Iterative Approximation Research Articles

Multivariate Approximation Using Symmetrized and Perturbed Hyperbolic Tangent-Activated Multidimensional Convolution-Type Operators

In this article, we introduce, for the first time, multivariate symmetrized and perturbed hyperbolic tangent-activated convolution-type operators in three forms. We present their approximation properties, that is, their quantitative convergence to the unit operator via the multivariate modulus of continuity. We continue with the multivariate global smoothness preservation of these operators. We present, in detail, the related multivariate iterative approximation, as well as, multivariate simultaneous approximation, and their combinations. Using differentiability in our research, we produce higher rates of approximation, and multivariate simultaneous global smoothness preservation is also achieved.

Self-navigated subspace reconstruction for real-time MR imaging of the vocal tract

Purpose: Real-time MRI offers a continuous and dynamic view of the object being imaged. Researchers have applied real-time MRI to speech production, which allows for the visualization of the vocal tract during speech.Methods: This study proposed applying self-navigated subspace reconstruction for real-time vocal tract imaging. We performed experiments on a clinical 3 T MRI using standard RF coils and rapid acquisition. Additionally, 1000 frames were compressed during reconstruction to a few principal components, and iterative low-rank approximation was performed on compressed k-space, in conjunction with the orthogonal basis estimation for the subspace.Results: The simulation study involving a 32-time acceleration showed that the proposed method produced a reasonably small root mean square error (RMSE) of 0.159, compared to 0.278 for sliding window reconstruction, 0.2527 for SToRM and 0.294 for low-rank reconstruction. The study also presented in vivo images of a typical sagittal image with a temporal resolution of 7 ms/frame or 21 ms/frame for the three-slice scan.Conclusion: Our study presented a subspace reconstruction technique that does not require a navigator echo, which can be used for real-time MRI, particularly in speech imaging applications.

Iterative Numerical Approximation Technique for 3D Eddy Current Models in Harmonic Regime Based on the Electromagnetic T-Formulation and the Finite Element Method

In this paper, an iterative numerical approximation technique is used for analyzing the distribution of eddy currents density in conductive materials plates by using the electromagnetic T-formulation and the Biot-Savart law, solved by the finite element method. The proposed approach allows for the meshing of the different parts of the studied system separately, including the sensor and the conductive plate, without the need for an air region. Firstly, this approach reduces the number of unknown variables by avoiding the air region of the system's mesh. Secondly, it simplifies the consideration of the sensor's motion without the need to remesh the system. For this purpose, a calculation code has been developed for solving an electromagnetic three-dimensional non-destructive testing model. This latter permits the resolution of JSEAM # 6 Benchmark problem to validate the proposed method. The impedance variation due to the presence of a defect are evaluated. The obtained results are compared with the experimental ones found in the literature. These results reveal a good agreement, which proves the validity of the proposed method.

Novel Accelerated Cyclic Iterative Approximation for Hierarchical Variational Inequalities Constrained by Multiple-Set Split Common Fixed-Point Problems

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed-point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, with special cases being able to be found in many important engineering practical applications, such as image recognizing, signal processing, and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results.

Weak solvability of elliptic variational inequalities coupled with a nonlinear differential equation

In this paper we establish existence, uniqueness, and boundedness results for an elliptic variational inequality coupled with a nonlinear ordinary differential equation. Under the general framework, we present a new application modeling the antiplane shear deformation of a static frictional adhesive contact problem. The adhesion process has been extensively studied, but it is usual to assume a priori that the intensity of adhesion is bounded by introducing truncation operators. The aim of this article is to remove this restriction.The proof is based on an iterative approximation scheme showing that the problem has a unique solution. A key ingredient is finding uniform a priori bounds for each iterate. These are obtained by adapting versions of the Moser iteration to our system of equations.

Reconstruction of Z-pinch emission spectra in the wavelength range of less than 10Å using a crystal X-ray spectrograph.

