Approximation Problem Research Articles

Robust Matrix Completion Method Based on TNNR and Total Row Difference for Recovering Optical Image

Matrix completion aims to recover a matrix from an incomplete matrix with many unknown elements and has wide applications in optical image recovery and machine learning, in which the popular method is to formulate it as a general low-rank matrix approximation problem. However, the traditional optimization model for matrix completion is less robust. This paper proposes a robust matrix completion method in which the truncated nuclear norm regularization (TNNR) is used as the approximation of the rank function and the sum of absolute values of the row difference, which is called as the total row difference, is used to constrain the oscillations of the missing matrix. By minimizing the value of total row difference in the objective, the proposed model controls the oscillation and reduces the impact of missing part in the process of matrix completion continuously. Furthermore, we propose a two-step iterative algorithm framework and design an ADMM algorithm for subproblem model that includes minimizing the total row difference. Experiments show that the proposed algorithm has more stable performance and better recovery effect and obviously reduces the sensitivity of the traditional TNNR models to the truncated rank parameter.

Development and research of a neural network alternate incremental learning algorithm

In this paper, the relevance of developing methods and algorithms for neural network incremental learning is shown. Families of incremental learning techniques are presented. A possibility of using the extreme learning machine for incremental learning is assessed. Experiments show that the extreme learning machine is suitable for incremental learning, but as the number of training examples increases, the neural network becomes unsuitable for further learning. To solve this problem, we propose a neural network incremental learning algorithm that alternately uses the extreme learning machine to correct the only output layer network weights (operation mode) and the backpropagation method (deep learning) to correct all network weights (sleep mode). During the operation mode, the neural network is assumed to produce results or learn from new tasks, optimizing its weights in the sleep mode. The proposed algorithm features the ability for real-time adaption to changing external conditions in the operation mode. The effectiveness of the proposed algorithm is shown by an example of solving the approximation problem. Approximation results after each step of the algorithm are presented. A comparison of the mean square error values when using the extreme learning machine for incremental learning and the developed algorithm of neural network alternate incremental learning is made.

Energy Minimization for UAV Swarm-Enabled Wireless Inland Ship MEC Network With Time Windows

This paper establishes a new unmanned aerial vehicle (UAV) swarm-enabled wireless inland ship mobile edge computing (MEC) network with time windows, where UAVs are deployed to support time-constrained and low-resource unmanned surface vehicles (USVs) communication and computation. We aim to minimize UAV swarm energy consumption subject to numerous constraints. To tackle the challenging formulated problem, we decompose the original challenging problem into a list of sub-problems according to block coordinate descent (BCD) method and propose a heuristic algorithm to solve subproblems in an iterative manner. Specifically, the optimization of USVs offloading decisions is formulated as mixed-integer non-linear programming and then a modified deferred acceptance algorithm is proposed. The joint consideration of UAVs flight speed and UAV swarm stability is converted into a single-objective optimization problem after determining UAV control system’s latency requirements, and then Lagrangian multiplier method with Karush-Kuhn-Tucker (KKT) conditions is proposed. Successive convex approximations (SCA)-based algorithm is proposed to convert the non-convex constraints to convex ones, and then UAVs hovering coordinates can be obtained by solving the convex approximation problem. Numerical results verify the effectiveness of jointly considering USVs offloading decisions, UAVs flight speed, UAVs horizontal hovering coordinates and the number of UAVs for energy efficiency in comparison with several advanced baseline algorithms without optimization design.

An adaptive metamodel-based global approximation method for black-box type problems

An adaptive metamodel-based global approximation (AMGA) method for solving the global approximation problem of black-box models in large design domain is proposed in this study. The method employs the RBF model to compute the Hessian matrix and a heuristic direct search algorithm DIRECT to find the maximum curvature point of the metamodel surface, through which the design domain is split to obtain additional sampling points and to achieve fast update and fast valuation of the metamodel. The initial design domain is split into a series of sub-design domains by continuous iterations, and the metamodels built within the various sub-design domains achieve the global approximate model of the entire design domain. To demonstrate the final effect of the global approximation model and design domain splitting, six common two-dimensional test functions are chosen. The AMGA method is further tested using seven typical test functions and compared to other sampling and metamodel updating methods, with the findings demonstrating the usefulness of the proposed method in the low-dimensional scenario (less than four variables). Finally, the AMGA method is applied to a sophisticated electric car model, yielding good results.

