Traditional Approximation Research Articles

AI-Powered Approaches for Hypersurface Reconstruction in Multidimensional Spaces

The present article explores the possibilities of using artificial neural networks to solve problems related to reconstructing complex geometric surfaces in Euclidean and pseudo-Euclidean spaces, examining various approaches and techniques for training the networks. The main focus is on the possibility of training a set of neural networks with information about the available surface points, which can then be used to predict and complete missing parts. A method is proposed for using separate neural networks that reconstruct surfaces in different spatial directions, employing various types of architectures, such as multilayer perceptrons, recursive networks, and feedforward networks. Experimental results show that artificial neural networks can successfully approximate both smooth surfaces and those containing singular points. The article presents the results with the smallest error, showcasing networks of different types, along with a technique for reconstructing geographic relief. A comparison is made between the results achieved by neural networks and those obtained using traditional surface approximation methods such as Bézier curves, k-nearest neighbors, principal component analysis, Markov random fields, conditional random fields, and convolutional neural networks.

Modeling of the Black Sea circulation using equations of heat and salt advection–diffusion having discrete nonlinear invariants

In this work the accuracy of reconstructing the Black Sea circulation by using new approximations of nonlinear terms in the transport equations, ensuring the conservation of temperature and salinity to a power greater than two, is analyzed based on the results of forecast calculations. Three numerical experiments with differences in the schemes for calculating temperature and salinity are carried out. In the first experiment – traditional schemes are used to conserve of temperature and salinity in the first and second degrees; in the second one – the temperature is conserved in the first and fifth degrees, salinity in the first and third; in the third one – the temperature in the first and third, salinity in the first and fifth degrees. Calculations are performed on the basis of the MHI model with a resolution of 1.6 km and taking into account realistic atmospheric forcing for 2016. Validation of the results is carried out based on comparison of model fields with in-situ and satellite measurements of temperature and salinity in 2016. Analysis of mean and root mean square errors showed that new schemes for the advection-diffusion equations of heat and salt, ensuring the conservation of predictive parameters to a power greater than two, improve the accuracy of reconstructing the salinity in the Black Sea upper 100m layer throughout the year compared with traditional approximation. The root mean square errors in the salinity field are reduced by 15–20%, the thickness of the upper mixed layer in winter and the depth of the upper boundary of the thermocline layer in summer in the central part of the sea are modeled approximately 10% more accurately. Based on the results of three experiments, the smallest deviations from observational data are obtained when using approximations that ensure the conservation of temperature to the third power and salinity to the fifth power.

Direct Entrainment as a Measure of Dilution

Abstract The turbulent transport of mass, energy, moisture, and momentum between the clouds and the surrounding environment plays a central role in determining the vertical structure of the troposphere. This article investigates the connection between net entrainment, defined here as the net transport of air into individual clouds, and net dilution, defined as the tendencies of passive tracers such as static energy or total water mixing ratio. Entrainment and detrainment rates for 2.6 × 106 individual cloud samples are obtained from a large-eddy simulation of shallow convective boundary layer atmosphere that explicitly calculates the turbulent fluxes across the cloud boundaries. The equations describing the tendencies of cloud mass and tracer concentrations are derived as a function of directly calculated entrainment and detrainment rates of the individual clouds, and used to calculate net entrainment and dilution rates. Directly calculated net entrainment and dilution rates agree well with cloud mass and tracer tendencies and give a dilution time scale of 13 min. In contrast, the traditional bulk-plume approximation overestimates the effect of entrainment and detrainment on the dilution of cloud field properties due to the differential tracer transport through the moist shell surrounding the cloud. Using direct measures of entrainment and detrainment for individual clouds separates different processes that influence the turbulent mass transport between the clouds and their environment.