This work is devoted to the reconstruction of Z-pinch plasma emission spectra in the wavelength range of less than 10Å recorded by using a crystal x-ray spectrograph at the Angara 5-1 mega-ampere facility. The spectrograph JA-1 used in experiments has a cylindrical mica crystal with dimensions of 50 × 40mm2 and radius of curvature of 100mm. Registration of spectra is performed on the photographic film UF-4 with dimensions of 30 × 10mm2. To reconstruct the spectra, the previously developed method based on iterative approximation of a true spectrum shape while minimizing a residual between experimental and calculated spectrograms is used. The calculated spectrogram was obtained taking into account the instrumental function of the spectrograph. To define the instrumental function a virtual Monte-Carlo model in the Geant4 toolkit has been developed. This model takes into account the interaction of radiation with the mica crystal using dynamical theory of diffraction. A true spectrum of Z-pinch plasma radiation is reconstructed for a 16mm high load made of two nested cylindrical wire liners. External liner with a diameter of 12mm has 40 Al wires with a diameter of 18 μm. The internal liner with a diameter of 5mm has 4W wires with a diameter of 6 μm. The W wires have a sputtered layer of Rethat is 0.5 μm thick.

Distributed least-squares progressive iterative approximation for blending curves and surfaces

Least-squares progressive iterative approximation (LSPIA) is an effective tool for fitting data points with curves and surfaces in CAD/CAM, due to its intuitive geometric meaning and its suitability for handling mass data. However, the classic LSPIA method for blending curves and surfaces has a slow convergence rate and takes a long CPU execution time, since the spectral radius of its iteration matrix is near to 1. To achieve a reduction in CPU execution time, this paper presents a distributed least-squares progressive iterative approximation (DLSPIA) method by dividing the collocation matrix into some blocks, which are called processors. The proposed method distributes the computation of the control points progressively in a way that each processor is responsible for a block of the whole point set. Combining the information obtained from the previous processors with that of the current processor, the DLSPIA method can progressively and quickly approximate the least-squares fitting result of the original data set block by block via distributed computation. And the algorithm converges within a finite number of iterations. Furthermore, the iterative formulae for blending surface fitting are expressed in matrix form, which can avoid the computation of the matrix Kronecker product to reduce the CPU execution time. Several numerical examples are presented to demonstrate the superiority of the proposed method compared with the previous methods.

Iterative Chebyshev approximation method for optimal control problems

We present a novel numerical approach for solving nonlinear constrained optimal control problems (NCOCPs). Instead of directly solving the NCOCPs, we start by linearizing the constraints and dynamic system, which results in a sequence of sub-problems. For each sub-problem, we use finite number of Chebyshev polynomials to estimate the control and state vectors. To eliminate the errors at non-collocation points caused by conventional collocation methods, we additionally estimate the coefficient functions involved in the linear constraints and dynamic system by Chebyshev polynomials. By leveraging the characteristics of Chebyshev polynomials, the approximate sub-problem is changed into an equivalent nonlinear optimization problem with linear equality constraints. Consequently, any feasible point of the approximate sub-problem will satisfy the constraints and dynamic system throughout the entire time scale. To validate the efficacy of the new method, we solve three examples and assess the accuracy of the method through the computation of its approximation error. Numerical results obtained show that our approach achieves lower approximation error when compared to the Chebyshev pseudo-spectral method. The proposed method is particularly suitable for scenarios that require high-precision approximation, such as aerospace and precision instrument production.

A novel analysis method for tension estimation of cross-linked suspenders based on single vibration measurements

While the use of vibration measurements to detect axially loaded forces is a well-established scheme for the diagnosis of tensioned slender members, the traditional approaches are not suitable for suspenders which are attached with transverse fasteners. As a result, the evaluation process for linked suspenders generally requires unlocking and reconnecting of the fasteners during performing vibration tests. This paper presents an innovated idea of using single vibration measurements with simple analytical models to develop an effective technique to estimate tension of individual member for cross-linked suspenders. The development of the technique relies on the use of simplified string formulas such that iterative approximations with numerical coding can be effectively executed for both the backward estimation and forward verification stages of the assessment procedure. The feasibility of the proposed method has been numerically studied and verified with selected data from field tests. It is anticipated that the procedure can serve as a better on-site tool for the vibration tests of fastener- restrained suspenders and cables.

