Closed-form Solution Research Articles

Preliminary Implications of Viscoelastic Ray Theory for Anelastic Seismic Tomography Models

ABSTRACT The recent developments in general viscoelastic ray theory provide a rigorous mathematical framework for anelastic seismic tomography. They provide closed-form solutions of forward ray-tracing and simple inverse problems for anelastic horizontal and spherical layered media with material gradients. They provide ray-tracing computation algorithms valid for all angles of incidence that account for changes in wave speed, attenuation, and trajectory of anelastic P and S body waves induced by anelastic boundaries. They account for theoretical predictions that seismic waves refract as inhomogeneous waves across anelastic boundaries for all angles of incidence, which in turn accounts for energy carried by plane waves along seismic boundaries at head wave critical angles and wide-angle refracted (WAR) ray paths that are not predicted by elastic models. Exact viscoelastic ray-tracing numerical results for various models provide examples that illustrate the effects of anelastic boundaries on the travel times and amplitudes of seismic waves. They show the effects are strongly dependent on angle of incidence. For near-critical and wide angles of incidence the anelastic effects on travel times and amplitudes can be large and are not explained by elastic ray theory, but the effects on travel times can be relatively small and difficult to distinguish from those for elastic media for pre-near-critical angles of incidence. The results for some models indicate that reflected anelastic WAR waves may be observable at the surface and possibly account for some prominent seismic arrivals not explained by elasticity. These preliminary results suggest that the application of exact viscoelastic ray-tracing computation algorithms to exploration and teleseismic data sets can reveal new insights regarding the properties and distribution of anelastic materials in the Earth.

Cross-modal de-deviation for enhancing few-shot classification

Few-shot learning poses a critical challenge due to the deviation problem caused by the scarcity of available samples. In this work, we aim to address deviations in both feature representations and prototypes. To achieve this, we propose a cross-modal de-deviation framework that leverages class semantic information to provide robust prior knowledge for the samples. This framework begins with a visual-to-semantic autoencoder trained on the labeled samples to predict semantic features for the unlabeled samples. Then, we devise a binary linear programming model to match the initial prototypes with the cluster centers of the unlabeled samples. To circumvent potential mismatches between the cluster centers and the initial prototypes, we perform the label assignment process in the semantic space by transforming the cluster centers into semantic representations and utilizing the class ground truth semantic features as reference points. Moreover, we model a linear classifier with the concatenation of the refined prototypes and the class ground truth semantic features serving as the initial weights. Then we propose a novel optimization strategy based on the alternating least squares (ALS) model. From the ALS model, we can derive two closed-form solutions regarding to the features and weights, facilitating alternative optimization of them. Extensive experiments conducted on few-shot learning benchmarks demonstrate the competitive advantages of our CMDD method over the state-of-the-art approaches, confirming its effectiveness in reducing deviation. The code is available at: https://github.com/pmhDL/CMDD.git.

Real option pricing under the regime-switching model with jumps on a finite time horizon

We consider an irreversible investment decision problem on a finite time horizon where an instantaneous cash flow process of a firm follows a regime-switching jump-diffusion (RSJD) model. The value of a project can be derived from a partial integro-differential equation (PIDE) and then we obtain a closed-form solution of the PIDE. It is proved that the value of the project converges to the solution on an infinite time horizon problem as the lifetime of the project tends to infinity. Moreover, the value function of a real option with a finite expiration date can be evaluated by solving a linear complementarity problem (LCP) under the RSJD model. We apply a 2-step backward differentiation formula (BDF2 method) combined with an operator splitting method to solve the LCP and then it has the second-order convergence rate in the discrete ℓ2-norm with respect to the time and spatial variables. The stability of the proposed method to solve the LCP is also proved in the discrete ℓ2-norm. Finally, a variety of numerical experiments are carried out to determine the optimal investment time and the numerical results on the finite time horizon are analyzed.

Numerical modelling of reinforced fill over a void considering rate-dependent stiffness of the reinforcement

The problem of a reinforced fill over a void has been the subject of much research in the geosynthetics literature. Previous studies have mainly focused on finding closed-form solutions to predict the tensile loads and strains in the reinforcement layer once a void develops below the fill. In this paper, a 2D finite difference (FLAC) model that implements the hyperbolic isochronous load-strain model for the reinforcement by Bathurst and Naftchali (2021) is used to investigate the influence of the rate-dependent properties of polymeric geosynthetic reinforcement materials on reinforcement tensile strains and load, and overall system performance including vertical deformation at the reinforcement elevation and at the fill surface. The paper also investigates the influence of fill soil properties and constitutive model type, foundation condition, void geometry and fill height on system performance. The results of numerical modelling are compared to predictions made using the closed-form solution of Giroud et al. (1990) and in the BSI 8006-1 (2010) design code. The results of numerical modelling demonstrate that the choice of fill height to void width and the stiffness of the rate-dependent geosynthetic reinforcement layer are important to ensure that the maximum reinforcement strain, allowable strength and fill surface settlement criteria are not exceeded.

