Numerical Approximation Method Research Articles

A trust-region scheme for constrained multi-objective optimization problems with superlinear convergence property

In this paper, a numerical approximation method is developed to find approximate solutions to a class of constrained multi-objective optimization problems. All the functions of the problem are not necessarily convex functions. At each iteration of the method, a particular type of subproblem is solved using the trust region technique, and the step is evaluated using the notions of actual reduction and predicted reduction. A non-differentiable l ∞ penalty function restricts the constraint violations. An adaptive BFGS update formula is introduced. Global convergence of the proposed algorithm is established under the Mangasarian-Fromovitz constraint qualification and some mild assumptions. Furthermore, it is justified that the proposed algorithm displays a super-linear convergence rate. Numerical results are provided to show the efficiency of the algorithm in the quality of the approximated Pareto front.

Computation of Green’s Function in a Strongly Heterogeneous Medium Using the Lippmann–Schwinger Equation: A Generalized Successive Over-Relaxion plus Preconditioning Scheme

The computation of Green’s function is a basic and time-consuming task in realizing seismic imaging using integral operators because the function is the kernel of the integral operators and because every image point functions as the source point of Green’s function. If the perturbation theory is used, the problem of the computation of Green’s function can be transformed into one of solving the Lippmann–Schwinger (L–S) equation. However, if the velocity model under consideration has large scale and strong heterogeneity, solving the L–S equation may become difficult because only numerical or successive approximate (iterative) methods can be used in this case. In the literature, one of these methods is the generalized successive over-relaxation (GSOR) iterative method, which can effectively solve the L–S equation and obtain the desired convergent iterative series. However, the GSOR iterative method may encounter slow convergence when calculating the high-frequency Green’s function. In this paper, we propose a new scheme that utilizes the GSOR iterative with a precondition to solve the complex wavenumber L–S equation in a slightly attenuated medium. The complex wavenumber with imaginary components localizes the energy of the background Green’s function and reduces its singularity by enabling exponential decay. Introduction of the preconditioning operator can further improve the convergence speed of the GSOR iterative series. Then, we provide a preconditioned generalized successive over-relaxation (pre-GSOR) iterative format. Our numerical results show that if an appropriate damping factor and a proper preconditioning operator are selected, the method presented here outperforms the GSOR iterative for the real wavenumber L–S equation in terms of the convergence speed, accuracy, and adaptation to high frequencies.

Multi-material topology optimization considering both isotropy and anisotropy with fibre orientation optimization

Research on multi-material topology optimization (MMTO) considering both isotropic and anisotropic materials as candidate materials is limited. Previous studies required researchers to preselect the anisotropic material fibre orientations, which are not subject to change during the optimization. As the preselected orientations cannot change, better MMTO solutions may be overlooked. To address this issue, a novel MMTO algorithm incorporating the anisotropic material fibre orientation as a design variable is proposed, eliminating the need to preselect fibre orientations and leading to better MMTO solutions. A numerical approximation method named the carry-through method is also proposed. This allows the compliance sensitivity over anisotropic material fibre orientation to be calculated without the need for the strain–displacement matrix information, simplifying the MMTO method implementation. Comparative studies on three models demonstrate that the proposed MMTO method outperforms the existing MMTO method, achieving improvements of 7%, 10% and 14% on the objectives in these models.

New Lie Symmetries and Exact Solutions of a Mathematical Model Describing Solute Transport in Poroelastic Materials

A one-dimensional model for fluid and solute transport in poroelastic materials (PEMs) is studied. Although the model was recently derived and some exact solutions, in particular steady-state solutions and their applications, were studied, special cases occurring when some parameters vanish were not analysed earlier. Since the governing equations are nonintegrable in nonstationary cases, the Lie symmetry method and modern tools for solving ODE systems are applied in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with new Lie symmetries are identified. Some of them have a highly nontrivial structure that cannot be predicted from a physical point of view or using Lie symmetries of other real-world models. Applying the symmetries obtained, multiparameter families of exact solutions are constructed, including those in terms of elementary and special functions (hypergeometric, Whittaker, Bessel and modified Bessel functions). A possible application of the solutions obtained is demonstrated, and it is shown that some exact solutions can describe (at least qualitatively) the solute transport in PEM. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.

