Approximation Method Research Articles

EFFICIENT WRONG-WAY RISK MODELING FOR FUNDING VALUATION ADJUSTMENTS

Wrong-Way Risk (WWR) is an important component in Funding Valuation Adjustment (FVA) modeling. Yet, the standard assumption is independence between market risks and the counterparty defaults and funding costs. This typical industrial setting is our point of departure, where we aim to assess the impact of WWR without running a full Monte Carlo simulation with all credit and funding processes. We propose to split the exposure profile into two parts: an independent and a WWR-driven part. For the former, exposures can be re-used from the standard xVA calculation. We express the second part of the exposure profile in terms of the stochastic drivers and approximate these by a common Gaussian stochastic factor. Within the affine setting, the proposed approximation is generic, is an add-on to the existing xVA calculations and provides an efficient and robust way to include WWR in FVA modeling. Case studies for an interest rate swap and a representative multi-currency portfolio of swaps illustrate that the approximation method is applicable in a practical setting. We analyze the approximation error and use the approximation to compute WWR sensitivities, which are needed for risk management. The approach is equally applicable to other metrics such as Credit Valuation Adjustment.

Sequential Convex Programming for Nonlinear Optimal Control in UAV Trajectory Planning

Abstract: In this paper, an algorithm is proposed to solve the non-convex optimization problem using sequential convex programming. An approximation method was used to solve the collision avoidance constraint. An iterative approach was utilized to estimate the non-convex constraints, replacing them with their linear approximations. Through the simulation, we can see that this method allows for quadcopters to take off from a given initial position and fly to the desired final position within a specified flight time. It is guaranteed that the quadcopters will not collide with each other in different scenarios.

Neural Networks and the Nonlinear Feynman–Kac Theorem Applied to Financial Options Pricing

The classic financial market model proposed by Black and Scholes allows solving, under a series of simplifying assumptions, the problem of valuing financial derivatives. However, these assumptions deviate from the reality of global markets, particularly assuming perfect liquidity. In this work, a financial market model is proposed in which the illiquidity of the risky asset is stochastic, described by a mean-reversion process. The dynamics of the asset price in this context are established alongside the valuation PDE for derivatives. Additionally, as an approximation method to the solution of this PDE, the implementation of the extension of the Feynman–Kac representation theorem to the nonlinear case using deep neural networks is proposed.

A Comparison of Bayesian Approximation Methods for Analyzing Large Spatial Skewed Data

A Comparison of Bayesian Approximation Methods for Analyzing Large Spatial Skewed Data

E-PINN: extended physics informed neural network for the forward and inverse problems of high-order nonlinear integro-differential equations

Physics informed neural network (PINN) is a new deep learning paradigm, which embeds the physical information delineated by PDEs in the loss function and optimizes the weights in the neural network. Based on PINN, an extended PINN(E-PINN) is proposed, which is a mixture of the polynomial function approximation method and PINN's learning framework. A preprocess layer is added before the classical PINN, using Legendre polynomials as the polynomial basis function. Therefore, E-PINN not only has the excellent approximation ability of the polynomial basis function, but also inherits the learning framework of the neural network method. In numerical experiments, the proposed E-PINN algorithms have high accuracy in solving 1D, 2D high-order nonlinear Fredholm equations and equations system, including the forward and inverse problems.

Fast approximation of fiber reinforced injection molding processes using eikonal equations and machine learning

Injection molding is a popular production process for short fiber reinforced components. The mechanical properties of such components depend on process-induced fiber orientations which are commonly predicted via numerical simulations. However, high computational costs prevent process simulations from being used in iterative procedures, such as topology optimization or finding optimal injection locations. We propose a fast approximation method that extracts nodal features and train a regression model to predict fill states, cooling times, volumetric shrinkage, and fiber orientations. The features are determined by solving eikonal equations with a fast iterative method and computing spatial moments to characterize node-adjacent material distributions. Subsequently, we use these features to train feed forward neural networks and gradient boosted regression trees with simulation data of a large dataset of geometries. This approach is significantly faster than conventional methods, providing 20x speed-up for single simulations and more than 200x speed-up in gate location optimization. It generalizes to arbitrary geometries and injection locations.