Orthogonal subspace exploration for matrix completion

Matrix completion, aiming at restoring a low-rank matrix from observed entries, indicates the connection with the subspace clustering due to the low-rank property. However, it is expensive to incorporate subspace learning into the pervasive surrogate of matrix completion, the nuclear norm. In this paper, we design an orthogonal subspace exploration model for matrix completion, which can be easily integrated due to the succinct formulation. Then, we propose a non-convex surrogate with tractable solutions for low-rank matrix completion, so that the subspace exploration can be performed simultaneously. Compared with the existing surrogates (e.g., nuclear norm, Schatten-p norm, max norm, etc.), the proposed surrogate is differential such that the optimization is still simple even after the subspace exploration is incorporated. Although the surrogate is non-convex, a parameter-free algorithm that is proved to converge into the global optimum is developed. The optimization consists of closed-form solutions so that the orthogonal subspace exploration will not distinctly bring additional costs and the algorithm empirically converges within dozens of iterations. Experiments illustrate the efficiency and superiority of our model.

An extreme worst-case risk measure by expectile

Abstract Expectiles have received increasing attention as a risk measure in risk management because of their coherency and elicitability at the level $\alpha\geq1/2$ . With a view to practical risk assessments, this paper delves into the worst-case expectile, where only partial information on the underlying distribution is available and there is no closed-form representation. We explore the asymptotic behavior of the worst-case expectile on two specified ambiguity sets: one is through the Wasserstein distance from a reference distribution and transforms this problem into a convex optimization problem via the well-known Kusuoka representation, and the other is induced by higher moment constraints. We obtain precise results in some special cases; nevertheless, there are no unified closed-form solutions. We aim to fully characterize the extreme behaviors; that is, we pursue an approximate solution as the level $\alpha $ tends to 1, which is aesthetically pleasing. As an application of our technique, we investigate the ambiguity set induced by higher moment conditions. Finally, we compare our worst-case expectile approach with a more conservative method based on stochastic order, which is referred to as ‘model aggregation’.

Influence of Chemical and Radiation on an Unsteady MHD Oscillatory Flow using Artificial Neural Network (ANN)

This paper delves into the intricate interplay between chemical and thermal radiation in the context of an unstable magnetohydrodynamic(MHD) oscillatory flow through a porous medium. The fluid under investigation is presumed to be incompressible, electrically conductive, and radiating with the additional influence of a homogeneous magnetic field applied perpendicular to the channel’s plane. Analytical closedform solutions are derived for the momentum, energy, and concentration equations providing a comprehensive understanding of the system’s behavior. The investigation systematically explores the impact of various flow factors, presenting their effects through graphical representations. The governing partial differential equations (PDE) of the boundary layer are transformed into a set of coupled nonlinear ordinary differential equations (ODE) using a closed-form method. Subsequently, an artificial neural network (ANN) is applied to these ODEs, and the obtained results are validated against numerical simulations. The temperature profiles exhibit oscillatory behavior with changes in the radiation parameter (N), revealing insights into the system’s dynamic response. Furthermore, the paper uncovers that higher heat sources lead to increased temperature profiles. Additionally, concentration profiles demonstrate a decrease with escalating chemical reaction parameters, with a reversal observed as the Schmidt number (Sc) increases. This study highlights the efficacy of an ANN model in providing highly efficient estimates for heat transfer rates from an engineering standpoint. This innovative approach leverages the power of artificial intelligence to enhance our understanding of complex fluid magnetohydrodynamics and porous media flows.