Incremental Multilayer Broad Learning System With Stochastic Configuration Algorithm for Regression

Broad learning system (BLS) is a novel randomized learning framework which has a faster modeling efficiency. Although BLS with incremental learning has a better extendibility for updating model rapidly, the incremental mode of BLS lacks self-supervision mechanism which cannot adjust the structure adaptively. Learning from the idea of stochastic configuration network (SCN), a novel incremental multi-layer broad learning system based on stochastic configuration (SC) algorithm is proposed for regression, termed as IMLBLS-SC. Firstly, to improve the feature learning ability, SC algorithm is adopted to configure the parameters of enhancement nodes instead of random weights. Secondly, the multi-layer model with enhancement nodes can be added gradually according to supervision mechanism without human intervention. Thirdly, all the enhancement nodes and feature nodes are fully connected with output nodes. Finally, 2 function approximation problems and 8 classical datasets are selected to verify the regression performance of IMLBLS-SC, experimental results demonstrate that IMLBLS-SC outperforms the RVFLN, SCN, BLS and BSCN.

A dynamical method for optimal control of the obstacle problem

Abstract In this paper, we consider the numerical method for an optimal control problem governed by an obstacle problem. An approximate optimization problem is proposed by regularizing the original non-differentiable constrained problem with a simple method. The connection between the two formulations is established through some convergence results. A sufficient condition is derived to decide whether a solution of the first-order optimality system is a global minimum. The method with a second-order in time dissipative system is developed to solve the optimality system numerically. Several numerical examples are reported to show the effectiveness of the proposed method.

Applications of the duality theory of convex analysis to the complete electrode model of electrical impedance tomography

The duality theory of convex analysis is applied to the complete electrode model (CEM), which is a standard model in electrical impedance tomography (EIT). This results in a dual formulation of the CEM and a general error estimate. This new formulation of the CEM is written in terms of current fields and is shown to have a unique solution. Using this formulation, the general error estimate is proved, from which two a posteriori error estimates and a well known asymptotic result on CEM solutions are obtained. The first a posteriori error estimate assesses the accuracy of solutions to approximate problems, and the second one assesses the accuracy of approximate solutions. Numerical tests to apply this second estimate are performed, employing the finite element method to obtain approximate solutions.

Utility Preference Robust Optimization with Moment-Type Information Structure

Utility Preference Robust Optimization with Moment-Type Information Structure In some decision-making problems, information on the true utility function of the decision maker may be incomplete, which may bring potential modeling risk. In “Utility Preference Robust Optimization with Moment-Type Information Structure,” Guo, Xu, and Zhang propose a maximin utility preference robust optimization model where information about the DM’s preference is constructed by moment-type conditions. The authors propose a piecewise linear approximation approach to tackle the maximin problem, reformulate the approximate problem as a single mixed integer linear program, and derive error bounds for the approximate ambiguity set, the optimal value, and the optimal solutions. To examine the performance of the model and the computational schemes, they carry out extensive numerical tests and demonstrate the effectiveness of the model and efficiency of the computational methods.

Approximation of classes of Poisson integrals by Fejer means

The work is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, a many studies is devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer means have been widely studied in the last time. One of the important problems in this field is the study of asymptotic behavior of the upper bounds over a fixed classes of functions of deviations of the trigonometric polynomials. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals of continuous functions by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. The asymptotic behavior of the upper bounds on classes of Poisson integrals of periodic functions of the real variable of deviations of linear means of Fourier series, which are defined by applying the Fejer summation method is studied. The mentioned classes consist of analytic functions of a real variable, which are narrowing of bounded harmonic in unit disc functions of complex variable. In the work, asymptotic equalities for the upper bounds of deviations of Fejer means on classes of Poisson integrals were obtained.

Finite-Element Domain Approximation for Maxwell Variational Problems on Curved Domains

We consider the problem of domain approximation in finite element methods for Maxwell equations on general curved domains, i.e., when affine or polynomial meshes fail to exactly cover the domain of interest and an exact parametrization of the surface may not be readily available. In such cases, one is forced to approximate the domain by a sequence of polyhedral domains arising from inexact mesh. We deduce conditions on the quality of these approximations that ensure rates of error convergence between discrete solutions—in the approximate domains—to the continuous one in the original domain. Moreover, we present numerical results validating our claims.