A Novel Localized Fast Multipole Method for Computations With Spatially Correlated Observation Error Statistics in Data Assimilation

AbstractSeveral observation types (e.g., geostationary satellite and Doppler radar observations) have recently been found to exhibit strong spatial error correlations. Including these error statistics in data assimilation for numerical weather prediction can improve analysis quality and forecast skill. Moreover, it allows for increases in the spatial density of observations assimilated, which is needed for the provision of information on appropriate scales for high‐resolution forecasting. However, introducing correlated error statistics may increase the computational complexity and parallel communication costs of matrix‐vector products involving observation precision matrices (inverse observation error covariance matrices). Without new approaches, we cannot take full advantage of new observation uncertainty estimates. We develop a new numerical approximation method based on a particular type of fast multipole method and a domain localization approach. The basic idea is to divide the observation domain into boxes and then separate calculations of matrix‐vector products according to the partition. These calculations can be done in parallel with very low communication overheads. The new method is easy to implement and parallelize, and it is applicable to a wide variety of observation precision matrices. We applied the new method to a simple variational data assimilation problem and found that the computational cost of the variational minimization was dramatically reduced while preserving analysis accuracy across a range of scales. The new method has the potential to be used as an efficient technique for practical applications where a large number of observations with mutual error correlations need to be assimilated quickly.

Biased versus unbiased numerical methods for stochastic simulations

Approximate numerical methods are one of the most used strategies to extract information from many-interacting-agents systems. In particular, numerical approximations are of extended use to deal with epidemic, ecological and biological models, since unbiased methods like the Gillespie algorithm can become unpractical due to high CPU time usage required. However, the use of approximations has been debated and there is no clear consensus about whether unbiased methods or biased approach is the best option. In this work, we derive scaling relations for the errors in approximations based on binomial extractions. This finding allows us to build rules to compute the optimal values of both the discretization time and number of realizations needed to compute averages with the biased method with a target precision and minimum CPU-time usage. Furthermore, we also present another rule to discern whether the unbiased method or biased approach is more efficient. Ultimately, we will show that the choice of the method should depend on the desired precision for the estimation of averages.

Learning autonomous race driving with action mapping reinforcement learning

Autonomous race driving poses a complex control challenge as vehicles must be operated at the edge of their handling limits to reduce lap times while respecting physical and safety constraints. This paper presents a novel reinforcement learning (RL)-based approach, incorporating the action mapping (AM) mechanism to manage state-dependent input constraints arising from limited tire-road friction. A numerical approximation method is proposed to implement AM, addressing the complex dynamics associated with the friction constraints. The AM mechanism also allows the learned driving policy to be generalized to different friction conditions. Experimental results in our developed race simulator demonstrate that the proposed AM-RL approach achieves superior lap times and better success rates compared to the conventional RL-based approaches. The generalization capability of driving policy with AM is also validated in the experiments.

A new one-dimensional model of free-piston Stirling engine based on quasi-steady flow approximation and simi-implicit numerical method

Free-piston Stirling engine is a widely used device for thermal-to-electric conversion especially in aerospace applications. A dynamic model for free-piston Stirling engine is established in this paper for the purpose of modeling the free-piston Stirling engine more accurately and understanding its operation characteristics. This paper proposes a new one-dimension numerical model with quasi-steady flow approximation for free-piston Stirling engine with linear alternator, which can be categorized as third-order theoretical model for Stirling engine. The corresponding simi-implicit discretizationmethod is presented. In order to accurately obtain the thermal efficiency of the Stirling machine, various energy loss mechanisms are taken into consideration. A beta type free-piston Stirling engine, Component Test Power Convertor, is studied to demonstrate the correctness and accuracy of the numerical model established in this paper. The dynamic steady-state anticipated operation condition is simulated first. Then, the sensitivity analysis is carried out in which the effect of load resistance, heater temperature and cooler temperature, and charged pressure on the system operation parameter is studied. The predicted system parameters including frequency, displacer and piston amplitude, output power, and alternator output voltage and current agrees well with the designed value. Under the dynamic steady operation condition, the simulated indicated power is 15.51 kW and the electric power is 13.12 kW. The relative deviation between the predicted electric power and the measured output electric power of 12.78 kW is 2.66 %. The corresponding indicated efficiency and thermal-to-electric efficiency is 25.06 % and 21.20 % respectively. The amplitude of displacer and piston is 13.96 mm and 13.69 mm for this condition. Compared with the experimental data at different piston amplitude, the derivation of predicted engine parameters is within 10 %. The comparison with experimental results proves the accuracy of the proposed model in this paper. Compared with the previous model based on the adiabatic thermodynamic model, the third-order model presented in this paper shows higher accuracy on system parameter prediction for free-piston Stirling engine.