Closed-form solutions of consistency ratio in best worst method minmax optimization model: max of edge error matrix and minmax edge error determinant methods

The Best Worst Method (BWM), a reduced version of the AHP, is a recent multi-criteria decision-making tool based on pairwise comparisons with reference to the best and worst criteria. Consistency Ratio (CR) measurement for the rating quality and prioritizations is still a controversial topic. Firstly, the computation for the current CR of BWM must rely on a software optimization solver to find the optimal values, and the solver may not always guarantee the exact optimal solutions, especially if the computational cost settings are not large enough for higher number of criteria. Secondly, much effort to evaluate optimization algorithms is needed to find the best solutions with the least computational resources due to diverse solvers possibly leading to different results with different performances. Thirdly, optimization programming code is not trivial to be implemented for general BWM users. To address these issues, this paper presents the closed-form solutions, Max of Edge Error Matrix (MEEM) (Eq. (44) of Theorem 4) and Minmax Edge Error Determinant (MEED) (Algorithm 1), to replace the BWM optimization models to directly calculate the CR values. Two simulations have been performed with a basic laptop using a single process. One simulation of twenty thousand random pairs of vectors took 26.34 h to perform to verify that the approximate results are higher than or very close to the exact closed-form values of both methods when high computational cost is allocated for the solver to increase the precision. Another simulation of one million random pairs of vectors only took 1.27 h to perform to verify that the MEED and MEEM methods always produce the same results for the number of criteria up to nine. The computational time for the exact results is dramatically reduced when the solver is not needed. The advantages of the proposed solutions include the following: the software to solve the optimization model to obtain CR is unnecessary, and the proposed calculation is extremely efficient to obtain the exact accuracy. The two-step optimization model can preserve the fixed Minmax Edge Error to find the weights which add up to one, which is the condition to determine if the model reaches exact optimal solutions. As the CR optimization model produces multiple versions of weights, which are recommended not to be used, the new method does not need to compute the unnecessary weight values to get the Minmax Edge Error. With the provision of equations leading to closed forms, users can understand the properties of CR in much clearer perspectives. Due to the computational efficiency and explainability, the proposed closed forms can replace the CR optimization model to compute CR efficiently and accurately for all diverse applications using BWM.

Temperature change-induced linear and nonlinear axial responses of internal replacement pipe (IRP) systems for pipeline rehabilitation incorporating the effects of soil friction

The internal replacement pipe (IRP) is a developing trenchless system utilised for restoring buried steel and cast-iron legacy pipelines. It is crucial to ensure that this advanced system is appropriately designed to reinstate the functionality of damaged pipelines effectively and safely. The present paper investigates the structural response of IRP systems used in repairing pipelines with circumferential discontinuities subjected to seasonal temperature changes. Analytical and numerical approaches verified via experimental data and available closed-form solutions were implemented to analyse a total of 180 linear and nonlinear finite element (FE) simulations. A set of analytical expressions was developed to describe the loading and induced responses of the system. Based on an extensive FE parametric study, five modification factors were derived and applied to developed analytical expressions to characterise the structural response incorporating the effects of soil friction. Results showed that there is a major difference between the results of linear and nonlinear analyses highlighting the importance of including the material nonlinearities in the FE analysis. A significant difference was observed between the discontinuity openings with and without the consideration of soil friction implying that appropriate inclusion of soil friction in the FE model is crucial to get realistic system responses subjected to temperature change. Although the application of IRP holds immense promise as a trenchless solution for rehabilitating legacy pipelines, the lack of established design procedures and standards for these technologies has restricted their application in gas pipelines. Results obtained from numerical and analytical models developed in the present research will provide valuable insights for the design and development of safe and efficient IRP systems urgently needed in the pipeline industry.

Investigating peristaltic flow of Newtonian fluid in a permeable channel: Effects of nonlinear curvature and wall properties

This investigation delves into the peristaltic flow characteristics of a Newtonian fluid within a permeable channel, emphasizing the influence of nonlinear curvature and wall properties. The study employs wave trains of varying amplitudes and phases to induce an asymmetric configuration along the channel boundaries. The curvature effects are contingent upon the ratio of channel width to wavelength, and the accuracy of results is maintained up to the second order in δ. To facilitate the derivation of closed-form solutions at higher levels, a domain transformation is executed, converting a channel with a variable cross section into one with a uniform cross section. To impose constraints on parameters and achieve a singular flux in the presence of a specified pressure gradient, a distinctiveness criterion is formulated. This study investigates the interplay of inertia and curvature on pumping and trapping, examining both symmetric and asymmetric channels. The pressure difference escalates with wave number in both channel configurations. Moreover, in the absence of inertial forces and flux, the impact of curvature on the pressure difference demonstrates a direct proportionality. Conversely, an augmentation in the permeability of the porous channel correlates with a reduction in the pressure difference. An intriguing observation arises as phase differences increase: the visibility of the bolus diminishes, and the streamlines become more prominent. This establishes a favorable environment for fluid flow, enhancing our understanding of the intricate dynamics at play in peristaltic flow within permeable channels.

