-spline basis is a locally supported basis that forms a convex partition of unity. In this paper, we explore several alternative C1 cubic spline spaces over triangulations equipped with a B-spline basis. They are defined over a Powell–Sabin refined triangulation and present different types of C2 super-smoothness. The super-smooth B-splines are obtained through an extraction process, i.e., they are expressed in terms of less smooth basis functions. These alternative spline spaces maintain the same optimal approximation power as Clough–Tocher splines. This is illustrated with a selection of numerical examples in the context of least squares approximation and finite element approximation for second and fourth order boundary value problems.

Statistical physics of deep neural networks: Initialization toward optimal channels

In deep learning, neural networks serve as noisy channels between input data and its latent representation. This perspective naturally relates deep learning with the pursuit of constructing channels with optimal performance in information transmission and representation. While considerable efforts are concentrated on realizing optimal channel properties during network optimization, we study a frequently overlooked possibility that neural networks can be initialized toward optimal channels. Our theory, consistent with experimental validation, identifies primary mechanics underlying this unknown possibility and suggests intrinsic connections between statistical physics and deep learning. Unlike the conventional theories that characterize neural networks applying the classic mean-field approximation, we offer analytic proof that this extensively applied simplification scheme is not appropriate in studying neural networks as information channels. To fill this gap, we develop a restricted mean-field framework applicable for characterizing the limiting behaviors of information propagation in neural networks without strong assumptions on inputs. Based on it, we propose an analytic theory to prove that mutual information maximization is realized between inputs and propagated signals when neural networks are initialized at dynamic isometry, a case where information transmits via norm-preserving mappings. These theoretical predictions are validated by experiments on real neural networks, suggesting the robustness of our theory against finite-size effects. Finally, we analyze our findings with information bottleneck theory to confirm the precise relations among dynamic isometry, mutual information maximization, and optimal channel properties in deep learning. Our work may lay a cornerstone for promoting deep learning in terms of network initialization and suggest general statistical physics mechanisms underlying diverse deep learning techniques.

The shrinking target problem for matrix transformations of tori: Revisiting the standard problem

Let T be a d×d matrix with real coefficients. Then T determines a self-map of the d-dimensional torus Td=Rd/Zd. Let {En}n∈N be a sequence of subsets of Td and let W(T,{En}) be the set of points x∈Td such that Tn(x)∈En for infinitely many n∈N. For a large class of subsets (namely, those satisfying the so called bounded property (B) which includes balls, rectangles, and hyperboloids) we show that the d-dimensional Lebesgue measure of the shrinking target set W(T,{En}) is zero (resp. one) if a natural volume sum converges (resp. diverges). In fact, we prove a quantitative form of this zero-one criteria that describes the asymptotic behaviour of the counting function R(x,N):=#{1≤n≤N:Tn(x)∈En}. The counting result makes use of a general quantitative statement that holds for a large class measure-preserving dynamical systems (namely, those satisfying the so called summable-mixing property). We next turn our attention to the Hausdorff dimension of W(T,{En}). In the case the subsets En are balls, rectangles or hyperboloids we obtain precise formulae for the dimension. These shapes correspond, respectively, to the simultaneous, weighted and multiplicative theories of classical Diophantine approximation. The dimension results for balls generalises those obtained in [25] for integer matrices to real matrices. In the final section, we discuss various problems that stem from the results proved in the paper.

How Big Should Your Data Really Be? Data-Driven Newsvendor: Learning One Sample at a Time