A hybrid kernel-based meshless method for numerical approximation of multidimensional Fisher’s equation

We propose and analyze a meshless method of lines by considering some hybrid radial kernels. These hybrid kernels are constructed by linearly combining infinite smooth radial functions to piecewise smooth radial functions; which are then used for spatial approximation on trial spaces spanned by translates of positive definite radial functions. After spatial approximation, a high-order ODE solver is invoked for efficient and stable time-integration of the resultant semi-discrete system of ordinary differential equations (ODEs). Unlike the mesh-based method of lines, the proposed method works for arbitrary scattered data points and is equally effective for problems over non-rectangular domains. The proposed method is tested on one-, two- and three-dimensional reaction–diffusion Fisher equation for its numerical stability, accuracy, and efficiency against the contemporary meshless and mesh-based methods. The economical computational cost, improved accuracy, eigenvalues stability, and well-conditioning of system matrices are observed against RBF collocation and RBF-PS methods.

Щодо підвищення точності аналітичних та чисельних методів геодезичних та картометричних операцій

Purpose of this work is to determine and justify the use of analytical and numerical geodetic and cartometric methods on the reference ellipsoid, as well as their calculation accuracy in the geographic information environment. Methodology. The research compiled a register of geodetic and cartometric methods used in geodetic practice and implemented in modern geographic information systems. Standard tools in GIS often use approximate numerical methods, which affects the accuracy of models of geospatial objects in the GIS environment. Therefore, we have analysed and established for each operation of geodetic and cartometric methods a mathematical model that determines a particular cartometric property with maximum accuracy either by analytical or numerical methods with the number of terms in the binomial series of 6 or more. Results. The author proposed 10 operations of the geodetic method and 3 operations of the cartometric method, for which mathematical models and their accuracy were established and substantiated with their corresponding implementation in the MATLAB v. R2018a. The defined list of geodetic and cartometric operations made it possible to move away from the classification of distance lengths that influenced the further use of certain surfaces (map projection plane, sphere, spheroid, or reference ellipsoid) and mathematical models of operations. The presented mathematical models allow performing the relevant geodetic and cartometric methods with maximum accuracy using modern computer technologies. The mathematical models of geodetic and cartometric methods are investigated, which have practically no limitations for achieving the required accuracy, especially for large and ultra-large distances. The scientific novelty of the research is to define and justify a clear list of mathematical models of numerical and analytical geodetic and cartometric methods instead of cartometric methods on the map and standard methods of instrumental GIS; using the surface of the reference ellipsoid, and not just cartographic projections, spheroid or sphere. The practical significance of the research lies in the use of numerical and analytical geodetic and cartometric methods that significantly increase the accuracy of operations in these works, as well as in the creation/updating of digital topographic maps, navigation and route planning, etc. The research results can be concluded that well-founded mathematical models will ensure an increase in the accuracy of computational operations taking into account the curvature of the Earth in all sectors and areas of the economy, which will affect the quality of accounting and monitoring of relevant objects, integration and geospatial analysis of heterogeneous geospatial data, improve the quality (topological consistency) of geospatial data, etc

Solution to the schrödinger equation for bound states of polar molecules using shallow neural networks

The time independent Schrödinger equation can be solved analytically in very few cases. This is why scientists rely on approximate numerical methods. One that is yet to be a part of the everyday toolbox for physicists are neural networks. With it, it is possible to calculate eigenfunctions and the energy spectra of bound states. A double exponential central potential is used to describe the interaction between two electric dipoles that form through van der Waals interactions. These systems can be important when the electric dipoles are very large and differ from covalent, ionic, and other kinds of bonds, as is the case of the monomers M−C60, where M=Cs,Li or other alkali metals. Based in statistical physics, it has been shown that a dimer can be formed through a spontaneous and exothermic reaction. The ground state and the first two excited states of this system were calculated, also the solutions and their associated energies were plotted. The result for the ground state is compared to the shooting method and the direct variational method. The numerical value of the ground states obtained suggests that the dimers can be dissociated by electromagnetic waves with wavelengths in the far infrared regime, very close to microwaves.