A cabin assembly optimization method based on force and position coordination analysis

In existing automated docking assembly solutions for aerospace products, segmented docking surfaces suffer from machining errors and deformation, making assembly difficult. The aim of this study was to reduce the internal stresses in cabin assembly and maintain sufficient assembly accuracy. A cabin assembly optimization method was developed based on force–position coordination analysis. To address the interference problem of the cabin flange surface, a penalty-function-based reweighting iterative approximation method was constructed based on the square-hole characteristics of the docking. In the assembly verification, mechanism calibration and measurement field iterative compensation optimized the mechanism pose, and the accuracy of the pose adjustment component and the cabin assembly stress were tested. The experimental results show that the method proposed can reduce the interference problem during the assembly of a flange face and optimize the assembly stress while achieving assembly accuracy. The results for accuracy and interference force after flange face assembly were obtained; however, no feedback on the analytical method was obtained through the experimental results. Therefore, it is necessary to test further the propositions investigated. The preassembly analysis method for large cabin assemblies proposed optimizes the cabin assembly accuracy and internal stress under the deformation state of the assembly surface.

Newtonian Program Analysis of Probabilistic Programs

Due to their quantitative nature, probabilistic programs pose non-trivial challenges for designing compositional and efficient program analyses. Many analyses for probabilistic programs rely on iterative approximation. This article presents an interprocedural dataflow-analysis framework, called NPA-PMA, for designing and implementing (partially) non-iterative program analyses of probabilistic programs with unstructured control-flow, nondeterminism, and general recursion. NPA-PMA is based on Newtonian Program Analysis (NPA), a generalization of Newton's method to solve equation systems over semirings. The key challenge for developing NPA-PMA is to handle multiple kinds of confluences in both the algebraic structures that specify analyses and the equation systems that encode control flow: semirings support a single confluence operation, whereas NPA-PMA involves three confluence operations (conditional, probabilistic, and nondeterministic). Our work introduces ω-continuous pre-Markov algebras (ωPMAs) to factor out common parts of different analyses; adopts regular infinite-tree expressions to encode probabilistic programs with unstructured control-flow; and presents a linearization method that makes Newton's method applicable to the setting of regular-infinite-tree equations over ωPMAs. NPA-PMA allows analyses to supply a non-iterative strategy to solve linearized equations. Our experimental evaluation demonstrates that (i) NPA-PMA holds considerable promise for outperforming Kleene iteration, and (ii) provides great generality for designing program analyses.

Characterizing brain tau and cognitive decline along the amyloid timeline in Alzheimer's disease.

Recent longitudinal PET imaging studies have established methods to estimate the age at which amyloid becomes abnormal at the level of the individual. Here we recontextualized amyloid levels into the temporal domain to better understand the downstream Alzheimer's disease processes of tau neurofibrillary tangle (NFT) accumulation and cognitive decline. This cohort study included a total of 601 individuals from the Wisconsin Registry for Alzheimer's Prevention and Wisconsin Alzheimer's Disease Research Center that underwent amyloid and tau PET, longitudinal neuropsychological assessments and met clinical criteria for three clinical diagnosis groups: cognitively unimpaired (n = 537); mild cognitive impairment (n = 48); or dementia (n = 16). Cortical 11C-Pittsburgh compound B (PiB) distribution volume ratio (DVR) and sampled iterative local approximation were used to estimate amyloid positive (A+; global PiB DVR > 1.16 equivalent to 17.1 centiloids) onset age and years of A+ duration at tau PET (i.e. amyloid chronicity). Tau PET burden was quantified using 18F-MK-6240 standardized uptake value ratios (70-90 min, inferior cerebellar grey matter reference region). Whole-brain and region-specific approaches were used to examine tau PET binding along the amyloid timeline and across the Alzheimer's disease clinical continuum. Voxel-wise 18F-MK-6240 analyses revealed that with each decade of A+, the spatial extent of measurable tau spread (i.e. progressed) from regions associated with early to late NFT tau stages. Regional analyses indicated that tau burden in the entorhinal cortex was detectable, on average, within 10 years of A+ onset. Additionally, the entorhinal cortex was the region most sensitive to early amyloid pathology and clinical impairment in this predominantly preclinical sample. Among initially cognitively unimpaired (n = 472) individuals with longitudinal cognitive follow-up, mixed effects models showed significant linear and non-linear interactions of A+ duration and entorhinal tau on cognitive decline, suggesting a synergistic effect whereby greater A+ duration, together with a higher entorhinal tau burden, increases the likelihood of cognitive decline beyond their separable effects. Overall, the amyloid time framework enabled a spatiotemporal characterization of tau deposition patterns across the Alzheimer's disease continuum. This approach, which examined cross-sectional tau PET data along the amyloid timeline to make longitudinal disease course inferences, demonstrated that A+ duration explains a considerable amount of variability in the magnitude and topography of tau spread, which largely recapitulated NFT staging observed in human neuropathological studies. By anchoring disease progression to the onset of amyloid, this study provides a temporal disease context, which may help inform disease prognosis and timing windows for anti-amyloid therapies.

