Approximate Point Research Articles

Accurate simulations of moving flexible objects with an improved immersed boundary-lattice Boltzmann method

Accurately capturing boundaries is crucial for simulating fluid–structure interaction (FSI) problems involving flexible objects undergoing large deformations. This paper presents a coupling of the immersed boundary-lattice Boltzmann method with a node-based partly smoothed point interpolation method (NPS-PIM) to enhance the accuracy of simulating moving flexible bodies in FSI problems. The proposed method integrates a multiple relaxation time scheme and employs a force correction technique to address boundary capturing inaccuracies. The effect of virtual fluid is accounted for through a Lagrangian point approximation, ensuring precise FSI force calculations for unsteady solid motions. NPS-PIM is utilized as the solid solver, constructing a moderately softened model stiffness by combining the finite element method (FEM) with the node-based smoothed PIM (NS-PIM). Simulations of flow fields near flexible objects with large deformations demonstrate that the proposed approach reduces numerical errors, improves computational efficiency compared to traditional FSI models using FEM and NS-PIM, and accurately captures the behavior of moving flexible bodies and detailed flow fields.

Approximating Fixed Point of (α, β, γ)-Nonexpansive Mapping Using Picard-Thakur Iterative Scheme

In this paper, we explore the fixed point approximation of (α, β, γ)–nonexpansive mappings using the Picard-Thakur iterative scheme. We identify the efficiency of Picard-Thakur iterative scheme through a numerical example by comparing it with other iterative schemes. We also proved strong and weak fixed point convergence theorems for (α, β, γ)–nonexpansive mappings by using Picard-Thakur iterative scheme.

Grid Point Approximation for Distributed Nonparametric Smoothing and Prediction

Kernel smoothing is a widely used nonparametric method in modern statistical analysis. The problem of efficiently conducting kernel smoothing for a massive dataset on a distributed system is a problem of great importance. In this work, we find that the popularly used one-shot type estimator is highly inefficient for prediction purposes. To this end, we propose a novel grid point approximation (GPA) method, which has the following advantages. First, the resulting GPA estimator is as statistically efficient as the global estimator under mild conditions. Second, it requires no communication and is extremely efficient in terms of computation for prediction. Third, it is applicable to the case where the data are not randomly distributed across different machines. To select a suitable bandwidth, two novel bandwidth selectors are further developed and theoretically supported. Extensive numerical studies are conducted to corroborate our theoretical findings. Two real data examples are also provided to demonstrate the usefulness of our GPA method.

Co-design of an unmanned cable shovel for structural and control integrated optimization: A highly heterogeneous constrained multi-objective optimization algorithm

Unmanned cable shovel is an intelligence-based large machine used for excavating in open-pit coal mines, with significant implications for the security and development of energy resources. The co-design of unmanned cable shovels, considering elements from both structural and control systems, has demonstrated superior performance in obtaining optimal solutions compared to the traditional approaches that focus on individual systems. However, the inherent complexities of diverse systems often lead to heterogeneous function evaluations, posing challenges for the existing optimization algorithms. To address highly heterogeneous multi-objective optimization problems characterized by surrogate-approximated expensive functions and explicit inexpensive functions, this study proposes a highly heterogeneous constrained multi-objective optimization (HHCMO) algorithm. Leveraging NSGA-III as an evolutionary optimizer and the Kriging model for approximating expensive functions, HHCMO strategically addresses the challenges posed by highly heterogeneous objective evaluations. A Monte Carlo sampling-based expected hypervolume improvement (EHVI) criterion, employing a new nearest point approximation method in a unified sampling space, effectively handles the impact of heterogeneous objective evaluations on the sequential infill of the Kriging model. Thorough evaluations on both unconstrained and constrained multi-objective benchmark problems validate the correctness of HHCMO and its superiority relative to state-of-the-art algorithms. Finally, the effectiveness of the proposed method is successfully demonstrated in an engineering scenario of an unmanned cable shovel, where it adeptly handles expensive functions from the structural system and inexpensive functions from the control system.

