Differential Approximation Research Articles

Bayesian inference for Cox proportional hazard models with partial likelihoods, nonlinear covariate effects and correlated observations.

We propose a flexible and scalable approximate Bayesian inference methodology for the Cox Proportional Hazards model with partial likelihood. The model we consider includes nonlinear covariate effects and correlated survival times. The proposed method is based on nested approximations and adaptive quadrature, and the computational burden of working with the log-partial likelihood is mitigated through automatic differentiation and Laplace approximation. We provide two simulation studies to show the accuracy of the proposed approach, compared with the existing methods. We demonstrate the practical utility of our method and its computational advantages over Markov Chain Monte Carlo methods through the analysis of Kidney infection times, which are paired, and the analysis of Leukemia survival times with a semi-parametric covariate effect and spatial variation.

Optimal control approach for nonlinear chemical processes with uncertainty and application to a continuous stirred-tank reactor problem

Practical chemical process is usually a dynamic process including uncertainty. Stochastic constraints can be used to chemical process modeling, where constraints cannot be strictly satisfied or need not be fully satisfied. Thus, optimal control of nonlinear systems with stochastic constraints can be available to address practical nonlinear chemical process problems. This problem is hard to cope with due to the stochastic constraints. By introducing a novel smooth and differentiable approximation function, an approximation-based approach is proposed to address this issue, where the stochastic constraints are replaced by some deterministic ones. Following that, the stochastic constrained optimal control problem is converted into a deterministic parametric optimization problem. Convergence results show that the approximation function and the corresponding feasible set converge uniformly to that of the original problem. Then, the optimal solution of the deterministic parametric optimization problem is guaranteed to converge uniformly to the optimal solution of the original problem. Following that, a computation approach is proposed for solving the original problem. Numerical results, obtained from a nonlinear continuous stirred-tank reactor problem including stochastic constraints, show that the proposed approach is less conservative compared with the existing typical methods and can obtain a stable and robust performance when considering the small perturbations in initial system state.

Solutions for elastic moduli of three-phase composite with random distribution of coated-ellipse inclusions

Some solutions in this work are developed to estimate the elastic moduli of three-phase isotropic composite with random coated-ellipse inclusion in the matrix. Solutions to the macro-elastic moduli of materials in two-dimensional space using approximation and numerical methods including equivalent-inclusion (EI), polarization approximation (PA), differential approximations (DA), and fast Fourier transformation (FFT). In which, there is a combination of those methods to give approximations such as EI-PA, EI-DA, FFT-EI. The construction algebraic expressions can be directly applied to the random coated-ellipse model, in special cases it can be used for circular aggregate particles. The numerical solutions using FFT analysis will be compared with EI-PA, EI-DA, and Hashin–Shtrikman’s bounds. From this, it is possible to indicate the best solution that engineers can use to determine the elastic modulus of the coated-ellipse model.

AUC optimization for deep learning-based voice activity detection

Voice activity detection (VAD) based on deep neural networks (DNN) have demonstrated good performance in adverse acoustic environments. Current DNN-based VAD optimizes a surrogate function, e.g., minimum cross-entropy or minimum squared error, at a given decision threshold. However, VAD usually works on-the-fly with a dynamic decision threshold, and the receiver operating characteristic (ROC) curve is a global evaluation metric for VAD at all possible decision thresholds. In this paper, we propose to maximize the area under the ROC curve (MaxAUC) by DNN, which can maximize the performance of VAD in terms of the entire ROC curve. However, the objective of the AUC maximization is nondifferentiable. To overcome this difficulty, we relax the nondifferentiable loss function to two differentiable approximation functions—sigmoid loss and hinge loss. To study the effectiveness of the proposed MaxAUC-DNN VAD, we take either a standard feedforward neural network or a bidirectional long short-term memory network as the DNN model with either the state-of-the-art multi-resolution cochleagram or short-term Fourier transform as the acoustic feature. We conducted noise-independent training to all comparison methods. Experimental results show that taking AUC as the optimization objective results in higher performance than the common objectives of the minimum squared error and minimum cross-entropy. The experimental conclusion is consistent across different DNN structures, acoustic features, noise scenarios, training sets, and languages.

Analytical ray transfer matrix for the crystalline lens.

