Exponential Approximation Research Articles

A Neural Network Monte Carlo Approximation for Expected Utility Theory

This paper proposes an approximation method to create an optimal continuous-time portfolio strategy based on a combination of neural networks and Monte Carlo, named NNMC. This work is motivated by the increasing complexity of continuous-time models and stylized facts reported in the literature. We work within expected utility theory for portfolio selection with constant relative risk aversion utility. The method extends a recursive polynomial exponential approximation framework by adopting neural networks to fit the portfolio value function. We developed two network architectures and explored several activation functions. The methodology was applied on four settings: a 4/2 stochastic volatility (SV) model with two types of market price of risk, a 4/2 model with jumps, and an Ornstein–Uhlenbeck 4/2 model. In only one case, the closed-form solution was available, which helps for comparisons. We report the accuracy of the various settings in terms of optimal strategy, portfolio performance and computational efficiency, highlighting the potential of NNMC to tackle complex dynamic models.

On geometry of numbers and uniform rational approximation to the Veronese curve

Consider the classical problem of rational simultaneous approximation to a point in \({\mathbb {R}}^{n}\). The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and Moshchevitin in 2018. Recently Nguyen, Poels and Roy provided information on the best approximating rational vectors to the points where the gap is close to this minimal value. Combining the latter result with parametric geometry of numbers, we effectively bound the dual linear form exponents in the described situation. As an application, we slightly improve the upper bound for the classical exponent of uniform Diophantine approximation \({\widehat{\lambda }}_{n}(\xi )\), for even \(n\ge 4\). Unfortunately our improvements are small, for \(n=4\) only in the fifth decimal digit. However, the underlying method in principle can be improved with more effort to provide better bounds. We indeed establish reasonably stronger results for numbers which almost satisfy equality in the estimate by Marnat and Moshchevitin. We conclude with consequences on the classical problem of approximation to real numbers by algebraic numbers/integers of uniformly bounded degree.

Fully anisotropic hyperelasto-plasticity with exponential approximation by power series and scaling/squaring

Fully anisotropic hyperelasto-plasticity with exponential approximation by power series and scaling/squaring

A graph arising in the Geometry of Numbers

The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among classical exponents of simultaneous approximation can be guessed by a study of these graphs; in particular the so called regular graph is of major importance as it provides an extremal case for some of these inequalities. The aim of this paper is to define and construct an analogue of the regular graph in the case of weighted simultaneous approximation.

Temperature Measurement at Curved Surfaces Using 3D Printed Planar Resistance Temperature Detectors

In this article, novel 3D printed sensors for temperature measurement are presented. A planar structure of the resistive element is made, utilizing paths of a conductive filament embedded in an elastic base. Both electrically conductive and flexible filaments are used simultaneously during the 3D printing procedure, to form a ready–to–use measuring device. Due to the achieved flexibility, the detectors may be used on curved and irregular surfaces, with no concern for their possible damage. The geometry and properties of the proposed resistance detectors are discussed, along with a printing procedure. Numerical models of considered sensors are characterized, and the calculated current distributions as well as equivalent resistances of the different structures are compared. Then, a nonlinear influence of temperature on the resistance is experimentally determined for the exemplary planar sensors. Based on these results, using first–order and hybrid linear–exponential approximations, the analytical formulae are derived. Additionally, the device to measure an average temperature from several measuring surfaces is considered. Since geometry of the sensor can be designed utilizing presented approach and printed by applying fused deposition modeling, the functional device can be customized to individual needs.

Improvement of method for estimation of parameters of contact interaction of billet with roll tool in lines of continuous electric pipe-welded mills

A technique for calculating the parameters of the shape of the boundary and areas of contact indentations of a pipe billet with a roller tool of a forming mill, in which the boundaries of the shape of the contact indentation are described by rectilinear lines, and the shape is modeled by triangles on the plane of the pipe billet, is investigated. The results of an experiment on fixing the contact sections of a tubular billet with initial dimensions of 160×1.5×1800 mm from steel 09G2S with rolls in the section of open pass are presented. As a result of the experiment, the contact areas of the pipe billet with the lower and upper rolls were fixed. The boundary of the experimental contact areas are curved arcs. Comparison of the experimental data with the technique showed that the parameters of the boundaries of the contact area according to the technique lead to an overestimation of the values by 30-40 % compared to the experimental data. On the basis of experimental data, the development of the existing technique for determining contact areas is presented, in which the boundaries are defined in the form of arcuate curves. The boundaries of the entry contact sections for the lower and upper rolls are determined by the equations of exponential approximation. Taking into account the equations of the boundaries of the sections, a method is presented for determining the areas of contact sections of a pipe billet on the surfaces of the lower and upper rolls, the results of which do not exceed 6 % of the experimental data. The engineering mathematical software PTC Mathcad Prime 6.0 was used to calculate and construct the areas of contact areas.