Approximation Methods for a Class of Non-Lipschitz Mathematical Programs with Equilibrium Constraints

Approximation Methods for a Class of Non-Lipschitz Mathematical Programs with Equilibrium Constraints

On a Method for Calculating Bending and Shear Vibrations of a Porous Piezoelement in the Low-Frequency Region

Introduction. Devices for collecting and storing energy from the external environment are low-power sources of electric energy that are actively used. The autonomous devices for monitoring the damaged condition of various structures include them as well. The working element of these devices is a piezoelectric generator (PEG) — a converter of mechanical energy into electrical energy. The design of PEG is associated with the preliminary construction of their mathematical and computer models, with the help of which the calculation and optimization of structures is carried out. One of the ways to model and calculate PEG is to develop approximate calculation methods based on applied theories. The applied theories for calculating bending vibrations of multilayer piezoactive plates are known and previously developed in the literature. However, in the scientific literature there is not enough information about bending and shear vibrations as a tool for improving the efficiency of engineering calculations of the described structures. The objective of this work was to develop an applied method for calculating bending and shear vibrations of piezoceramic plates, including porous ones.Materials and Methods. Piezoceramics PZT-4, including porous ones, were used as the piezoactive material of the plate. When using porous ceramics, the rigidity of the structure decreased to a greater extent than the piezoelectric modules, which made it possible to obtain a more effective PEG under mechanical action. The mathematical formulation was carried out within the framework of the linear theory of electroelasticity with plate polarization in thickness. The sides of the plate were electrodated, the right side was fixed, and a smooth contact in the vertical wall was set on the left side. Steady-state vibrations of the plate were caused by pressure on the front surfaces of the plate or the difference in electrical potentials at the electrodes. To calculate the characteristics of PEG, the authors proposed an applied theory based on hypotheses about the distribution of characteristics of the stress-strain state and the electric field.Results. Transverse vibrations of a piezoceramic plate in the low-frequency region (below the first bending-shear resonance) were studied. Due to the fact that the mathematical formulation was considered within the framework of the linear theory of elasticity, the problem was divided into the sum of two. The first one took into account the mechanical effect: a distributed load and a transverse force at the left end acted on the front surfaces of the plate, and the potentials at the electrodes were zero. In the second task, there were no mechanical loads, but the potential difference was set at the electrodes. Based on hypotheses about the distribution of deformations, mechanical stresses and electric potential, both problems were reduced to a system of ordinary differential equations and boundary conditions. Comparison with the results of calculations by the finite element method in the ACELAN package showed the adequacy of the proposed applied theory in the low-frequency region.Discussion and Conclusion. Since the formulation of the problem was considered in the linear theory of electroelasticity, and the low-frequency region was studied, the work succeeded in dividing the problem of bending-shear vibrations of a porous piezoceramic plate into two: bending — with mechanical action at zero potentials, and shear — when setting the potential difference and zero mechanical action. The corresponding hypotheses about bending and shear were used. Two systems of ordinary differential equations and boundary conditions, which were solved analytically without the use of “heavy” finite element packages, were constructed. To compare the results and confirm the adequacy of the proposed method, the finite element modeling of such tasks was carried out in a specialized ACELAN package. The comparison showed that the error in determining displacements and electric potential when using this approach, in the case of setting mechanical loads and potential differences, did not exceed 6%. The method developed in the paper can be applied in the design of piezoelectric generators for energy storage in the low-frequency region.

Implementation of a Digital Model of Thermal Characteristics Based on the Temperature Field

Introduction. Computer modeling allows engineers to make valid design decisions by accurately assessing the thermal characteristics of design objects. The implementation of digital twin technology in the process of designing technical facilities is the current direction of scientific research and development. To do this, it is necessary to develop computer models whose accuracy meets the requirements for digital twins. However, the scientific literature does not widely present the results of research aimed at implementing digital twin technology in the design process. The general issues related to the use of digital twins in various industries are mainly considered. Therefore, the objective of this study was the development of a digital model and a comparative analysis of the accuracy of calculations of thermal characteristics of the design object.Materials and Methods. The main tool for conducting the research was the methodology proposed by the authors for developing a computer model of thermal characteristics for the implementation of digital twin technology. The numerical solution was implemented through constructing a thermal model for calculating the temperature field based on the finite element method in the ANSYS engineering analysis system from ANSYS, Inc. (USA). For the analytical solution, a computer model of thermal characteristics developed on the basis of the state-space method, implemented in the ANSYS Twin Builder module, was used. The state-space model was matched to the behavior of the original thermal model through approximating the transfer function to the stepwise response of the thermal load using the time domain vector approximation method. Verification of the constructed analytical model was carried out in the engineering calculation system MATLAB from the MathWorks company (USA). The research was carried out for a 400V machine model manufactured by NPO “Stankostroenie” LLC, Sterlitamak (Russia).Results. The developed digital model makes it possible to calculate the thermal characteristics of the design object with high accuracy. The results of the comparative analysis showed a high degree of correspondence between the values of thermal characteristics obtained using the proposed digital model and the results of numerical simulation. The maximum error in calculating thermal characteristics did not exceed 0.1ºC.Discussion and Conclusion. Computer modeling that combines numerical calculation methods and a scientific approach based on digital twin technology, provides obtaining the result as close as possible to the results of experiments. The digital model proposed in the study is an effective solution, since it provides performing calculations to evaluate thermal characteristics in real time, which is one of the most important requirements for the implementation of digital twin technology.