Efficiency high-order discontinuous Galerkin method for low speeds flows with time-derivative preconditioning

A local preconditioning high-order discontinuous Galerkin (DG) method is introduced to enhance the efficiency and accuracy of the density-based approach for low-speed flows. The method is based on the time-derivative preconditioning technique typically employed in finite volume methods. Traditional approximation methods that only precondition low-order derivatives lead to incompatibility of the equations. To extend the preconditioning technique to the DG method, we employ an analytic preconditioning mass matrix derived from the DG formulation of the preconditioned Navier–Stokes equations. Modified numerical flux functions and a self-adaptive local velocity truncation parameter are employed for DG systems with preconditioning. We demonstrate the efficiency and accuracy of the preconditioned DG method using explicit and implicit schemes for steady flows and a dual-time scheme for unsteady flows. The method has been implemented and used to compute a variety of flow problems, including inviscid and laminar flows, utilizing various degrees of polynomial approximations. Computations with and without preconditioning are performed to analyze the influence of spatial discretization on the accuracy and convergence of the DG solutions at low Mach numbers. Numerical results underscore the enhanced accuracy and convergence properties of the proposed method for low-speed flows, indicating significant performance improvements over traditional methods.

Interface stiffness identification of rough and weak bonded interface using developed ultrasonic reflection phase derivative spectrum

The thickness and interface roughness of coatings both affect the interface bonded quality. Existed ultrasonic testing methods based on traditional phase screen approximation or spring model assumption are difficult to simultaneously identify the interface roughness and stiffness of coating. This paper, a new method for integrated identifying coating thickness, interface roughness, and interface stiffness using developed ultrasonic reflection phase derivative spectrum (URPDS) is proposed. A phase-screen-approximated spring-model (PSASM) for ultrasound vertically propagating into rough and weak bonded interface is constructed. On basis of PSASM, a URPDS of coating/substrate structure is developed for identifying the interface stiffness and other parameters of coated parts. Cross-correlation analysis is used to eliminate the phase deviation of URPDS introduced by reference signal and initial phase of tested signal. Sensitivity analysis is used to determine the high-sensitivity regions of URPDS to interface roughness and interface stiffness. Genetic algorithm optimization is used to achieve integrated identification of coating thickness, interface roughness, and interface stiffness. The rationality of PSASM is verified through numerical simulation using a series of coating/substrate models with rough and weak bonded interface, and the relationship between the high-sensitivity regions and the high-precision measurement ranges of interface roughness Rq and interface stiffness Kn is clarified. Ultrasonic experiments are implemented on Nickel-coating samples and coated parts using plane wave probe. The coating thickness, interface roughness, and interface stiffness could be identified accurately, which shows that the proposed URPDS method can identify the interface stiffness of rough contacted dissimilar media or coated parts with rough interface.

Evaluating the Electromagnetic Field Problem Generated by the Power Transmission Line Using High Order Compact Method

This paper compares a finite difference method-based computational scheme for evaluating electromagnetic field problems in power transmission lines in cases of on surface and in Ground. The scheme uses Maxwell's partial differential equations to represent electric and magnetic field components and approximate boundary conditions. The High Order Compact Method (HOC) is applied to estimate the electric field (Ez )in ground and compares it with the standard central difference scheme. The HOC produces more accurate results than the traditional central approximation at 99.7% for the electric field Ez. It also calculates both of Electric and magnetic intensity to evaluate the effect of Electric and magnetic intensities in ground and on surface respect to distance.

The impact of radiative levitation on mode excitation of main-sequence B-type pulsators