Convergence of the Two Point Flux Approximation method and the fitted Two Point Flux Approximation method for options pricing with local volatility function

In this paper, we deal with numerical approximations for solving the Black–Scholes Partial Differential Equation (PDE) for European and American options pricing with local volatility. This PDE is well-known to be degenerated. Local volatility model is a model where the volatility depends locally of both stock price and time. In contrast to constant volatility or time-dependent volatility models for which analytical representations of the exact solution is known for European Call options, there is no analytical solution for local volatility. The space discretization is performed using the classical finite volume method with Two-Point Flux Approximation (TPFA) and a novel scheme called Fitted Two-Point Flux Approximation (FTPFA). The Fitted Two-Point Flux Approximation (FTPFA) combines the fitted finite volume method and the standard TPFA method. More precisely the fitted finite volume method is used when the stock price approaches zero with the goal to handle the degeneracy of the PDE while the TPFA method is used on the rest of space domain. This combination yields our fitted TPFA scheme. The Euler method is used for the time discretization. We provide the rigorous convergence proofs of the two fully discretized schemes. Numerical experiments to support theoretical results are provided.

Subleading analysis for S3 partition functions of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 holographic SCFTs

We investigate the 3-sphere partition functions of various 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 holographic SCFTs arising from the N stack of M2-branes in the ’t Hooft limit both analytically and numerically. We first employ a saddle point approximation to evaluate the free energy F = – log Z at the planar level, tracking the first subleading corrections in the large ’t Hooft coupling λ expansion. Subsequently, we improve these results by determining the planar free energy to all orders in the large λ expansion via numerical analysis. Remarkably, the resulting planar free energies turn out to take a universal form, supporting a prediction that these S3 partition functions are all given in terms of an Airy function even beyond the special cases where the Airy formulae were derived analytically in the literature; in this context we also present new Airy conjectures in several examples. The subleading behaviors we captured encode a part of quantum corrections to the M-theory path integrals around dual asymptotically Euclidean AdS4 backgrounds with the corresponding internal manifolds through holographic duality.

Метод определения эффективности вариантов организации перевозок барже-буксирными составами

A study has been conducted on the method of evaluating options for organizing cargo transportation by barge-tow trains based on determining their effectiveness. Using the methods of the classical theory of transport processes, this problem has no quantitative solution, since it is not possible to take into account the regrouping of the composition during the passage of the route. To overcome this problem, additional mathematical tools have been developed for constructing integral time and velocity matrices, an objective function and graphs of the ratios of the total volume of transported cargo to the time it was en route. The consistent implementation of the proposed tools ensures the coherence of the calculation methods for determining the effectiveness of a flight with and without regrouping on the route, and also makes it possible to select the most appropriate option for organizing transportation. The cumulative result allows us to reflect the method of evaluation, comparison and selection of the most effective option for organizing a flight / group of flights, taking into account the ability to set individual parameters for the movement of the train and cargo. An example of approximation of the coordinate points of the route from the Nizhne-Svirsky shipping lock to Lake Ladoga is illustrated, as well as an example of discretization of an inland waterway section according to a three-level model. The parameters of equalization of the objective function of determining the efficiency of cargo transportation are considered. It is concluded that the optimal transportation option can be carried out on the basis of a combined approach, when several parameters are taken into account or sequentially set in the process of modeling the operation of ships. Graphs have been built that demonstrate the differences between the ways of organizing flights.

Point Spread Function Approximation of High-Rank Hessians with Locally Supported Nonnegative Integral Kernels

Point Spread Function Approximation of High-Rank Hessians with Locally Supported Nonnegative Integral Kernels

Fixed Point Dynamics in a New Type of Contraction in b-Metric Spaces

Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the “open” balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity and fixed points. After doing so, we introduce new types of contractivities that extend the concept of Banach contraction. We study some of their properties, giving sufficient conditions for the existence of fixed points and common fixed points. Afterwards, we consider some iterative schemes in quasi-normed spaces for the approximation of these critical points, analyzing their convergence and stability. We apply these concepts to the resolution of a model of integral equation of Urysohn type. In the last part of the paper, we refine some results about partial contractivities in the case where the underlying set is a strong b-metric space, and we establish some relations between mutual weak contractions and quasi-contractions and the new type of contractivity.

A dimensional mass transference principle from ball to rectangles for projections of Gibbs measures and applications

Mass transference principle are tools designed to provide estimates for the Hausdorff dimension of points approximable at a certain rate by a specific sequence of points (for instance rationals). Usually, such results are established provided that ambiant space is equipped with an Ahlfors regular measure (see [7,28] for instance). In the case of balls, Barral and Seuret established that mass transference principles still holds provided that the ambiant measure is a Gibbs measure or a self-similar measure satisfying the open set condition ([3]). In this article, we establish a mass transference principle “from ball to rectangle” assuming that the ambiant measure is the projection of Gibbs measure on the dyadic grid. Such measures are not Ahlfors regular in general and this result extends in particular the mass transference principle from ball to rectangle in the case of the Lebesgue measure, established in [28], and partially the result of Barral and Seuret [3]. As an application, given b∈N, we estimate the dimension of weighted approximation of points of [0,1]2 approximable by points with rational coordinates whose expansion in base b has prescribed frequencies. These results are new and surprising in the sens that, although the Lebesgue measure “scales” naturally rectangles, this needs not be the case for projection of Gibbs measures.