Hermite–Padé approximation and integrability

We show that solution to the Hermite–Padé type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev–Petviashvili) system and of its adjoint linear problem. Our result explains the appearance of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorithms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite–Padé approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite–Padé problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.

CRT-Based Homomorphic Encryption over the Fraction

Homomorphic encryption technology is the holy grail of cryptography and has a wide range of applications in practice. This paper proposes a homomorphic encryption scheme over the fraction based on the Chinese remainder theorem (CRT) Dayan qiuyi rule. This homomorphic scheme performs encryption and decryption operations by forming congruence groups and has homomorphism. The solution in this paper first combines the traditional CRT algorithm with the Dayan qiuyi rule to obtain the CRTF algorithm that can be operated on the fraction field. Finally, in the decryption process, modulo arithmetic is used twice to obtain the correct plaintext components, restored to plaintext by CRTF. The scheme’s security is related to a decisional version of an approximate GCD problem. The proof of theoretical derivation shows that this paper’s homomorphic encryption scheme can realize the homomorphic addition operation in the fraction field. Compared with the CKKS scheme, efficiency is improved.

Corrigendum: Error bounds and asymptotic expansions for Toeplitz product functionals of unbounded spectra

We investigate error orders for integral limit approximations to traces of products of Toeplitz matrices generated by integrable functions on having some singularities at the origin. Even though a sharp error order of the above approximation is derived in Theorem 2 of Lieberman and Phillips (2004, Journal of Time Series Analysis, 25(5) 733–753), its proof contains an inaccuracy. In the present article, we reinvestigate error orders of the above trace approximation problem and rigorously validate the sharp error order derived in Lieberman and Phillips (2004, Journal of Time Series Analysis, 25(5) 733–753).

Unity Power Factor Operation in Microgrid Applications Using Fuzzy Type 2 Nested Controllers

The issue of low-power factor operation microgrids was reported for several layouts. Although numerous power factor improvement strategies have been applied and tested, various concerns remain to be addressed such as transient performance, simplicity of implementation, and satisfying the power-quality standards. The presented research aimed to design and implement controllers that can improve the transient response of microgrids due to changes in the load demand and achieve a near-unity power factor at the AC grid side, to which the DC microgrid is connected. Due to the nonlinear nature of microgrids, as they rely on power electronics converters, a Fuzzy type 2 controller was designed, implemented, and tested. The focus was given to improving the power factor of the DC microgrids. The validation of the proposed technique was verified by comparing its performance with Fuzzy type 1 and autotuned conventional PI controllers. To achieve the set aims, two nested control loops were designed with an inner current loop and an outer voltage loop. Besides MATLAB/Simulink simulations, a 10 kHz-sampling dSPACE platform was used to implement the suggested system. Two operational scenarios were tested: (1) a step change in the DC link voltage and (2) a change in the AC load (increase and decrease) at the output of the power inverter, connected to the DC grid. The simulation and experimental results confirmed that the proposed Fuzzy type 2 controller performed better than the other two techniques regarding the dynamic response, steady-state error, and compliance with power quality standards. Conventional approaches develop controllers using a linearized model, which limits the model accuracy and ignores higher-order variability. The method employs the nonlinear model. Fuzzy type 2 can better approximate high-precision problems than Fuzzy type 1.

JARNÍK TYPE THEOREMS ON MANIFOLDS

AbstractLet $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $ -approximable points and for the set of inhomogeneous multiplicative $\psi $ -approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. reine angew. Math.740 (2018), 63–76] and Beresnevich and Velani [‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s-Hausdorff measure, where $s\in \mathbb R$ , to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.

Exploring the optimality of approximate state preparation quantum circuits with a genetic algorithm

We study the approximate state preparation problem on noisy intermediate-scale quantum (NISQ) computers by applying a genetic algorithm to generate quantum circuits for state preparation. The algorithm can account for the specific characteristics of the physical machine in the evaluation of circuits, such as the native gate set and qubit connectivity. We use our genetic algorithm to optimize the circuits provided by the low-rank state preparation algorithm introduced by Araujo et al., and find substantial improvements to the fidelity in preparing Haar random states with a limited number of CNOT gates. Moreover, we observe that already for a 5-qubit quantum processor with limited qubit connectivity and significant noise levels (IBM Falcon 5T), the maximal fidelity for Haar random states is achieved by a short approximate state preparation circuit instead of the exact preparation circuit. We also present a theoretical analysis of approximate state preparation circuit complexity to motivate our findings. Our genetic algorithm for quantum circuit discovery is freely available at https://github.com/beratyenilen/qc-ga.