We study the classical newsvendor problem in which the decision maker must trade off underage and overage costs. In contrast to the typical setting, we assume that the decision maker does not know the underlying distribution driving uncertainty but has only access to historical data. In turn, the key questions are how to map existing data to a decision and what type of performance to expect as a function of the data size. We analyze the classical setting with access to past samples drawn from the distribution (e.g., past demand), focusing not only on asymptotic performance but also on what we call the transient regime of learning, that is, performance for arbitrary data sizes. We evaluate the performance of any algorithm through its worst-case relative expected regret, compared with an oracle with knowledge of the distribution. We provide the first finite sample exact analysis of the classical sample average approximation (SAA) algorithm for this class of problems across all data sizes. This allows one to uncover novel fundamental insights on the value of data: It reveals that tens of samples are sufficient to perform very efficiently but also that more data can lead to worse out-of-sample performance for SAA. We then focus on the general class of mappings from data to decisions without any restriction on the set of policies and derive an optimal algorithm (in the minimax sense) and characterize its associated performance. This leads to significant improvements for limited data sizes and allows to exactly quantify the value of historical information. This paper was accepted by David Simchi-Levi, data science. Supplemental Material: The data files and online appendix are available at https://doi.org/10.1287/mnsc.2023.4725 .

The joy of mathematical discovery: To the 120th anniversary of Academician A.N. Kolmogorov

This article is dedicated to the memory of the outstanding mathematician of the 20th century, Andrei Nikolaevich Kolmogorov, whose 120th birthday is celebrated this year. The author describes in detail the formation of Kolmogorov as a scientist, the significance of Moscow State University in the scientific life and pedagogical activity of the famous mathematician. Kolmogorov’s contribution to such branches of mathematics as classical analysis, topology, geometry, approximation theory, functional analysis, and probability theory is analyzed, and the contribution of his scientific school to Russian science, including not only mathematics and physics but also the humanities, is shown.

Virtual Elements on polyhedra with a curved face

We revisit classical Virtual Element approximations on polygonal and polyhedral decompositions. We also recall the treatment proposed for dealing with decompositions into polygons with curved edges. In the second part of the paper, we introduce a couple of new ideas for the construction of Virtual Element Method (VEM)-approximations on domains with curved boundary, both in two and three dimensions. The new approach looks promising, although sound numerical tests should be made to validate the efficiency of the method.

INFLUENCE OF ELECTROMAGNETIC RADIATION ON RESISTANCE OF SEMICONDUCTOR DEVICES

The subject matter is the processes of analysis and mechanisms of appearance of instabilities of natural vibrations of semiconductor structures., due to their interaction with flows of charged particles under the influence of external electromagnetic radiation. It is shown that the influence of pulsed electromagnetic radiation is accompanied by the appearance of currents in the conductive elements of products that are capable of exciting natural oscillations of semiconductor components and are the cause of failures of radio products. The aim is to develop the theory of collisionless damping of surface polaritons in the classical approximation, as well as to study the mechanisms of collisionless damping of surface plasmons at the boundaries of semiconductor components of radio products under conditions when the temperature of carriers of conducting solids is much lower than the plasmon energy (quantum approximation). The objectives are: a kinetic equation describing the change in the number of surface plasmons as a result of their interaction with conduction electrons; obtaining its solution, which determine the decrement of oscillations and the power of spontaneous emission of particles. The methods used are: a method of successive approximations for solving the kinetic equations of the charged particle flux - semiconductor structure system within the framework of the quantum approach, when the interaction of waves and particles is in the nature of random collisions and is described by the method of secondary quantization of the system (representation of occupation numbers). The following results are obtained: Expressions are obtained for the decrements of surface plasmons in the presence of an infinitely high and infinitely small potential barrier at the interface between media. A kinetic equation is obtained that describes the change in the number surface plasmons as a result of their interaction with conduction electrons; his decisions are given, which determine oscillation decrement. Expressions are obtained for the decrements of surface plasmons at presence of infinitely high and infinitely small potential barrier at the interface between A physical model of the occurrence of reversible failures (effects induced by electromagnetic radiation currents per volt - ampere characteristics of semiconductor devices). The ranges of parameters of the external electromagnetic radiation at which this physical model is realized. Conclusion. Calculated relations are obtained that relate the parameters of semi-conductor structures: the concentration of free carriers, permittivity, carrier temperature with the value oscillation decrement in the classical and quantum approximations. The comparative analysis of quantitative estimates of the increments of oscillation instabilities carried out in the work makes it possible to solve the problems of optimizing the operating characteristics of active devices in the microwave range. The results of the work can be used in the development of microwave devices designed to amplify, generate and convert electromagnetic oscillations in the millimeter and submillimeter ranges.

Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions

We study the approximation by tensor networks (TNs) of functions from classical smoothness classes. The considered approximation tool combines a tensorization of functions in $L^p([0,1))$, which allows to identify a univariate function with a multivariate function (or tensor), and the use of tree tensor networks (the tensor train format) for exploiting low-rank structures of multivariate functions. The resulting tool can be interpreted as a feed-forward neural network, with first layers implementing the tensorization, interpreted as a particular featuring step, followed by a sum-product network with sparse architecture. In part I of this work, we presented several approximation classes associated with different measures of complexity of tensor networks and studied their properties. In this work (part II), we show how classical approximation tools, such as polynomials or splines (with fixed or free knots), can be encoded as a tensor network with controlled complexity. We use this to derive direct (Jackson) inequalities for the approximation spaces of tensor networks. This is then utilized to show that Besov spaces are continuously embedded into these approximation spaces. In other words, we show that arbitrary Besov functions can be approximated with optimal or near to optimal rate. We also show that an arbitrary function in the approximation class possesses no Besov smoothness, unless one limits the depth of the tensor network.

Comparative study of convection characteristics forasystem with deformable interface based on the Navier-Stokes and Oberbeck-Boussinesq equations.

The right choice of the appropriate mathematical model is crucial for evaluating the physical plausibility of modelling results. The issue of the correct application of the classical Boussinesq approximation for studying the heat and mass transfer in fluidic systems with a deformable boundary is a subject of scientific discussions despite the good agreement of numerous theoretical and numerical results obtained within the convection models based on the Oberbeck-Boussinesq equations with the data of physical experiments and observations. A comparative analysis of the results of numerical simulations in the framework of two-sided models based on the Navier-Stokes equations, and their Boussinesq approximation, is performed in the context of a convection problem in a locally heated two-phase system with a deformable interface. It is demonstrated that the application of the standard Boussinesq approximation allows one to give a consistent description of the effect of interface deformations on combined buoyant-thermocapillary driven fluid motions. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Nonlinear function-on-scalar regression via functional universal approximation.

We consider general nonlinear function-on-scalar (FOS) regression models, where the functional response depends on multiple scalar predictors in a general unknown nonlinear form. Existing methods either assume specific model forms (e.g., additive models) or directly estimate the nonlinear function in a space with dimension equal to the number of scalar predictors, which can only be applied to models with a few scalar predictors. To overcome these shortcomings, motivated by the classic universal approximation theorem used in neural networks, we develop a functional universal approximation theorem which can be used to approximate general nonlinear FOS maps and can be easily adopted into the framework of functional data analysis. With this theorem and utilizing smoothness regularity, we develop a novel method to fit the general nonlinear FOS regression model and make predictions. Our new method does not make any specific assumption on the model forms, and it avoids the direct estimation of nonlinear functions in a space with dimension equal to the number of scalar predictors. By estimating a sequence of bivariate functions, our method can be applied to models with a relatively large number of scalar predictors. The good performance of the proposed method is demonstrated by empirical studies on various simulated and realdatasets.

Modeling the thermal conductivity of carbon nanotube (CNT) nanofluids and nanocomposites – a fresh restart

After two decades of discussion it seems that the model-based prediction of the thermal conductivity of carbon nanotube (CNT) nanofluids and nanocomposites has come to a dead end and needs a fresh restart, since misleading statements and claims and repeated over and over again, mainly in the current literature on CNT nanofluids. In this paper a simple but rigorous model is presented (in two variants) for predicting the thermal conductivity of CNT nanofluids and nanocomposites. The model is based on Fricke's classical Maxwell-type effective medium approximation for systems with spheroidal particles, but – in contrast to all previous models – takes into account the Molyneux bounds for describing the random orientation of the CNTs. We show that the predictions of our model are significantly lower than the corresponding predictions via Nan's model and will thus turn out to be closer to experimental data reported in the literature.