Neural network approximation: Three hidden layers are enough

A three-hidden-layer neural network with super approximation power is introduced. This network is built with the floor function (⌊x⌋), the exponential function (2x), the step function (1x≥0), or their compositions as the activation function in each neuron and hence we call such networks as Floor-Exponential-Step (FLES) networks. For any width hyper-parameter N∈N+, it is shown that FLES networks with width max{d,N} and three hidden layers can uniformly approximate a Hölder continuous function f on [0,1]d with an exponential approximation rate 3λ(2d)α2−αN, where α∈(0,1] and λ>0 are the Hölder order and constant, respectively. More generally for an arbitrary continuous function f on [0,1]d with a modulus of continuity ωf(⋅), the constructive approximation rate is 2ωf(2d)2−N+ωf(2d2−N). Moreover, we extend such a result to general bounded continuous functions on a bounded set E⊆Rd. As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of ωf(r) as r→0 is moderate (e.g., ωf(r)≲rα for Hölder continuous functions), since the major term to be concerned in our approximation rate is essentially d times a function of N independent of d within the modulus of continuity. Finally, we extend our analysis to derive similar approximation results in the Lp-norm for p∈[1,∞) via replacing Floor-Exponential-Step activation functions by continuous activation functions.

Threshold selection in univariate extreme value analysis

Threshold selection plays a key role in various aspects of statistical inference of rare events. In this work, two new threshold selection methods are introduced. The first approach measures the fit of the exponential approximation above a threshold and achieves good performance in small samples. The second method smoothly estimates the asymptotic mean squared error of the Hill estimator and performs consistently well over a wide range of processes. Both methods are analyzed theoretically, compared to existing procedures in an extensive simulation study and applied to a dataset of financial losses, where the underlying extreme value index is assumed to vary over time.

Efficient difference method for time-space fractional diffusion equation with Robin fractional derivative boundary condition

In this paper, a numerical method is proposed to solve the time-space fractional diffusion equation with Robin fractional derivative boundary condition. Under the weak regularity assumptions of solution, we present a numerical scheme based on the L1 method for time discretization on graded mesh and the Grunwald-Letnikov formula for spatial discretization on uniform mesh. And a fast scheme for the considered problem is constructed based on the exponential summation approximation of the kernel function t−α. Meanwhile, a detailed analysis of stability and convergence is given. Then, the extrapolation method is applied to the space direction to make it reach the second-order accuracy. Finally, numerical experiments show that the proposed method is effective.

Z-Transform Exponential Approximation of One-Dimensional Functions: Theory and Applications

Z-Transform Exponential Approximation of One-Dimensional Functions: Theory and Applications

A method for mobile device positioning using a sensor network of BLE beacons, approximation of the RSSI value and artificial neural networks

The paper considers the development of a method for positioning a mobile device using a sensor network of BLE-beacons, the approximation of RSSI values and artificial neural networks. The aim of the work is to develop a method for positioning small-scale industrial mechanization equipment for building unmanned systems for product movement tracking. The work is divided into four main parts: data synthesis, signal filtering, selection of BLE beacons, translation of the RSSI values into a distance, and multilateration. A simplified Kalman filter is proposed for filtering the input signal to suppress Gaussian noise. A description of two approaches to translating the RSSI value into a distance is given: an exponential approximation function with a coefficient of determination of 0.6994 and an artificial feedforward neural network. A comparison of the results of these approaches is carried out on several test samples: a training one, a test sample at a known distance (0–50 meters) and a test sample at an unknown distance (60–100 meters). The artificial neural network is shown to perform better in all experiments, except for the test sample at a known distance (0–50 meters), for which the r.m.s. error is higher by 0.02 m 2 than that for the approximation function, which can be neglected. An algorithm for positioning a mobile device based on the multilateration method is proposed. Experimental studies of the developed method have shown that the positioning error does not exceed 0.9 meters in a 5×5.5 m room under monitoring. The positioning accuracy of a mobile device using the proposed method in the experiment is 40.9 % higher. Experimental studies are also conducted in a 58.4×4.5 m room, showing more accurate results compared to similar studies.