Context. Numerical computations of stellar oscillations for models representative of B-type stars predict fewer modes to be excited than observations reveal from modern space-based photometric data. One shortcoming of state-of-the-art evolution models of B-type stars that may cause a lack of excited modes is the absence of microscopic diffusion in most such models. Aims. We investigate whether the inclusion of microscopic diffusion in stellar models of B-type stars, notably radiative levitation experienced by isotopes, leads to extra mode driving by the opacity mechanism compared to the case of models that do not include microscopic diffusion. Methods. We consider the case of slowly to moderately rotating stars and use non-rotating equilibrium models, while we account for (uniform) rotation in the computations of the pulsation frequencies. We calculate 1D stellar structure and evolution models with and without microscopic diffusion and examine the effect of radiative levitation on mode excitation, for both low-radial order pressure and gravity modes and for high-radial order gravity modes. As is common practice in asteroseismology, rotation is included in the pulsation computations according to the mode’s frequency regime. For modes having frequencies below twice the rotation frequency, that is, modes in the sub-inertial regime, we adopt the traditional approximation of rotation. For modes in the super-inertial regime with frequency above twice the rotation frequency, rotation is treated perturbatively up to first order in the rotation. We consider macroscopic envelope mixing induced by internal gravity waves to compute the modes and study its effect on the surface abundances. Results. We find systematically more modes to be excited for the stellar models including microscopic diffusion compared to those without it, in agreement with observational findings of pulsating B-type dwarfs. Furthermore, the models with microscopic diffusion predict that excited modes occur earlier on in the evolution compared to modes without it. In order to maintain realistic surface abundances during the main sequence, we include macroscopic envelope mixing by internal gravity waves. Along with microscopic diffusion, such macroscopic envelope mixing ensures both more excited modes and surface abundances consistent with spectroscopic studies of B-type stars. Conclusions. While radiative levitation has so far largely been neglected in stellar evolution computations of B-type stars for computational convenience, it impacts mode excitation predictions for stellar models of such stars. We conclude that the process of radiative levitation is able to reduce the discrepancy between predicted and observed excited pulsation modes in B-type stars.

Non-linear three-mode coupling of gravity modes in rotating slowly pulsating B stars

Context. Slowly pulsating B (SPB) stars display multi-periodic variability in the gravito-inertial mode regime with indications of non-linear resonances between modes. Several have undergone asteroseismic modeling in the past few years to infer their internal properties, but only in a linear setting. These stars rotate fast, so that rotation is typically included in the modeling by means of the traditional approximation of rotation (TAR). Aims. We aim to extend the set of tools available for asteroseismology, by describing time-independent (stationary) resonant non-linear coupling among three gravito-inertial modes within the TAR. Such coupling offers the opportunity to use mode amplitude ratios in the asteroseismic modeling process, instead of only relying on frequencies of linear eigenmodes, as has been done so far. Methods. Following observational detections, we derive expressions for the resonant stationary non-linear coupling between three gravito-inertial modes in rotating stars. We assess selection rules and stability domains for stationary solutions. We also predict non-linear frequencies and amplitude ratio observables that can be compared with their observed counterparts. Results. The non-linear frequency shifts of stationary couplings are negligible compared to typical frequency errors derived from observations. The theoretically predicted amplitude ratios of combination frequencies match with some of their observational counterparts in the SPB targets. Other, unexplained observed ratios could be linked to other saturation mechanisms, to interactions between different modes, or to different opacity gradients in the driving zone. Conclusions. For the purpose of asteroseismic modeling, our non-linear mode coupling formalism can explain some of the stationary amplitude ratios of observed resonant mode couplings in single SPB stars monitored during 4 years by the Kepler space telescope.

Local reduced subspaces of subchannel‐inspired subdomains

AbstractIn this article, we study an update of the traditional subchannel approximation utilizing local reduced order bases. Through employing the symmetries and periodicity of a 7‐pin bundle, the global domain is decomposed into numerous repeating subdomains following several dividing strategies. We locally study the reduced basis generated by proper orthogonal decomposition. We analyze the similarities, assessing the truncation error and the distance between the linear subspaces spanned by the reduced bases. We focus on the first stage of building a reduced order model, the generation of the reduced subspace, which is usually not regarded in detail in our application problem. Our assessment related to flow blockage in liquid metal‐cooled nuclear reactors, a postulated high‐risk accident that results in potential fuel damage.

Distributionally Robust Observable Strategic Queues

This paper presents an extension of Naor’s analysis on the join-or-balk problem in observable M/M/1 queues. Although all other Markovian assumptions still hold, we explore this problem assuming uncertain arrival rates under the distributionally robust settings. We first study the problem with the classical moment ambiguity set, where the support, mean, and mean-absolute deviation of the underlying distribution are known. Next, we extend the model to the data-driven setting, where decision makers only have access to a finite set of samples. We develop three optimal joining threshold strategies from the perspectives of an individual customer, a social optimizer, and a revenue maximizer such that their respective worst-case expected benefit rates are maximized. Finally, we compare our findings with Naor’s original results and the traditional sample average approximation scheme. Funding: This research was supported by the National Science Foundation [Grants 2342505 and 2343869].