Energy Efficient and Differentially Private Federated Learning via a Piggyback Approach

This paper aims to develop a differential private federated learning (FL) scheme with the least artificial noises added while minimizing the energy consumption of participating mobile devices. By observing that some communication efficient FL approaches and even the nature of wireless communications contribute to the differential privacy (DP) preservation of training data on mobile devices, in this paper, we propose to jointly leverage gradient compression techniques (i.e., gradient quantization and sparsification) and additive white Gaussian noises (AWGN) in wireless channels to develop a piggyback DP approach for FL over mobile devices. Even with the piggyback DP approach, information distortion caused by gradient compression and noise perturbation may slow down FL convergence, which in turn consumes more energy of mobile devices for local computing and model update communications. Thus, we theoretically analyze FL convergence and formulate an energy efficient FL optimization under piggyback DP, transmission power, and FL convergence constraints. Furthermore, we propose an efficient iterative algorithm where closed-form solutions for artificial DP noise and power control are derived. Extensive simulation and experimental results demonstrate the effectiveness of the proposed scheme in terms of energy efficiency and privacy preservation.

Thermal tides in neutrally stratified atmospheres: Revisiting the Earth’s Precambrian rotational equilibrium

Rotational dynamics of the Earth, over geological timescales, have profoundly affected local and global climatic evolution, probably contributing to the evolution of life. To better retrieve the Earth’s rotational history, and motivated by the published hypothesis of a stabilized length of day during the Precambrian, we examined the effect of thermal tides on the evolution of planetary rotational motion. The hypothesized scenario is contingent upon encountering a resonance in atmospheric Lamb waves, whereby an amplified thermotidal torque cancels out the opposing torque generated by the oceans and solid interior, driving the Earth into rotational equilibrium. With this scenario in mind, we constructed an ab initio model of thermal tides on rocky planets describing a neutrally stratified atmosphere. The model takes into account dissipative processes with Newtonian cooling and diffusive processes in the planetary boundary layer. We retrieved, from this model, a closed-form solution for the frequency-dependent tidal torque, which captures the main spectral features previously computed using 3D general circulation models. In particular, under longwave heating, diffusive processes near the surface and the delayed thermal response of the ground prove to be responsible for attenuating, and possibly annihilating, the accelerating effect of the thermotidal torque at the resonance. When applied to the Earth, our model prediction suggests the occurrence of the Lamb resonance in the Phanerozoic, but with an amplitude that is insufficient for the rotational equilibrium. Interestingly, though our study was motivated by the Earth’s history, the generic tidal solution can be straightforwardly and efficiently applied in exoplanetary settings.

Closed-Form Solutions of Miss Distance for Higher-Order Guidance System

Closed-Form Solutions of Miss Distance for Higher-Order Guidance System

An elliptical compressible liquid inclusion in an infinite nonlinearly coupled thermoelectric matrix

We study the plane strain problem associated with an elliptical thermoelectrically insulated and compressible liquid inclusion embedded in an infinite homogeneous and isotropic nonlinearly coupled thermoelectric matrix under uniform remote electric current density and energy flux. With the aid of conformal mapping and analytic continuation, an exact closed-form solution to the problem is derived. In particular, an elementary expression for the internal uniform hydrostatic tension within the compressible liquid inclusion is obtained. Our result shows that the internal uniform hydrostatic tension is unaffected by the remote energy flux and is proportional to the square of the two components of the remote electric current density vector. Also presented is the hoop stress distribution along the elliptical interface on the matrix side.

Two Different Analytical Approaches for Solving the Pantograph Delay Equation with Variable Coefficient of Exponential Order

The pantograph equation is a basic model in the field of delay differential equations. This paper deals with an extended version of the pantograph delay equation by incorporating a variable coefficient of exponential order. At specific values of the involved parameters, the exact solution is obtained by applying the regular Maclaurin series expansion (MSE). A second approach is also applied on the current model based on a hybrid method combining the Laplace transform (LT) and the Adomian decomposition method (ADM) denoted as (LTADM). Although the MSE derives the exact solution in a straightforward manner, the LTADM determines the solution in a closed series form which is theoretically proved for convergence. Further, the accuracy of such a closed-form solution is examined through various comparisons with the exact solution. For validation, the residual errors are calculated and displayed in graphs. The results show that the solution obtained utilizing the LTADM is in full agreement with the exact solution using only a few terms of the closed-form series solution. Moreover, it is found that the residual errors tend to zero, which reflects the effectiveness of the LTADM. The present approach may merit further extension by including other types of linear delay differential equations with variable coefficients.