Solver-free reduced order homogenization for nonlinear periodic heterogeneous media

Reduced-order homogenization (ROH) and related methods are important computational tools for simulating the material behavior of composites. These methods generally sacrifice accuracy in exchange for superior computational efficiency, relative to methods such as classical computational homogenization (CCH). In this study, building on the recently developed solver-free CCH, we propose a fine-scale solver-free reduced order homogenization approach that avoids solving the fine-scale equilibrium equations and approximates the phase-average eigenstrains by sampling the fine-scale eigenstrain at a small number of points. The proposed method, which we call solver-free ROH, works by pre-computing history-dependent eigenstrain influence function tensors and sampling point contribution factors based on training data from a small set of CCH simulations. Then, during the online stage of the computation, phase-average eigenstrains are computed from sampling point eigenstrains and used to compute homogenized stresses and strains. Focusing on small-deformation problems, this paper formulates and verifies the solver-free ROH approach for nonlinear periodic heterogeneous media. First, in the formulation, we delineate a sampling point approximation of the phase eigenstrains and describe the use of this approximation within the coarse-scale stress update. Next, we verify the solver-free ROH using loading cases outside the training data set. Finally, we use the proposed method to simulate a multilayer composite plate in three point bending (3pt-bend) and open hole tension (OHT), demonstrating the method’s efficiency and accuracy relative to the CCH.

Results of coupled iterative fixed point approximation

Let X be a Banach space and F a subset of X. In this paper, we showed weak and strong convergence theorems for a Krasnoselskij-type iterative method to approximate coupled solution of nonexpansive operator E: FxF-> F, where F is nonempty, depending on two variables. Furthermore, we obtained a fixed point for a (b, \theta, L)-almost contraction operator E: X +X-> X using the Krasnoselskij iteration {xn}n=0infty.

Fixed point approximation via a new faster iteration process in Banach spaces with an application

Fixed point approximation via a new faster iteration process in Banach spaces with an application

A new fixed point approximation method for solving third-order BVPs based on Green's function

A new fixed point approximation method for solving third-order BVPs based on Green's function

Coincidence point theorems in some generalized metric spaces

"Let (X, d) be a complete dislocated metric space, (Y, ρ) be a semimetric space and f, g : X → Y be two mappings. Several coincidence point results are obtained for singlevalued and multivalued mappings. Keywords: Dislocated metric space, semimetric space, singlevalued and multivalued mapping, comparison function, comparison pair, lower semi-continuity, coincidence point displacement functional, iterative approximation of coincidence point, weakly Picard mapping, pre-weakly Picard mapping."

Path integral factorization and the gravitational effective action

We discuss the factorization and continuity properties of fields in the Euclidean gravitational path integral with higher dimension operators constructed from powers of the Riemann tensor. We construct the boundary terms corresponding to the microcanonical ensemble and show that the saddle point approximation to the path integral with a quasilocal energy constraint generally yields a saddle point with discontinuous temperature. This extends a previous result for the Euclidean Schwarzschild–de Sitter geometry in Einstein gravity and shows that it is robust against at least some types of quantum corrections from heavy fields. As an application, we compute the entropy of SdS in D = 4 using the BTZ method. Our result matches the entropy calculated using Wald’s formula.

An abstract approach to Marcinkiewicz-Zygmund inequalities for approximation and quadrature in modulation spaces

We study the approximation and quadrature error of points that satisfy Marcinkiewicz-Zygmund inequalities. First, we investigate the use of Marcinkiewicz-Zygmund inequalities in an abstract Hilbert space for an abstract approximation and quadrature rule. The setting is then specified to Sobolev spaces induced by Freud weights e − 2 σ | x | α e^{-2\sigma |x|^\alpha } with α > 1 \alpha >1 and σ > 0 \sigma >0 , and we derive specific bounds for the approximation and quadrature error. For the Gaussian weight e − 2 π x 2 e^{-2\pi x^2} , we verify that the Sobolev spaces essentially coincide with a specific class of modulation spaces that are well known in (time-frequency) analysis.