Nonlinear iterative approximation of steady incompressible chemically reacting flows

Nonlinear iterative approximation of steady incompressible chemically reacting flows

Monotone Iterative Technique for a Kind of Nonlinear Fourth-Order Integro-Differential Equations and Its Application

In this paper, we consider the existence and iterative approximation of solutions for a class of nonlinear fourth-order integro-differential equations (IDEs) with Navier boundary conditions. We first prove the existence and uniqueness of analytical solutions for a linear fourth-order IDE, which has rich applications in engineering and physics, and then we establish a maximum principle for the corresponding operator. Based upon the maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to obtain iterative solutions for the nonlocal nonlinear problem under certain conditions. Some examples are presented to illustrate the main results.

GLS‐PIA: n‐Dimensional Spherical B‐Spline Curve Fitting based on Geodesic Least Square with Adaptive Knot Placement

AbstractDue to the widespread applications of curves on n‐dimensional spheres, fitting curves on n‐dimensional spheres has received increasing attention in recent years. However, due to the non‐Euclidean nature of spheres, curve fitting methods on n‐dimensional spheres often struggle to balance fitting accuracy and curve fairness. In this paper, we propose a new fitting framework, GLS‐PIA, for parameterized point sets on n‐dimensional spheres to address the challenge. Meanwhile, we provide the proof of the method. Firstly, we propose a progressive iterative approximation method based on geodesic least squares which can directly optimize the geodesic least squares loss on the n‐sphere, improving the accuracy of the fitting. Additionally, we use an error allocation method based on contribution coefficients to ensure the fairness of the fitting curve. Secondly, we propose an adaptive knot placement method based on geodesic difference to estimate a more reasonable distribution of control points in the parameter domain, placing more control points in areas with greater detail. This enables B‐spline curves to capture more details with a limited number of control points. Experimental results demonstrate that our framework achieves outstanding performance, especially in handling imbalanced data points. (In this paper, “sphere” refers to n‐sphere (n ≥ 2) unless otherwise specified.)

Asynchronous progressive iterative approximation method for least squares fitting

For large data fitting, the least squares progressive iterative approximation (LSPIA) methods have been proposed by Lin and Zhang (2013) and Deng and Lin (2014), in which a constant step size is used. In this paper, we further accelerate the LSPIA method in terms of a Chebyshev semi-iterative scheme and present an asynchronous LSPIA (denoted by ALSPIA) method. The control points in ALSPIA are updated by using an extrapolated variant in which an adaptive step size is chosen according to the roots of Chebyshev polynomial. Our convergence analysis shows that ALSPIA is faster than the original LSPIA method in both singular and non-singular least squares fitting cases. Numerical examples show that the proposed algorithm is feasible and effective.

High-precision modelling of deformation of sandwich structures under bilateral symmetric and oblique-symmetric loading

The analysis and assessment of the stress-strain state (STS) of multilayer plates with orthotropic layers under the action of stationary transverse and tangential loads is an urgent task. It includes calculations on the strength and deformability of various homogeneous and multi-layered plates with layers of constant thickness but arbitrary structure according to the thickness of the plate. Combining materials with isotropic and transversely isotropic physical characteristics into a multilayer package allows you to create multifunctional structures. The SSS of such structures, due to their structural heterogeneity and relatively low transverse stiffness of the individual layers, is significantly associated with the influence of transverse shear deformations and transverse compression deformations. Therefore, the problem of refined modeling of SSS plates, which would take into account these types of deformations, is urgent. Based on the decomposition of the SSS plate into flexural and unflexural components, it is proposed to optimize the design scheme of deformation of a rectangular multilayer plate. This significantly simplifies its modeling. For unflexural and exclusively flexural SSS, a two-dimensional, high-degree iterative approximation, but three-dimensional models of deformation of multilayer rectangular plate on a rigid foundation with isotropic, transverse-isotropic and orthotropic layers are constructed in an elastic formulation. That models takes full account deformation of transverse shear and transverse compression at transverse and the tangential loading of a plate. The model is continuous, that is, the number of equations and the order of differentiation of the calculation system of equations does not depend on the number of layers in the slab. This order of differentiation and the number of calculation equations can depend only on the order of iterative approximation of the model. A way of precise satisfaction of all defining correlations of the layers of material under keeping the conditions of their contact is found, while in the known continual models the dependence between the cross normal stress and cross deformation is only integral.The results of the analytical solution of the problem of deformation of a rectangular plate under boundary conditions of the Navier type under the action of transverse and tangential loads are given. By solving the test problems of the deformation of two-layer plates with transversally isotropic layers and three-layer plates with orthotropic layers and comparing the solutions with the exact three-dimensional solutions of these problems obtained by known methods, an assessment of the accuracy of the proposed refined models is given. The limits of admissible parameters of elastic characteristics of transversely isotropic and orthotropic plates for application of the offered models are established.