Application of a time-fractal fractional derivative with a power-law kernel to the Burke-Shaw system based on Newton's interpolation polynomials

This paper proposes some updated and improved numerical schemes based on Newton's interpolation polynomial. A Burke-Shaw system of the time-fractal fractional derivative with a power-law kernel is presented as well as some illustrative examples. To solve the model system, the fractal-fractional derivative operator is used. Under Caputo's fractal-fractional operator, fixed point theory proves Burke-Shaw's existence and uniqueness. Additionally, we have calculated the Lyapunov exponent (LE) of the proposed system. This method is illustrated with a numerical example to demonstrate the applicability and efficiency of the suggested method. To analyze this system numerically, we use a fractal- fractional numerical scheme with a power-law kernel to analyze the variable order fractal- fractional system. Furthermore, the Atangana-Seda numerical scheme, based on Newton polynomials, has been used to solve this problem. This novel method is illustrated with numerical examples. Simulated and analytical results agree. We contribute to real-world mathematical applications. Finally, we applied a numerical successive approximation method to solve the fractional model.•The purpose of this section is to define a mathematical model to study the dynamic behavior of the Burke-Shaw system.•With the Danca algorithm [1,2] and Adams-Bashforth-Moulton numerical scheme, we compute the Lyapunov exponent of the system, which is useful for studying Dissipativity.•In a generalized numerical method, we simulate the solutions of the system using the time-fractal fractional derivative of Atangana-Seda.

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. Non-local nonlinear PDE models can accurately capture these phenomena, but traditional numerical approximation methods are infeasible when the considered non-local PDE is high-dimensional. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions. In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman–Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the non-local term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

Sequential Monte Carlo with cross-validated neural networks for complexity of hyperbolic black hole solutions in 4D

This paper investigates the self-similar solutions of the Einstein-axion-dilaton configuration from type IIB string theory and the global SL(2,R) symmetry. We consider the Continuous Self Similarity (CSS), where the scale transformation is controlled by an SL(2, R) boost or hyperbolic translation. The solutions stay invariant under the combination of space-time dilation with internal SL(2,R) transformations. We develop a new formalism based on Sequential Monte Carlo (SMC) and artificial neural networks (NNs) to estimate the self-similar solutions to the equations of motion in the hyperbolic class in four dimensions. Due to the complex and highly nonlinear patterns, researchers typically have to use various constraints and numerical approximation methods to estimate the equations of motion; thus, they have to overlook the measurement errors in parameter estimation. Through a Bayesian framework, we incorporate measurement errors into our models to find the solutions to the hyperbolic equations of motion. It is well known that the hyperbolic class suffers from multiple solutions where the critical collapse functions have overlap domains for these solutions. To deal with this complexity, for the first time in literature on the axion-dilaton system, we propose the SMC approach to obtain the multi-modal posterior distributions. Through a probabilistic perspective, we confirm the deterministic α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} solutions available in the literature and determine all possible solutions that may occur due to measurement errors. We finally proposed the penalized Leave-One-Out Cross-validation (LOOCV) to combine the Bayesian NN-based estimates optimally. The approach enables us to determine the optimum weights while dealing with the co-linearity issue in the NN-based estimates and better predict the critical functions corresponding to multiple solutions of the equations of motion.

Predictive Direct Yaw Moment Control Based on the Koopman Operator

In this brief, we propose a predictive algorithm for direct yaw moment control (DYC) in which a vehicle model is identified by a finite-dimensional approximation of the Koopman operator. The Koopman operator is a linear predictor for nonlinear dynamical systems based on raising the nonlinear dynamics into a higher-dimensional space where its evolution is linear. A novel method for the finite-dimensional numerical approximation of the Koopman operator is proposed, called enhanced extended dynamic mode decomposition (E <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> DMD). This method allows the reduction of the basis dimension, determined by a user-defined dictionary of observable functions, to achieve a trade-off between model complexity and accuracy. The E <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> DMD Koopman vehicle model was obtained from the dataset generated by simulating different scenarios using the nonlinear vehicle model and was then used to develop a Koopman operator model predictive control (KMPC) algorithm. KMPC was compared to a linear time variant (LTV) and a nonlinear model predictive control (NMPC), which are widely used in the literature, and showed better performance in some cases and a reduction in computational complexity in all cases.

Galerkin–Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds

A new numerical approximation method for a class of Gaussian random fields on compact connected oriented Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace–Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin–Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin–Chebyshev approximation are shown and confirmed through numerical experiments.