We present the formulation of a paraxial ray transfer or ABCD matrix for onion-type GRIN lenses. In GRIN lenses, each iso-indicial surface (IIS) can be considered a refracting optical surface. If each IIS is a shell or layer, the ABCD matrix of a GRIN lens is computed by multiplying a typically high number of translation and refraction matrices corresponding to the K layers inside the lens. Using a differential approximation for the layer thickness, this matrix product becomes a sum. The elements A, B, C, and D of the approximated GRIN ray transfer matrix can be calculated by integrating the elements of a single-layer matrix. This ABCD matrix differs from a homogeneous lens matrix in only one integration term in element C, corresponding to the GRIN contribution to the lens power. Thus the total GRIN lens power is the sum of the homogeneous lens power and the GRIN contribution, which offers a compact and simple expression for the ABDC matrix. We then apply this formulation to the crystalline lens and implement both numerical and analytical integration procedures to obtain the GRIN lens power. The analytical approximation provides an accurate solution in terms of Gaussian hypergeometric functions. Last, we compare our numerical and analytical procedures with published ABCD matrix methods in the literature, and analyze the effect of the iso-indicial surface's conic constant (Q) and inner curvature gradient (G) on the lens power for different lens models.

Generalized implicit solution of ITZ percolation threshold and its effect on the diffusivity of concrete: Influence of aggregate shape- and size-polydispersities

Concrete is usually assumed to be a three-phase particulate composite composed of aggregate, cement paste and interfacial transition zone (ITZ) around the aggregate. Due to the high porosity of the ITZ, the transport of aggressive media in the ITZ is more rapid than that in the cement paste. However, it is still unclear how concrete components, like ITZ thickness and aggregate shape- and size-polydispersities affect the ITZ percolation threshold. This paper develops a generalized implicit solution to establish the quantitative relationship between the ITZ percolation threshold and concrete components using an analogy between lattice and continuum percolation. The reliability of the generalized implicit solution is confirmed by comparing the theoretically calculated ITZ percolation thresholds with the numerical results in this work and other literature. The concrete diffusivity is also predicted by combining the obtained ITZ percolation threshold with the differential effective medium approximation and the multi-component percolation-based effective medium model.

Differential Operator Approximation Based Tightly Coupled Multiphysics Solver Using Cascaded Fourier Network

AbstractCalculating the unknown coupled multiphysics fields from the known boundary condition is of great significance in computational physics. Existing classical algorithms are usually time consuming and resource demanding. Explosive growth in deep learning (DL) has provided a substitutive way to accelerate the calculation process by fully making use of the parallel computing capability of the graphics processing unit. In this work, a cascaded DL framework is presented to compute the coupled multiphysics fields related to electrical potential, temperature, velocity, etc. The whole framework is comprised of an input module and several Fourier modules, which achieves to acquire the coupled fields from the explicit boundary conditions. Compared with conventional networks, the Fourier network emerges consistent average error among different resolutions which saves training costs. As a result, a fully trained network can obtain the expected physics fields in real time, offering rosy prospects for practical scenarios.

Peculiarities of overhead crane movement mechanism operation when powered by a basbar's trolley at action of higher current harmonics

Purpose. Analysis and research of interconnected electromagnetic processes between electric drives of overhead crane moving mechanism and trolleys under conditions of higher current harmonics, taking into account design features, nonlinearity of magnetic and electrophysical properties of materials, proximity effects, surface and external surface effects. Methodology. The research was carried out using methods of electromagnetic field theory, electrical circuit theory, mathematical physics, solving a system of first order differential equations, interpolation, approximation and regression analysis. Findings. A simulating model of interrelated electromagnetic processes between electric drives of an overhead crane mechanism and current-conducting elements of the power supply system of an overhead crane has been developed. Investigations of electromagnetic processes between the electric drives of the overhead crane mechanisms and the trolleys have been carried out. The laws of voltage losses in the busbar's trolleys from the torque of the asynchronous electric drive have been established. This allowed to determine the maximum permissible distance of overhead crane movement to the feeding point of the busbar's trolley section, at which trouble-free operation is ensured as well as to determine the number of feeding points of the busbar's trolley sections and the distance between them. This will ensure the same level of voltage losses and active losses in the trolleys at the fundamental harmonic, and taking into account the higher harmonics of the load current. Originality. .A simulating model of interrelated electromagnetic processes between electric drives of an overhead crane mechanism and current-conducting elements of the power supply system of an overhead crane has been developed. which allows to use integrated parameters of busbar trolleys according to field calculation taking into account skin effect, the shape and location of tires (trolleys) in the busbar, the distances between them and the number of phases of the busbar, depending on the peculiarities of the modes of operation of the electric drive and the distance of the bridge crane to the power point of busbar trolley sections Practical value. Based on the theoretical research, recommendations to reduce voltage losses and active losses in the trolley from the action of higher harmonic loads to the level of voltage losses and active losses at the fundamental harmonic current have been developed.