Generalized multilevel B-spline approximation for scattered data interpolation in image processing

This paper proposes a Generalized Multilevel B-spline Approximation (GMBA) method, which addresses scattered data interpolation problems in image processing. Mathematically, the GMBA provides a better solution for the B-spline control lattice by superimposing identical level B-splines compared with traditional Multilevel B-spline Approximation (MBA). Specifically, the GMBA allows the spacing of next control lattice to be subdivided arbitrarily or remained unchanged, which is determined by a predefined spacing set or the current error level. These improvements bring higher approximation accuracy and more flexibility for algorithm design to avoid over-fitting. In this paper, basic GMBA algorithm and its refined algorithm are compiled for image processing. Finally, six relevant cases are involved to test the GMBA, including surface approximation, image enlargement, image completion, and Salt-and-Pepper (SAP) noise removal. The experimental results show that the GMBA has better performance than the MBA in surface approximation and image processing, performs comparatively fast with the best performance on more than half of the standard test images compared with traditional algorithms, and has partially better performance even than deep learning algorithms. The GMBA can effectively recover meaningful details in images contaminated with even extremely high SAP noise level (up to 99%).

Approximating a Function with a Jump Discontinuity—The High-Noise Case

This paper presents a novel deep-learning network designed to detect intervals of jump discontinuities in single-variable piecewise smooth functions from their noisy samples. Enhancing the accuracy of jump discontinuity estimations can be used to find a more precise overall approximation of the function, as traditional approximation methods often produce significant errors near discontinuities. Detecting intervals of discontinuities is relatively straightforward when working with exact function data, as finite differences in the data can serve as indicators of smoothness. However, these smoothness indicators become unreliable when dealing with highly noisy data. In this paper, we propose a deep-learning network to pinpoint the location of a jump discontinuity even in the presence of substantial noise.

A dynamic approach to three-dimensional radiative transfer in subkilometer-scale numerical weather prediction models: the dynamic TenStream solver v1.0

Abstract. The increasing resolution of numerical weather prediction models makes inter-column three-dimensional (3D) radiative transport more and more important. However, 3D radiative-transfer solvers are still computationally expensive, largely preventing their use in operational weather forecasting. To address this issue, Jakub and Mayer (2015) developed the TenStream solver. It extends the well-established two-stream method to three dimensions by using 10 instead of 2 streams to describe the transport of radiative energy through Earth's atmosphere. Building upon this method, this paper presents the dynamic TenStream solver, which provides a further acceleration of the original TenStream model. Compared to traditional solvers, this speedup is achieved by utilizing two main concepts. First, radiation is not calculated from scratch every time the model is called. Instead, a time-stepping scheme is introduced to update the radiation field, based on the result from the previous radiation time step. Secondly, the model is based on incomplete solves, with just the first few steps of an iterative scheme towards convergence performed every time it is called. Essentially, the model thereby just uses the ingoing fluxes of a grid box to update its outgoing fluxes. Combined, these two approaches move radiative transfer much closer to the way advection is handled in the dynamical core of a numerical weather prediction (NWP) model, as both use previously calculated results to update their variables and thereby just require access to the neighboring values of an individual grid box, facilitating model parallelization. To demonstrate the feasibility of this new solver, we apply it to a precomputed shallow-cumulus-cloud time series and test its performance in terms of both speed and accuracy. In terms of speed, the dynamic TenStream solver is shown to be about 3 times slower than a traditional 1D δ-Eddington approximation but noticeably faster than currently available 3D solvers. To evaluate the accuracy of the dynamic TenStream solver, we compare its results as well as calculations carried out using a 1D δ-Eddington approximation and the original TenStream solver, to benchmark calculations performed with the 3D Monte Carlo solver MYSTIC. We demonstrate that at the grid box level, dynamic TenStream is able to calculate heating rates and net irradiances at domain boundaries that are very close to those obtained by the original TenStream solver, thus offering a much better representation of the MYSTIC benchmark than the 1D δ-Eddington results. By calling the dynamic TenStream solver less frequently than the δ-Eddington approximation, we furthermore show that our new solver produces significantly better results than a 1D δ-Eddington approximation carried out with a similar computational demand. At these lower calling frequencies, however, the incomplete solves in the dynamic TenStream solver also lead to a buildup of bias with time, which becomes larger the lower the calling frequency is.