Transformer-Based Parameter Estimation in Statistics

Parameter estimation is one of the most important tasks in statistics, and is key to helping people understand the distribution behind a sample of observations. Traditionally, parameter estimation is done either by closed-form solutions (e.g., maximum likelihood estimation for Gaussian distribution) or by iterative numerical methods such as the Newton–Raphson method when a closed-form solution does not exist (e.g., for Beta distribution). In this paper, we propose a transformer-based approach to parameter estimation. Compared with existing solutions, our approach does not require a closed-form solution or any mathematical derivations. It does not even require knowing the probability density function, which is needed by numerical methods. After the transformer model is trained, only a single inference is needed to estimate the parameters of the underlying distribution based on a sample of observations. In the empirical study, we compared our approach with maximum likelihood estimation on commonly used distributions such as normal distribution, exponential distribution and beta distribution. It is shown that our approach achieves similar or better accuracy as measured by mean-square-errors.

Deconstructable variable stiffness shear connectors in FRP deck resting on multigirder bridge system: Analytical model

Deconstructable structures offer great flexibility and reduce material waste. Fiber Reinforced-Polymer (FRP) decks resting on steel girders combine high durability, lightweight, fast construction, and deconstructablility. Composite action between the decks and girders is achieved using shear connectors. Until now, most shear connectors use adhesives or grouts, limiting their maintenance, reuse, or recycling. This paper presents a closed-form solution to analyze the behavior of an FRP deck connected to multi-girders using variable sequential stiffness shear connectors considering partial Degree of Composite Action (DCA). The analytical model, which is verified using Finite Element model and prior experimental studies, is further used to study stress and deflection for various multigirder configurations. Finally, the DCA considering variable stiffness shear connectors is presented and linked to effective width ratios, paving the way for adopting variable stiffness in bridge standards.

An Objective Function-Based Clustering Algorithm with a Closed-Form Solution and Application to Reference Interval Estimation in Laboratory Medicine

Clustering algorithms are usually iterative procedures. In particular, when the clustering algorithm aims to optimise an objective function like in k-means clustering or Gaussian mixture models, iterative heuristics are required due to the high non-linearity of the objective function. This implies higher computational costs and the risk of finding only a local optimum and not the global optimum of the objective function. In this paper, we demonstrate that in the case of one-dimensional clustering with one main and one noise cluster, one can formulate an objective function, which permits a closed-form solution with no need for an iteration scheme and the guarantee of finding the global optimum. We demonstrate how such an algorithm can be applied in the context of laboratory medicine as a method to estimate reference intervals that represent the range of “normal” values.

Optical second harmonic generation in anisotropic multilayers with complete multireflection of linear and nonlinear waves using ♯SHAARP.ml package

Optical second harmonic generation (SHG) is a nonlinear optical effect widely used for nonlinear optical microscopy and laser frequency conversion. Closed-form analytical solution of the nonlinear optical responses is essential for evaluating materials whose optical properties are unknown a priori. A recent open-source code, ♯SHAARP.si, can provide such closed form solutions for crystals with arbitrary symmetries, orientations, and anisotropic properties at a single interface. However, optical components are often in the form of slabs, thin films on substrates, and multilayer heterostructures with multiple reflections of both the fundamental and up to ten different SHG waves at each interface, adding significant complexity. Many approximations have therefore been employed in the existing analytical approaches, such as slowly varying approximation, weak reflection of the nonlinear polarization, transparent medium, high crystallographic symmetry, Kleinman symmetry, easy crystal orientation along a high-symmetry direction, phase matching conditions and negligible interference among nonlinear waves, which may lead to large errors in the reported material properties. To avoid these approximations, we have developed an open-source package named Second Harmonic Analysis of Anisotropic Rotational Polarimetry in Multilayers (♯SHAARP.ml). The reliability and accuracy are established by experimentally benchmarking with both the SHG polarimetry and Maker fringes using standard and commonly used nonlinear optical materials as well as twisted 2-dimensional heterostructures.

Solution of Laplace Equation by Modified Differential Transform Method

In this paper, we applied the modified two-dimensional differential transform method to solve Laplace equation. Laplace equation is one of Elliptic partial differential equations. These kinds of differential equations have specific applications models of physics and engineering. We consider four models with two Dirichlet and two Neumann boundary conditions. The simplicity of this method compared to other iteration methods is shown here. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.