Listwise Learning to Rank Based on Approximate Rank Indicators

We study here a way to approximate information retrieval metrics through a softmax-based approximation of the rank indicator function. Indeed, this latter function is a key component in the design of information retrieval metrics, as well as in the design of the ranking and sorting functions. Obtaining a good approximation for it thus opens the door to differentiable approximations of many evaluation measures that can in turn be used in neural end-to-end approaches. We first prove theoretically that the approximations proposed are of good quality, prior to validate them experimentally on both learning to rank and text-based information retrieval tasks.

Novel Exploit Feature-Map-Based Detection of Adversarial Attacks

In machine learning (ML), adversarial attack (targeted or untargeted) in the presence of noise disturbs the model prediction. This research suggests that adversarial perturbations on pictures lead to noise in the features constructed by any networks. As a result, adversarial assaults against image categorization systems may present obstacles and possibilities for studying convolutional neural networks (CNNs). According to this research, adversarial perturbations on pictures cause noise in the features created by neural networks. Motivated by adversarial perturbation on image pixel attacks observation, we developed a novel exploit feature map that describes adversarial attacks by performing individual object feature-map visual description. Specifically, a novel detection algorithm calculates each object’s class activation map weight and makes a combined activation map. When checked with different networks like VGGNet19 and ResNet50, in both white-box and black-box attack situations, the unique exploit feature-map significantly improves the state-of-the-art in adversarial resilience. Further, it will clearly exploit attacks on ImageNet under various algorithms like Fast Gradient Sign Method (FGSM), DeepFool, Projected Gradient Descent (PGD), and Backward Pass Differentiable Approximation (BPDA).

Closed-Form Evaluation of Mixed Potential Shielded Layered Media Green’s Functions With Spectral Differential Equation Approximation Method

A novel, robust, and computationally efficient approach is proposed for the evaluation of the Michalski–Zheng’s mixed-potential Green’s functions of general shielded layered media. A high-order variant of the spectral differential equation approximation method (SDEAM) is used to cast the spectra of Green’s functions’ components into pole-residue forms. The latter allows for closed-form evaluation of the Sommerfeld integrals (SIs) defining spatial components of the mixed-potential Green’s functions required by Method-of-Moment (MoM) discretization of different types of integral equations. The method produces Green’s functions for all elevations of interest at a fixed cost, making it substantially faster than popular alternative approaches. Numerical results validate the developed high-order SDEAM scheme and demonstrate its extended range of validity compared to the discrete complex image method (DCIM).

Internal structures and magnetic moments of rocky planets

Context. For a planet to be considered habitable on its surface, it is an important advantage for it to have a magnetic field that protects its atmosphere from stellar winds as well as cosmic rays. The magnetic protection of potentially habitable planets plays a key role in determining the chances of detecting atmospheric biosignatures. In making an estimate of a planet’s magnetic field, its internal structure must first be known. Aims. This study proposes to use the Preliminary Reference Earth Model (PREM) internal structure as the base of a numerical model, as it is considered the best available option for estimating the internal structure of rocky exoplanets. With this model, we estimate the magnetic properties of dry and water-rich Earth-like and Super-Earth-like planets and we apply it to the first 176 planets of this kind confirmed by the Transiting Exoplanet Survey Satellite exoplanet mission (TESS). Methods. Using PREM as a reference, we estimated the internal structure of dry and water-rich rocky planets. This model provides an estimation of the average density and core size of the planet, which are the key ingredients for estimating its magnetic moment, depending on whether it is tidally locked or not. Our model estimates the thermodynamic variables as a function of pressure and including saltwater as a component of water-rich exoplanets. In addition, we did not use the perfect layer differentiation approximation. We validated our model based on the selection of planets and satellites in the Solar System with similar characteristics. Results. Using our model, we estimated the internal structure and magnetic moment of selected dry and water-rich rocky planets and satellites in the Solar system. The differences with the observed values in the internal structure characteristics, mass, average density, moment of inertia factor, and local Rossby number are remarkably low or even negligible. The estimated magnetic moments are also very similar to the observed ones. We applied the model to the first dry and water-rich rocky planets confirmed by TESS, 176 in total, and we find, from an astrobiological perspective, that TOI-700 d and TOI-2257 b are the most interesting examples, as they are located in the habitable zone. However, we note that their magnetic moments are only about 0.01 of the Earth’s magnetic moment.

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.

An Improved Model for Light Transport in the Color Conversion Element of Light-Emitting Diodes

Light transport with fluorescence in a phosphor layer based on radiative transfer theory is an efficient tool for understanding the performance of a phosphor-converted light-emitting diode. In this work, the models based on radiative transfer theory including light conversion of phosphor particles are developed for calculating the light transport in planar remote phosphor layers. The models developed include an ordinary differential approximation and an improved model based on the differential approximation and an integral formulation of the transmission light flux for a phosphor layer with Fresnel boundaries. We calculate the light extraction efficiency (LEE) of a phosphor layer as a function of various parameters, such as the thickness of the layer and the concentration of phosphor. The present models are validated by comparing the results obtained by the present methods, the double spherical harmonics method of order one and a Monte Carlo method. Comparing the results obtained by those methods, one can see that the improvement based on the differential approximation and integral formulation of transmission light flux for calculating the LEE can yield better results for optically thin cases.