Sequential method conformal mappings

The well-known, very important Schwarz–Christoffel integral does not yet completely solve the problem of mapping a half-plane onto a predetermined polygon. This integral includes parameters (inverse images of the polygon), the relationship of which with the lengths of the edges of the polygon is not known in advance. The main difficulty in using the Schwarz–Christoffel formula lies in determining these parameters. If this difficulty can be overcome in some effective way, then the Schwarz–Christoffel formula expands the range of conformally mapped regions so much that it can be considered “universal”, given that the curvilinear boundary of the mapped region can be approximated by a broken line. Thus, together with the Schwarz–Christoffel formula, a new direction in the theory of functions arises - numerical methods of conformal mapping. However, these approximate numerical methods were developed independently of the Schwarz–Christoffel formula.

THE RESEARCH OF IMPLEMENTATION OF NUMERICAL RIGOROUS MATHEMATICAL METHODS WITH THE PARAMETER OF THE MEMBERS NUMBER IN THE TAYLOR SERIES

The current level of development of geoinformation technologies makes it possible to determine the cartometric properties of terrain objects with a user-defined accuracy. The author of this paper proposes a mathematical model that allows to implement the function of converting geodetic coordinates into rectangular flat Gauss-Kruger projections in a software environment based on the setting of the parameter of the number of members in the Taylor series, adjusting the accuracy of the coordinate transformation and their dimension. The purpose of this work is to implement a mathematical model of a numerical approximate computer method of converting geodetic coordinates into flat rectangular Gauss-Kruger projections. The results of this study will also be used to create functions for converting coordinates from rectangular flat Gauss-Kruger projections to geodetic coordinates and from one 6-degree zone of the Gauss-Kruger projection to another, taking into account the parameter of the number of members in the Taylor series. The proposed mathematical models are recommended to be implemented in the MATLAB or PhyCharm using the necessary libraries. The following studies will focus on determining the areas and lengths of objects located in two or more 6-degree zones of the Gauss-Kruger projection.

Your Labels are Selling You Out: Relation Leaks in Vertical Federated Learning

Vertical federated learning (VFL) is an emerging privacy-preserving paradigm that enables collaboration between companies. These companies have the same set of users but different features. One of them is interested in expanding new business or improving its current service with others' features. For instance, an e-commerce company, who wants to improve its recommendation performance, can incorporate users' preferences from another corporation such as a social media company through VFL. On the other hand, graph data is a powerful and sensitive type of data widely used in industry. Their leakage, e.g., the node leakage and/or the relation leakage, can cause severe privacy issues and financial loss. Therefore, protecting the security of graph data is important in practice. Though a line of work has studied how to learn with graph data in VFL, the privacy risks remain underexplored. In this paper, we perform the first systematic study on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">relation inference attacks</i> to reveal VFL's risk of leaking samples' relations. Specifically, we assume the adversary to be a semi-honest participant. Then, according to the adversary's knowledge level, we formulate three kinds of attacks based on different intermediate representations. Particularly, we design a novel numerical approximation method to handle VFL's encryption mechanism on the participant's representations. Extensive evaluations with four real-world datasets demonstrate the effectiveness of our attacks. For instance, the AUC of relation inference can reach more than 90%, implying an impressive relation inference capability. Furthermore, we evaluate possible defenses to examine our attacks' robustness. The results show that their impacts are limited. Our work highlights the need for advanced defenses to protect private relations and calls for more exploration of VFL's privacy and security issues.

Theoretical study on dynamic stress redistribution around circular tunnel with different unloading paths

This study presents a solution for stress redistribution induced by dynamic excavation of a rock tunnel in hydrostatic and non-hydrostatic stress states. Considering rock as an elastic medium, the analytical solution of dynamic excavation unloading was derived in the Laplace transform space utilizing the wave function expansion method and mode decomposition, and transformed into the time domain using the approximate numerical inversion method. The effects of the unloading path (linear, cosine, and exponential), unloading rate, and initial stress state on stress redistribution were analyzed. The analytical results indicated that a high unloading rate led to a high stress concentration and oscillations around the tunnel, and instantaneous dynamic unloading caused 20–30% stress concentration. When dynamic unloading is complete, the stress state around the tunnel converges to the Kirsch solution. The 3D numerical results indicated that deformation of the tunnel boring face increased as the initial stress increased. Moreover, the accumulation and release of strain energy and stress state change paths were analyzed and discussed. The results of this study provide a theoretical basis for understanding the stress adjustment and failure of surrounding rock induced by excavation of deep-buried openings in hydrostatic and non-hydrostatic stress states.