Potential Well in Poincaré Recurrence

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For -mixing and -mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

Multilevel joint modeling of hospitalization and survival in patients on dialysis

More than 720,000 patients with end‐stage renal disease in the United States require life‐sustaining dialysis treatment. In this population of typically older patients with a high morbidity burden, hospitalization is frequent at a rate of about twice per patient‐year. Aside from frequent hospitalizations, which is a major source of death risk, overall mortality in dialysis patients is higher than other comparable populations, including Medicare patients with cancer. Thus, understanding patient‐ and facility‐level risk factors that jointly contribute to longitudinal hospitalizations and mortality is of interest. Towards this objective, we propose a novel methodology to jointly model hospitalization, a binary longitudinal outcome, and survival, based on multilevel data from the United States Renal Data System (USRDS), with repeated observations over time nested in patients and patients nested in dialysis facilities. In our approach, the outcomes are modeled through a common set of multilevel random effects. In order to accommodate the USRDS data structure, we depart from the literature on joint modeling of longitudinal and survival data by including multilevel random effects and multilevel covariates, at both the patient and facility levels. An approximate Expectation‐Maximization algorithm is developed for estimation and inference where fully exponential Laplace approximations are utilized to address computational challenges.

Deep Neural Network Approximation Theory

This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relation between the complexity of the function (class) to be approximated and the complexity of the approximating network in terms of connectivity and memory requirements for storing the network topology and the associated quantized weights. The theory we develop establishes that deep networks are Kolmogorov-optimal approximants for markedly different function classes, such as unit balls in Besov spaces and modulation spaces. In addition, deep networks provide exponential approximation accuracy-i.e., the approximation error decays exponentially in the number of nonzero weights in the network-of the multiplication operation, polynomials, sinusoidal functions, and certain smooth functions. Moreover, this holds true even for one-dimensional oscillatory textures and the Weierstrass function-a fractal function, neither of which has previously known methods achieving exponential approximation accuracy. We also show that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.

Vibrational motion of a vortex lattice induced near the surface of a type-II superconductor by an external low-frequency ac magnetic field

The theoretical description and experimental study of a reversible vibrational vortex motion induced in a type-II superconductor by a weak low-frequency ac magnetic field are discussed. The case where the external magnetic field and vortex lines are parallel to the surface of the superconductor is considered. The equilibrium positions of vortices near a surface, the amplitude of their vibrations, and the contribution of this process to the dynamic permeability ${\ensuremath{\mu}}_{v}$ are calculated. In these calculations, the interactions of vortices with each other, with the surface, and with pinning centers are considered within the discrete vortex lattice model. The calculations in the London approximation are compared with the results obtained within a more accurate model, including the spatial variation of the order parameter in the vortex core by means of a variational function. It is shown that the structural parameters of the vortex lattice near the surface depend on the accuracy of the description of the intervortex interactions, but the contribution ${\ensuremath{\mu}}_{v}$ appears to be model independent. It is important that this conclusion is valid in the case of the exponential approximation for the interaction between vortex rows. In this approximation, the problem under consideration is analytically examined, and formulas describing the dependences of the parameters of the vortex lattice and the contribution ${\ensuremath{\mu}}_{v}$ on the external dc magnetic field and the characteristics of a superconductor are derived. The theoretical results are compared with experimental data for a $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{y}$ high-${T}_{c}$ superconducting single-crystal plate. The magnetic field is parallel to the plate, i.e., to the crystallographic $ab$ plane. A strong anisotropy of the properties and shape of the crystal is considered when analyzing the experimental data. A procedure for the determination of the experimental dependence ${\ensuremath{\mu}}_{v}(H)$ is described. It is shown that the reversible vibrations of vortices occur near the planes of the single-crystal plate in the direction of the crystallographic $c$ axis, whereas vortices enter its lateral ends and leave them (along the structural layers) even at a small magnetic field amplitude of about 1 Oe. It is found that the developed theoretical description reproduces the experimental data ${\ensuremath{\mu}}_{v}(H)$ in the entire studied range of temperatures and magnetic fields under the assumptions that pinning is isotropic and the length of interaction of vortices with pinning centers is close to the coherence length.