Polytropic spherical astrocosmic fluid stability in the EiBI gravity with strong collective correlative effects

We herein present a dynamical analysis of the Jeans (gravitational) instability in a viscoelastic polytropic astrocloud within the Eddington-inspired Born-Infeld (EiBI) gravity framework in spherical geometry. The zeroth-order material density curvature corrections to the effective self-gravitational potential are properly incorporated. The applied spherical normal mode analysis yields a unique form of monic cubic dispersion relation with multiparametric coefficients dependent on the diverse equilibrium astrofluidic conditions without invoking any kind of traditional quasi-classic approximation. It is seen that the positive EiBI gravity parameter (χ>0) acts as a stabilizing agent and a negative EiBI gravity parameter (χ<0) acts as a destabilizing agent against the inward non-local self-gravity action. Additionally, we find that larger astroclouds are more prone to gravitational collapse, irrespective of the χ-polarity, and vice-versa. It is further investigated that the cloud temperature plays a stabilizing role at larger wavenumbers and destabilizing at smaller wavenumbers for χ>0. However, for χ<0, the fluid temperature consistently exhibits a destabilizing effect. The viscoelastic relaxation time stabilizes the astrofluid against the self-gravity irrespective of the χ-polarity. The fluid density acts as a stabilizer for χ>0, destabilizer for χ<0, and so forth. This study could be potentially applicable in the EiBI-modified gravitational theory in exploring gravity-driven large-scale astrocosmic phenomena towards structure formation mechanism via the canonical Jeans condensation in diverse astrocosmic conditions.

Direct detection Φ-OTDR based on UWFBG array using linear-phase-modulated double-pulse

We propose what we believe to be a novel direct detection phase-sensitive optical time-domain reflectometry (Φ-OTDR) based on ultra-weak fiber Bragg grating (UWFBG) array to achieve distributed vibration measurements with exceptional sensitivity and remarkable stability. Our system employs a pulse modulator to generate a double pulse and achieves linear phase modulation of one pulse by one cycle through a phase modulator. The phase change can be quantitatively demodulated using our proposed N-step phase-shifted demodulation algorithm. This method effectively mitigates the influence of phase noise of the laser and the pulse modulator, while also eliminating fluctuations in the half-voltage of the phase modulator. Compared with the existing phase modulation methods, our method avoids stringent requirements for the stability and precision of phase modulation. Moreover, we propose a phase-shifted approximation method, breaking the limitation of sensing length on the traditional differential approximation method and improving the accuracy significantly. The technique's effectiveness is experimentally demonstrated on a 1 km UWFBG array with a reflectivity of −40 dB to −45 dB and a spatial resolution of 10 m. Vibrations with different amplitudes are measured quantitatively with good linearity. The low-frequency self-noise is greatly suppressed and the overall self-noise is −54.3 dB rad2/Hz.

An extended finite element method for the Nernst-Planck-Poisson equations

The Nernst-Planck-Poisson (NPP) system of equations is used to study the ion diffusion mechanism in materials, commonly used in storage batteries and solid oxide fuel cells. In particular, NPP equations are used to predict the concentration of charged defects and electric potential. Depending on the size of the computational domain, the distributions exhibit steep gradients near the boundaries. The traditional finite element method, when employed, requires extremely refined mesh to capture the steep gradient as they employ simple polynomials. To alleviate the mesh dependence, in this work, we propose to augment the traditional finite element approximation space with a suitable ansatz to capture the steep gradient within the framework of the extended finite element method. The robustness and the accuracy of the proposed framework is demonstrated by comparing it with an overkill finite element solution.