Позиционирование автономного необитаемого подводного аппарата с одновременной обработкой текущих и сохраненных измерений дальностей от менее чем трех гидроакустических маяков

The paper presents a recursive algorithm for determining the coordinates of an autonomous underwater vehicle (AUV) using time-different ranges to the acoustic beacons with log and heading indicator aiding with unknown a priori coordinates of the underwater vehicle. Any number of beacons can be used simultaneously, but at least at one moment the measurements from minimum three beacons are available. At this moment the initial linearization point is obtained using an approximate analytical method. At all the previous moments the measurements are saved for further processing. The algorithm uses two filters parallely processing the saved and current measurements. The solutions of the forward-time filter are updated by the backward-time filter data using the dummy measurements. The results of modeling and test this real data are presented, confirming the effectiveness of the developed algorithm.

Rare event approximation between subdistribution hazard ratio and cause-specific hazard ratio in survival analysis with competing risks.

BackgroundDespite the fact that competing risks are inevitable in epidemiological and clinical studies, distinctions between the hazard ratio estimated by handling competing risks as censoring and the subditribution hazard ratio are often overlooked.MethodsWe derive quantitative relationships between subdistribution hazard ratio and cause-specific hazard ratio, and derive an approximate calculation method to transform the two into each other. Numerical examinations of hypothetical six scenarios and published information of a randomized clinical trial of cholesterol-lowering therapy and a registry of acute myeloid leukemia were provided.ResultsGeneral and approximate relationships under rare event assumptions between the two types of hazard ratio were given. The approximation formula is based on a survival ratio and has two possible applications. First, one can calculate a subdistribution hazard ratio from published information. Second, this formula allows sample size estimation that takes the presence of competing risks into account.ConclusionsThe distinction between the two types of hazard ratio can be addressed by focusing on two quantities. One is how the event of interest and competing risk is rare, and the other is the survival ratio.

The Uchuu-glam BOSS and eBOSS LRG lightcones: exploring clustering and covariance errors

ABSTRACT This study investigates the clustering and bias of Luminous Red Galaxies (LRG) in the BOSS-LOWZ, -CMASS, -COMB, and eBOSS samples, using two types of simulated lightcones: (i) high-fidelity lightcones from UchuuN-body simulation, employing SHAM technique to assign LRG to (sub)haloes, and (ii) 16 000 covariance lightcones from GLAM-UchuuN-body simulations, including LRG using HOD data from Uchuu. Our results indicate that Uchuu and glam lightcones closely replicate BOSS/eBOSS data, reproducing correlation function and power spectrum across scales from redshifts 0.2 to 1.0, from 2 to $150 \,h^{-1}\,\mathrm{Mpc}$ in configuration space, from 0.005 to $0.7\, h\,\mathrm{Mpc}^{-1}$ in Fourier space, and across different LRG stellar masses. Furthermore, comparing with existing MD-Patchy and EZmock BOSS/eBOSS lightcones based on approximate methods, our GLAM-Uchuu lightcones provide more precise clustering estimates. We identify significant deviations from observations within $20 \,h^{-1}\,\mathrm{Mpc}$ scales in MD-Patchy and EZmock, with our covariance matrices indicating that these methods underestimate errors by between 10 per cent and 60 per cent. Lastly, we explore the impact of cosmology on galaxy clustering. Our findings suggest that, given the current level of uncertainties in BOSS/eBOSS data, distinguishing models with and without massive neutrino effects on large-scale structure (LSS) is challenging. This paper highlights the Uchuu and GLAM-Uchuu simulations’ robustness in verifying the accuracy of Planck cosmological parameters, providing a strong foundation for enhancing lightcone construction in future LSS surveys. We also demonstrate that generating thousands of galaxy lightcones is feasible using N-body simulations with adequate mass and force resolution.