The Spiral Galaxies Flat Rotational Velocity Curve Explained by the Constant Group Velocity of a Nonlinear Density Wave

Abstract The rotation velocity curves of stars in galaxies, the motions of pairs of galaxies, and the behavior of galaxies in clusters and super-clusters all indicate that there is a lack of mass on different scales in the universe. In this paper, we derive the expression for rotational velocity using the nonlinear density wave theory considering only stellar components and we show that such theory can support the observed flat rotational velocity curve due to the main property of the soliton wave, which is a constant group velocity of the wave. The surface mass density (SMD) function, used to derive gravitational potential gradient and rotational velocity, is not assumed but rather derived as a solution of the nonlinear Srödinger equation, on the contrary to the widely used, in the literature, exponential disk approximation. Three parameters relevant to the curve shape are the intensities of equilibrium SMD, the amplitude of the wave, and total angular velocity or differential rotation, equivalently. Since the shape of the rotational velocity is highly sensitive to the mentioned parameters, this theory eventually provides a method for a very accurate estimation of galaxy mass and angular velocity as well.

Relative error stability and instability of matrix exponential approximations for stiff numerical integration of long-time solutions

We study the relative error in the numerical integration of the long-time solution of a linear ordinary differential equation y′(t)=Ay(t),t≥0, where A is a normal matrix. The numerical long-time solution is obtained by using at any step an approximation of the matrix exponential. This paper analyzes the relative error in the stiff situation and it shows that, in this situation, some A-stable approximants exhibit instability with respect to perturbations in the initial value of the long-time solution.

АНАЛИЗ БОЛЬШИХ ДАННЫХ НАДЕЖНОСТИ НЕВОССТАНАВЛИВАЕМЫХ МНОГОКАНАЛЬНЫХ СИСТЕМ

Introduction: methods for analyzing big data of reliability of multichannel systems with a loaded reserve with nonrecoverable elements are considered. Big data contains information on the operating time to failure of elements, obtained by monitoring the operation of similar systems. The main problem that arises when analyzing big data is related to its variety and veracity. Reliability data of system elements correspond to different operating conditions and different laws of failure distribution. The exponential approximation of failure distributions greatly simplifies reliability analysis. However, it can lead to significant errors and requires a separate justification. Purpose: development of approaches to the analysis of heterogeneous big data of reliability of system elements characterized by different distributions of failures. Derivation of estimates of the accuracy of approximation of the distribution of failures by exponential laws and criteria for the possibility of such an approximation. Results: methods for calculating the mean time to failure of systems with a monotonic structure are described. An estimate of the error of exponential approximation of the distributions of failures of elements of a multichannel system is obtained. The relationship between the error of exponential approximation and the coefficient of variation of non-exponential distribution of failures is shown. The case of two channel systems is investigated in detail. For uniform, lognormal, gamma and Weibull distributions, the dependences of the average operating time to rejection of the coefficient of variation are plotted. Areas of variation of the coefficient of variation of these distributions are constructed, for which exponential approximations are justified. An algorithm for constructing the mean time to failure in the analysis of large reliability data of a non-recoverable two-channel system is presented. Practical relevance: analysis of reliability data obtained from monitoring the operation of similar systems eliminates costly reliability tests. The relationship between the sample coefficient of variation and the error of exponential approximation of failure distributions in the analysis of big data of system reliability is shown. This connection forms the basis of the criterion for the possibility of such an approximation.

On the Exponential Approximation of Type II Error Probability of Distributed Test of Independence

This paper studies distributed binary test of statistical independence under communication (information bits) constraints. While testing independence is very relevant in various applications, distributed independence test is particularly useful for event detection in sensor networks where data correlation often occurs among observations of devices in the presence of a signal of interest. By focusing on the case of two devices because of their tractability, we begin by investigating conditions on Type I error probability restrictions under which the minimum Type II error admits an exponential behavior with the sample size. Then, we study the finite sample-size regime of this problem. We derive new upper and lower bounds for the gap between the minimum Type II error and its exponential approximation under different setups, including restrictions imposed on the vanishing Type I error probability. Our theoretical results shed light on the sample-size regimes at which approximations of the Type II error probability via error exponents became informative enough in the sense of predicting well the actual error probability. We finally discuss an application of our results where the gap is evaluated numerically, and we show that exponential approximations are not only tractable but also a valuable proxy for the Type II probability of error in the finite-length regime.

Precise asymptotics of some meeting times arising from the voter model on large random regular graphs

We consider two independent stationary random walks on large random regular graphs of degree k≥3 with N vertices. On these graphs, the exponential approximations of the meeting times are known to follow from existing methods and form a basis for the voter model’s diffusion approximations. The main result of this note improves the exponential approximations to an explicit form such that the first moments are asymptotically equivalent to N(k−1)∕[2(k−2)].