A (2+1) -dimensional evolution model of Rossby waves and its resonance Y-type soliton and interaction solutions

In this paper, we derive a Kadomtsev-Petviashvili equation by using the multi-scale expansion and perturbation method, which is a model from the potential vorticity equation in the traditional approximation and describes the Rossby wave propagation properties. The N-soliton solutions, resonance Y-type soliton solutions, resonance X-type soliton solutions and interaction solutions of the equation are obtained with the help of dependent-variable transformation. In addition, the composite graphs are given to view the resonance phenomenon of Rossby waves. The results better enrich the research of Rossby waves in ocean dynamics and atmospheric dynamics.

Approximate and Memorize (A&M) : Settling opposing views in replay-based continuous unsupervised domain adaptation

Continuous Unsupervised Domain Adaptation (CUDA) can alleviate deep learning models’ performance degradation on out-of-distribution data. However, low stability, the erosion of past knowledge while adapting to new domains, remains a major challenge. Traditional proposed solutions such as (1) Approximation, or (2) Memorization fail to harmonize system’s computational load, memory resources and stability. In response, we introduce Approximate and Memorize (A&M). Compared to traditional approximation methods, A&M utilizes factorized generative models to mitigate modal collapse, offering improved computational efficiency and training stability. Moreover, compared to traditional memorization approaches, A&M boosts memory efficiency by learning compressed representations of past information rather than storing it in the raw format. A&M shows 50% less forgetting for problems with constrained memory and an extended number of domains, making A&M ideal for real-world systems due to its scalable and parallel design.

Making quantum chemistry compressive and expressive: Toward practical ab‐initio simulation

AbstractAb‐initio quantum chemistry simulations are essential for understanding electronic structure of molecules and materials in almost all areas of chemistry. A broad variety of electronic structure theories and implementations has been developed in the past decades to hopefully solve the many‐body Schrödinger equation in an approximate manner on modern computers. In this review, we present recent progress in advancing low‐rank electronic structure methodologies that rely on the wavefunction sparsity and compressibility to select the important subset of electronic configurations for both weakly and strongly correlated molecules. Representative chemistry applications that require the many‐body treatment beyond traditional density functional approximations are discussed. The low‐rank electronic structure theories have further prompted us to highlight compressive and expressive principles that are useful to catalyze idea of quantum learning models. The intersection of the low‐rank correlated feature design and the modern deep neural network learning provides new feasibilities to predict chemically accurate correlation energies of unknown molecules that are not represented in the training dataset. The results by others and us are discussed to reveal that the electronic feature sets from an extremely low‐rank correlation representation, which is very poor for explicit energy computation, are however sufficiently expressive for capturing and transferring electron correlation patterns across distinct molecular compositions, bond types and geometries.This article is categorized under: Electronic Structure Theory &gt; Ab Initio Electronic Structure Methods Software &gt; Quantum Chemistry Software &gt; Simulation Methods

A Generalized Slab Model

Abstract We construct a generalized slab model to calculate the ocean’s linear response to an arbitrary, depth-variable forcing stress profile. To introduce a first-order improvement to the linear stress profile of the traditional slab model, a nonlinear stress profile, which allows momentum to penetrate into the transition layer (TL), is used [denoted mixed layer/transition layer (MLTL) stress profile]. The MLTL stress profile induces a twofold reduction in power input to inertial motions relative to the traditional slab approximation. The primary reduction arises as the TL allows momentum to be deposited over a greater depth range, reducing surface currents. The secondary reduction results from the production of turbulent kinetic energy (TKE) beneath the mixed layer (ML) related to interactions between shear stress and velocity shear. Direct comparison between observations in the Iceland Basin, the traditional slab model, the generalized slab model with the MLTL stress profile, and the Price–Weller–Pinkel (PWP) model suggest that the generalized slab model offers improved performance over a traditional slab model. In the Iceland Basin, modeled TKE production in the TL is consistent with observations of turbulent dissipation. Extension to global results via analysis of Argo profiling float data suggests that on the global, annual mean, ∼30% of the total power input to near-inertial motions is allocated to TKE production. We apply this result to the latest global, annual-mean estimates for near-inertial power input (0.27 TW) to estimate that 0.08 ± 0.01 TW of the total near-inertial power input are diverted to TKE production.