System capacity model and algorithm for urban multimodal transport network with transfer

ABSTRACT This paper proposes a method for calculating transport networks capacity in dealing with multimodal transfers. Multimodal networks are represented by a modified ‘supernetwork’, while the passenger’s travels are defined as ‘superpaths’. Within this framework, the relation between the travel demand from O-D matrices and the resulting link flows in the supernetwork is modelled as a relationship matrix to describe urban mobility by using a logit-based stochastic user equilibrium. Based on this relationship matrix, an approximate iteration algorithm (AIA) is developed. Our numerical results show that the AIA performs better than the sensitivity analysis-based algorithm (SAB) and genetic algorithm (GA) regarding the execution-time, and that the capacity of multimodal transport networks can be underestimated if the combined travels are neglected.

An Efficiency Boost for Genetic Algorithms: Initializing the GA with the Iterative Approximate Method for Optimizing the Traveling Salesman Problem—Experimental Insights

The genetic algorithm (GA) is a well-known metaheuristic approach for dealing with complex problems with a wide search space. In genetic algorithms (GAs), the quality of individuals in the initial population is important in determining the final optimal solution. The classic GA using the random population seeding technique is effective and straightforward, but the generated population may contain individuals with low fitness, delaying convergence to the best solution. On the other side, heuristic population seeding strategies provide the advantages of producing individuals with high fitness and encouraging rapid convergence to the optimal solution. Using background information on the problem being solved, researchers have developed several population seeding approaches. In this paper, to enhance the genetic algorithm efficiency, we propose a new method for the initial population seeding based on a greedy approach. The proposed method starts by adding four extreme cities to the route, creating a loop, and then adding each city to the route through a greedy strategy that calculates the cost of adding every city to different locations along the route. This method identifies the best position to place the city as well as the best city to add. Employing local constant permutations improves the resultant route even more. Together with the suggested approach, we examine the GA’s effectiveness while employing conventional population seeding methods like nearest-neighbor, regression-based, and random seeding. Utilizing some of the well-known Traveling Salesman Problem (TSP) examples from the TSPLIB, the standard library for TSPs, tests were conducted. In terms of the error rate, average convergence, and time, the experimental results demonstrate that the GA that employs the suggested population seeding technique performs better than other GAs that use conventional population seeding strategies.

Accelerating Cutting-Plane Algorithms via Reinforcement Learning Surrogates

Discrete optimization belongs to the set of N P-hard problems, spanning fields such as mixed-integer programming and combinatorial optimization. A current standard approach to solving convex discrete optimization problems is the use of cutting-plane algorithms, which reach optimal solutions by iteratively adding inequalities known as cuts to refine a feasible set. Despite the existence of a number of general-purpose cut-generating algorithms, large-scale discrete optimization problems continue to suffer from intractability. In this work, we propose a method for accelerating cutting-plane algorithms via reinforcement learning. Our approach uses learned policies as surrogates for N P-hard elements of the cut generating procedure in a way that (i) accelerates convergence, and (ii) retains guarantees of optimality. We apply our method on two types of problems where cutting-plane algorithms are commonly used: stochastic optimization, and mixed-integer quadratic programming. We observe the benefits of our method when applied to Benders decomposition (stochastic optimization) and iterative loss approximation (quadratic programming), achieving up to 45% faster average convergence when compared to modern alternative algorithms.