VNIR-SWIR using OreXpress spectroradiometer as a spectral approximation method to differentiate epithermal mineralization alteration phases. A case study from Jacinto and Big Golden Hill, Cuba

VNIR-SWIR using OreXpress spectroradiometer as a spectral approximation method to differentiate epithermal mineralization alteration phases. A case study from Jacinto and Big Golden Hill, Cuba

Gauss–Bonnet Theorem Related to the Semi-Symmetric Metric Connection in the Heisenberg Group

In this paper, we introduce the notion of the semi-symmetric metric connection in the Heisenberg group. Moreover, by using the method of Riemannian approximations, we define the notions of intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on a surface, and the intrinsic Gaussian curvature of the surface away from characteristic points in the Heisenberg group with the semi-symmetric metric connection. Finally, we derive the expressions of those curvatures and prove the Gauss–Bonnet theorem related to the semi-symmetric metric connection in the Heisenberg group.

Improving the accuracy of halo mass based statistics for fast approximate N-body simulations

ABSTRACT Approximate N-body methods, such as fastpm and cola, have been successful in modelling halo and galaxy clustering statistics, but their low resolution on small scales is a limitation for applications that require high precision. Full N-body simulations can provide better accuracy but are too computationally expensive for a quick exploration of cosmological parameters. This paper presents a method for correcting distinct haloes identified in fast N-body simulations, so that various halo statistics improve to a percent level accuracy. The scheme seeks to find empirical corrections to halo properties such that the virial mass is the same as that of a corresponding halo in a full N-body simulation. The modified outer density contour of the corrected halo is determined on the basis of the fastpm settings and the number of particles inside the halo. This method only changes some parameters of the halo finder, and does not require any extra CPU-cost. We demonstrate that the adjusted halo catalogues of fastpm simulations significantly improve the precision of halo mass-based statistics from redshifts $z=0.0$ to 1.0, and that our calibration can be applied to different cosmologies without needing to be recalibrated.

A 3-D extension of the Multiscale Control Volume method for the simulation of the single-phase flow in anisotropic and heterogeneous porous media

The level of detail on modern geological models requires higher resolution grids that may render the simulation of multiphase flow in porous media intractable. Moreover, these models may comprise highly heterogeneous media with phenomena taking place in different scales. Scale transferring techniques allow for the solution of such problems in a lower resolution scale at reduced computational cost. Among these techniques, the Multiscale Finite Volume (MsFV) method constructs a set of numerical operators in order to map quantities from the fine-scale mesh to a coarser one and vice-versa while maintaining flux conservation on both scales. However, the MsFV formulation, as originally stated, is only consistent on k-orthogonal grids since it uses a linear Two-point Flux Approximation (TPFA) method and may struggle to generate consistent primal-dual coarse grids pairs on unstructured grids. The Multiscale Restriction Smoothed-Basis method (MsRSB) improves on the MsFV by introducing a new iterative procedure to find the multiscale operators and the concept of support regions which reduces the method's complexity when applied to unstructured fine and coarse grids. The original version was only consistent on k-orthogonal fine grids due to the TPFA discretization, but filtering methods have been developed to also enable consistent multipoint schemes on the fine scale. Meanwhile, the Multiscale Control Volume method (MsCV) replaces the TPFA by the Multipoint Flux Approximation with a Diamond stencil (MPFA-D) scheme on the fine-scale while further enhancing the generation of the geometric entities to allow unstructured grids on the fine and coarse scales for two-dimensional simulations. In this work we propose an extension to three-dimensional geometries of both the MsCV and the algorithm to obtain the multiscale geometric entities based on the concept of a background grid, a coarser grid used as a proxy for the primal coarse grid. We modify the original MPFA-D method to use the very robust Global Least Squares (GLS) interpolation technique to obtain the required auxiliary nodal unknowns. We also introduce an enhanced version of the 3-D MsCV with the incorporation of the enhanced MsRSB (E-MsRSB) to enforce M-matrix properties and improve convergence. Finally, we employ the MsCV operators in a two-stage smoother to show how it can be used as a good iterative procedure to recover the fine-scale solution. We show that the 3-D MsCV method produces good results employing unstructured grids on both scales to handle the simulation of the single-phase flow in anisotropic, and heterogeneous porous media and that the smoothing procedure is able to accurately retrieve the fine-scale solution within a few iterations.