Analysis of the interior acoustic wave propagation problems using the modified radial point interpolation method (M-RPIM)

Although the radial point interpolation method (RPIM), which is a typical meshless numerical technique, usually behaves much better than the conventional FEM in addressing the numerical dispersion error issue for acoustic computation and more accurate solutions can generally be yielded with the identical node distributions, the related numerical error still can not be completely removed and further improvements are still required. In this work, we proposed a modified RPIM (M-RPIM) to enhance the abilities of the original RPIM in suppressing the numerical dispersion error. In this M-RPIM, a simple and straightforward scheme is employed to ensure that the integrands in performing the numerical integration are continuously differentiable, while in the original RPIM the quadrature cells usually do not align with the shape function supports and then results in the discontinuously differentiable numerical approximation in the quadrature cells, hence considerable numerical integration error can be generated. Since the discontinuously differentiable entities in the system stiffness matrix can be completely avoided in the present M-RPIM, it is found that the numerical dispersion can be markedly suppressed and more accurate numerical solutions can be yielded than the original RPIM in solving acoustic problems. It should be pointed out that the numerical treatments and conclusions in this work are also applicable for most of other meshfree approximations which are similar to the RPIM.

8 b 10 MS/s differential SAR ADC in 28 nm CMOS for precise energy measurement

In this article an 8-bit differential Successive Approximation Register (SAR) Analog-to-Digital Converter (ADC), designed in the 28 nm CMOS process is presented. It is aimed at pixelated radiation imaging detectors. It allows to distinguish 256 levels of energy and is capable of converting 10 MS/s. The measured INL and DNL are ±0.5 LSB and ±0.3 LSB, respectively. Importantly, the proposed ADC’s comparator offset voltage correction is realized in a time domain allowing to shift the transfer characteristics within the 12 LSB range with no conversion rate degradation. The core of the ADC occupies only 30 µm × 60 µm and the power consumption is 45 µW.

On Differential Geometric Formulations of Slow Invariant Manifold Computation: Geodesic Stretching and Flow Curvature

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.

FFT, DA, and Mori-Tanaka approximation to determine the elastic moduli of three-phase composites with the random inclusions

In this work, some solutions such as Mori-Tanaka approximation (MTA), Differential approximations (DA), and Fast Fourier transformation method (FFT) were applied to estimate the elastic bulk and shear modulus of three-phase composites in 2D. In which two different sizes of circular inclusions are arranged randomly non-overlapping in a continuous matrix. The numerical solutions using FFT analysis were compared with DA, MTA, and Hashin-Strikman's bounds. The MTA and DA reasonably agreeable solution with the FFT solution shows the effectiveness of the approximation methods, which makes MTA, DA useful with simplicity and ease of application.

Construction of Key-dependent S-box for Secure Cloud Storage

As cloud storage systems have developed and been applied in complex environments, their data security has become more prevalent in recent years. The issue has been approached through many models. Data is encrypted and stored in these models. One of the most widely used encryption methods is the Advanced Encryption Standard (AES). In AES, the Substitution box(S-box) is playing a significant part in imparting the job of confusion. The security of the entire cryptosystem depends on its nonlinearity. In this work, a robust and secure S-box is constructed using a novel method, i.e., fingerprint features-based permutation function. Two stages are considered to construct a strong S-box. Firstly, random numbers are generated from the fingerprint features such as bifurcation and ridge endings of the user transmitting data. Subsequently, the permutation function is adapted on the random numbers (developed in the first stage) to augment the strength of the S-box. National Institute of Standards and Technology (NIST) STS 800-22 test suite is considered to evaluate the randomness of the enhanced fingerprint-based S-box. Also, the robustness of the constructed S-box is tested using cryptographical properties, namely Strict Avalanche Criterion (SAC), Nonlinearity (NL), Differential Approximation (DA) probability and output Bits Independence Criterion (BIC). Later, the cryptographical properties of the proposed S-box are compared with several existing S-boxes. After analyzing the characteristics of the proposed scheme, it is revealed that the newly constructed S-box is powerful, robust, and safe against linear and differential assaults.

Joint Differential Evolution and Successive Convex Approximation in UAV-Enabled Mobile Edge Computing

Joint Differential Evolution and Successive Convex Approximation in UAV-Enabled Mobile Edge Computing