STUDY OF A DISTRICT HEATING SYSTEM WATER BOILER AS A OBJECT OF CONTROL


 
 The analysis of the operating conditions of a typical district heating system, in particular, a hot water gas boiler, was carried out. It is shown that the boiler, as the main control object, operates under conditions of constant change in external heat load, which, due to their random nature, leads to a number of uncertainties. The expediency of the mathematical description of uncertainties using the stochastic method as the most tested in practical conditions is substantiated. According to the results of the passive experiment on the hot water gas boiler KVG6.5-150 of the district heating system of one of the districts of Kharkov, an array of hourly experimental data was obtained, reflecting the main performance indicators of the hot water boiler. As a result of data processing by the least squares method, a mathematical model of the boiler was obtained in the form of a linear regression equation, reflecting the relationship between the temperature of the coolant at the boiler outlet and the ambient air temperature, the temperature of the coolant at the inlet to the boiler and with the flow rates of natural gas and coolant to the boiler. The obtained regression equation was verified by Student's t-test, which confirmed the significance of all coefficients of the regression model. The practical significance of the multiple regression equation was assessed using the coefficient of determination. The quality of the multiple regression equation as a whole was assessed using Fisher's F-test. Since parallel surveys were not conducted, instead of checking the adequacy, the quality of the approximation of the experimental points by the accepted regression equation was assessed, that is, it was checked whether this equation makes sense. Such a test was carried out by comparing the residual variance and the variance relative to the mean. The calculation results showed that the value of the coefficient of determination significantly exceeds the allowable value, and the actual value of the Fisher criterion significantly exceeds the table value. The obtained indicators led to the conclusion that the relationship between the variables in the regression model is significant, and the proposed stochastic method and the multiple linear regression equation can be used to make decisions in the process of synthesizing the technical structure of a computer-integrated control system for objects of a district heating system.
 
 
 
 
 

Assessment of the partial saddle point approximation in field-theoretic polymer simulations.

Field-theoretic simulations are numerical treatments of polymer field theory models that go beyond the mean-field self-consistent field theory level and have successfully captured a range of mesoscopic phenomena. Inherent in molecularly-based field theories is a "signproblem" associated with complex-valued Hamiltonian functionals. One route to field-theoretic simulations utilizes the complex Langevin (CL) method to importance sample complex-valued field configurations to bypass the signproblem. Although CL is exact in principle, it can be difficult to stabilize in strongly fluctuating systems. An alternate approach for blends or block copolymers with two segment species is to make a "partial saddle point approximation" (PSPA) in which the stiff pressure-like field is constrained to its mean-field value, eliminating the signproblem in the remaining field theory, allowing for traditional (real) sampling methods. The consequences of the PSPA are relatively unknown, and direct comparisons between the two methods are limited. Here, we quantitatively compare thermodynamic observables, order-disorder transitions, and periodic domain sizes predicted by the two approaches for a weakly compressible model of AB diblock copolymers. Using Gaussian fluctuation analysis, we validate our simulation observations, finding that the PSPA incorrectly captures trends in fluctuation corrections to certain thermodynamic observables, microdomain spacing, and location of order-disorder transitions. For incompressible models with contact interactions, we find similar discrepancies between the predictions of CL and PSPA, but these can be minimized by regularization procedures such as Morse calibration. These findings mandate caution in applying the PSPA to broader classes of soft-matter models and systems.

Probing supersymmetric black holes with surface defects

It has long been conjectured that the large N deconfinement phase transition of mathcal{N} = 4 SU(N) super-Yang-Mills corresponds via AdS/CFT to the Hawking-Page transition in which black holes dominate the thermal ensemble, and quantitative evidence of this has come through the recent matching of the superconformal index of frac{1}{16} -BPS states to the supersymmetric black hole entropy. We introduce the half-BPS Gukov-Witten surface defect as a probe of the superconformal index, which also serves as an order parameter for the deconfinement transition. This can be studied directly in field theory as a modification of the usual unitary matrix model or in the dual description as a D3-brane probe in the background of a (complex) supersymmetric black hole. Using a saddle point approximation, we determine our defect index in the large N limit as a simple function of the chemical potentials and show independently that it is reproduced by the renormalized action of the brane in the black hole background. Along the way, we also comment on the Cardy limit and the thermodynamics of the D3-brane in the generalized ensemble. The defect index sharply distinguishes between the confining and the deconfining phases of the gauge theory and thus is a supersymmetric non-perturbative order parameter for these large N phase transitions which deserves further investigation. Finally, our work provides an example where the properties of a black hole coupled to an external system can be analyzed precisely.