A modified perturbation method for global dynamic analysis of generalized mixed Rayleigh–Liénard oscillator with cubic and quintic nonlinearities

Deriving analytical relationship between the system parameters and amplitude of the limit cycle is a meaningful and challenging task. Currently, numerous existing analytical approximate methods struggle to achieve this goal when expressions of restoring force or nonlinear damping is complicated. To overcome this shortcoming, this study proposes a modified generalized harmonic function perturbation method. Using the proposed method, a generalized mixed RayleighLiénard oscillator with cubic and quintic nonlinearities was investigated. The analytical relationships between the system parameters and amplitude of the limit cycle, as well as the expression of its characteristic quantity, were derived. By employing these analytical relationships, the existence, stability, number, position, and amplitude of each limit cycle are quantitatively analysed. The homoclinic and heteroclinic bifurcations were also predicted using the above analytical relationships. Additionally, analytical approximate solutions for this oscillator were calculated using the proposed method. All results obtained in this study were subsequently confirmed numerically to demonstrate their feasibility and validity. Consequently, the proposed method can be considered an effective supplement to perturbation-based methods. This also implies that the work presented in this paper has a certain theoretical significance and application value in the research area of quantitative analysis methods for strongly nonlinear oscillators.

Coverage Path Planning with Adaptive Hyperbolic Grid for Step-Stare Imaging System

Step-stare imaging systems are widely used in aerospace optical remote sensing. In order to achieve fast scanning of the target region, efficient coverage path planning (CPP) is a key challenge. However, traditional CPP methods are mostly designed for fixed cameras and disregard the irregular shape of the sensor’s projection caused by the step-stare rotational motion. To address this problem, this paper proposes an efficient, seamless CPP method with an adaptive hyperbolic grid. First, we convert the coverage problem in Euclidean space to a tiling problem in spherical space. A spherical approximate tiling method based on a zonal isosceles trapezoid is developed to construct a seamless hyperbolic grid. Then, we present a dual-caliper optimization algorithm to further compress the grid and improve the coverage efficiency. Finally, both boustrophedon and branch-and-bound approaches are utilized to generate rotation paths for different scanning scenarios. Experiments were conducted on a custom dataset consisting of 800 diverse geometric regions (including 2 geometry types and 40 samples for 10 groups). The proposed method demonstrates comparable performance of closed-form path length relative to that of a heuristic optimization method while significantly improving real-time capabilities by a minimum factor of 2464. Furthermore, in comparison to traditional rule-based methods, our approach has been shown to reduce the rotational path length by at least 27.29% and 16.71% in circle and convex polygon groups, respectively, indicating a significant improvement in planning efficiency.

Adaptive Fuzzy Control for Fractional-Order Networked Control Systems with Input Time Delay and Data Loss

This paper considers the adaptive fuzzy control of fractional-order nonlinear networked control systems subjected to network-induced input delay and data loss. To approximate unknown functions, fuzzy logic systems are employed. Furthermore, the Pade approximation method and an intermediate variable are introduced to eliminate the impact of input delay, and an adaptive fuzzy controller is designed using backstepping technology. Based on fractional-order Lyapunov stability theory, the proposed method can ensure that all signals are uniformly ultimately bounded, and the tracking error can converge to a small region of the origin. Two simulation examples are provided to verify the viability of the control method.

Study of the timelike electromagnetic form factors of the Λc

In this paper, the reaction of electron-positron annihilation into Λc+Λ¯c− is investigated. The Λc+Λ¯c− scattering amplitudes are obtained by solving the Lippmann-Schwinger equation. The contact, annihilation, and two pseudoscalar-exchange potentials are taken into account in the spirit of the chiral effective field theory. The amplitudes of e+e−→Λc+Λ¯c− are constructed by the distorted wave Born approximation method, with the final state interactions of the Λc+Λ¯c− rescattering implemented. By fitting to the experimental data, the unknown couplings are fixed, and high-quality solutions are obtained. With these amplitudes, the individual electromagnetic form factors in the timelike region, GEΛc, GMΛc, and their ratio, GEΛc/GMΛc, are extracted. Both modulus and phases are predicted. These individual electromagnetic form factors reveal new insights into the properties of the Λc. The separated contributions of the Born term, contact, annihilation, as well as the two pseudoscalar-exchange potentials to the electromagnetic form factors are isolated. It is found that the Born term dominates the whole energy region. The contact term plays a crucial role in the enhancement near the threshold, and the annihilation term is essential in generating the fluctuation of the electromagnetic form factors. Published by the American Physical Society 2024