Neural network approximation of deformation curves under uniaxial tension of steel and silumin specimens

The purpose of the study is developing and testing of the new computational technique for approximation of deformation curves of steel and silumin specimens under uniaxial tension. A scheme of testing steel and silumin specimens for uniaxial tensile is presented. The experiment was carried out in the mechanical testing laboratory of the Department of Applied Mathematics and Mechanics of the Voronezh State Technical University. The experimental deformation curve of a steel specimen was approximated by P. Ludwig’s equation. Prediction of the true stress from the logarithmic strain using a pretrained artificial neural network with a multilayer perceptron architecture is discussed. The neural network model was trained using the RProp (resilient backpropagation) method. The software implementation of the neural network approximation was carried out in a framework of the open source for data analysis — Knime Analytics Platform. A scheme for the implementation of a multilayer perceptron that solves the approximation problem is considered. The simulation results are compared by the values of the mean squared error (MSE) of the approximation. The neural network approximation is turned out to be an order of magnitude more accurate for the steel specimen than the approximation by the P. Ludwig equation. The neural network approximation provided even a smaller MSE value for a silumin specimen than that or a steel specimen. It is revealed that changing the architecture of an artificial neural network affects the quality of modeling. With an increase in the number of hidden layers, the accuracy of the approximation increases. Neural network approximation is an effective approach to solving the problem of the analytical description of experimental deformation curves and leaves the possibility of using a universal technique for a variety of materials and different types of tests.

An Approximation Method for Variational Inequality with Uncertain Variables

In this paper, a Stieltjes integral approximation method for uncertain variational inequality problem (UVIP) is studied. Firstly, uncertain variables are introduced on the basis of variational inequality. Since the uncertain variables are based on nonadditive measures, there is usually no density function. Secondly, the expected value model of UVIP is established after the expected value is discretized by the Stieltjes integral. Furthermore, a gap function is constructed to transform UVIP into an uncertain constraint optimization problem, and the optimal value of the constraint problem is proved to be the solution of UVIP. Finally, the convergence of solutions of the Stieltjes integral discretization approximation problem is proved.

Nonlinear independent component analysis for discrete-time and continuous-time signals

We study the classical problem of recovering a multidimensional source signal from observations of nonlinear mixtures of this signal. We show that this recovery is possible (up to a permutation and monotone scaling of the source’s original component signals) if the mixture is due to a sufficiently differentiable and invertible but otherwise arbitrarily nonlinear function and the component signals of the source are statistically independent with ‘nondegenerate’ second-order statistics. The latter assumption requires the source signal to meet one of three regularity conditions which essentially ensure that the source is sufficiently far away from the nonrecoverable extremes of being deterministic or constant in time. These assumptions, which cover many popular time series models and stochastic processes, allow us to reformulate the initial problem of nonlinear blind source separation as a simple-to-state problem of optimisation-based function approximation. We propose to solve this approximation problem by minimizing a novel type of objective function that efficiently quantifies the mutual statistical dependence between multiple stochastic processes via cumulant-like statistics. This yields a scalable and direct new method for nonlinear Independent Component Analysis with widely applicable theoretical guarantees and for which our experiments indicate good performance.

A dual objective global optimization algorithm based on adaptive weighted hybrid surrogate model for the hydrogen fuel utilization in hydrogen fuel cell vehicle

Current engineering optimizations mainly use surrogate models (SMs) to approximate complex black-box problems. However, SMs with different approximate characteristics may make the designers unable to accurately judge which type of SMs is more suitable for the actual optimization design. A reasonable combination of different SMs might be one of the solutions. To this end, a global optimization algorithm based on an adaptive weighted hybrid surrogate model (GOA-AWHS) is proposed. In each iteration, a hybrid model based on kriging and RBF is first constructed by adaptively selecting weight coefficients. Next, two objectives consisting of predicted objective, root mean square error and distance parameters are optimized to generate the Pareto frontier. Finally, further selection of data points on the Pareto front yields multiple promising optimal solutions. A series of standard numerical functions and hydrogen fuel utilization in hydrogen fuel cell vehicles are tested to demonstrate the effectiveness and robustness of the GOA-AWHS method.