Multilevel Algorithm Research Articles

Efficient Solution of Electromagnetic Scattering Problems Using Multilevel Adaptive Cross Approximation and LU Factorization

The multilevel adaptive cross approximation (MLACA), previously described in the literature, extends the single-level ACA with a recursive multilevel algorithm that significantly improves compression of off-diagonal matrix blocks resulting from electromagnetic integral equations (IE) discretized via the method of moments (MoM). In this article, the MLACA approach is extended and applied to a direct solution of the MoM matrix system via LU factorization. It will be shown through numerical experiments that the off-diagonal LU blocks are also compressible using MLACA, yielding a compression rate superior to the single-level ACA and a memory complexity of O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4/3</sup> log N). In addition, the MLACA LU block updates are performed in rank-reduced form, yielding a very efficient software implementation via a Level 3 BLAS optimized for the CPU or GPU.

An adaptive filtering algorithm of multilevel resolution point cloud

The existing filtering methods for airborne LiDAR point cloud have low accuracy. An adaptive filtering algorithm is proposed which is improved based on multilevel resolution algorithm. First double index structure of Octree and KDtree is established. Then the initial reference surface is constructed by ground seed points. According to the slope fluctuation situation, the grid resolution of the ground referential surface is adjusted in an adaptive way. Finally, the refined surface is formed gradually by multilevel renewing resolution to provide filtered point cloud with high accuracy. Experimental results show that the error of Type II can be effectively reduced, the average Kappa coefficient increases by 0.53% and the average total error decreases by 0.44% compared with multiresolution hierarchical classification algorithm. The result tested by practically measured data shows that Kappa coefficient can reach 90%. Especially, it maintains advantages of high accuracy under complex topographic environment.

A Critical Survey of the Multilevel Method in Complex Networks

Multilevel optimization aims at reducing the cost of executing a target network-based algorithm by exploiting coarsened, i.e., reduced or simplified, versions of the network. There is a growing interest in multilevel algorithms in networked systems, mostly motivated by the urge for solutions capable of handling large-scale networks. Notwithstanding the success of multilevel optimization in a multitude of application problems, we were unable to find a representative survey of the state-of-the-art, or consistent descriptions of the method as a general theoretical framework independent of a specific application domain. In this article, we strive to fill this gap, presenting an extensive survey of the literature that contemplates a systematic overview of the state-of-the-art, a panorama of the historical evolution and current challenges, and a formal theoretical framework of the multilevel optimization method in complex networks. We believe our survey provides a useful resource to individuals interested in learning about multilevel strategies, as well as to those engaged in advancing theoretical and practical aspects of the method or in developing solutions in novel application domains.

Ultrasound Elasticity Imaging Based Multilevel Estimation Using Radiofrequency Data

An ultrasound elastography introduced to differentiate hard tumor inclusion embedded in soft tissue background based on similarity measurement of before and after deformation. In this study, freehand elastography has considered to localize hard inclusion embedded in soft tissue of phantom breast, where deformation generated by applying gentile compression using probe physical surface of ultrasound machine. Radiofrequency data of before and after deformation acquired and then processed off-line. A non-ability of refinement operation to regularize displacement estimation outliers at correlation window length of 2λ is addressed, where a multilevel processing algorithm has proposed to reinforce refinement operation by producing smooth elastography. In the first level of the processing, displacement field has estimated at correlation window length of 3λ, where global stretching as re-correlation operation and refinement operation as spatial regulation are included. While at second level, production of displacement estimation outliers at correlation window length of 2λ are regularized based on replacement of estimated cells with that interpolated one at first level. Results show an ability of multilevel algorithm to cope the issues that encountered previously proposed algorithm of refinement on estimation outlier free displacement field at an axial resolution of 2λ, and produces differential strain field.

A multilevel approach for trace system in HDG discretizations

We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and high-order characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating and/or limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly h-coarser edges/faces. We then combine the coarse solver with a block-Jacobi fine scale solver to form a two-level solver/preconditioner. Numerical experiments indicate that the performance of the resulting two-level solver/preconditioner depends on the smoothness of the solution and can offer significant speedups and memory savings compared to the nested dissection direct solver. While the proposed algorithms are developed within the HDG framework, they are applicable to other hybrid(ized) high-order finite element methods. Moreover, we show that our multilevel algorithms can be interpreted as a multigrid method with specific intergrid transfer and smoothing operators. With several numerical examples from Poisson, pure transport, and convection-diffusion equations we demonstrate the robustness and scalability of the algorithms with respect to solution order. While scalability with mesh size in general is not guaranteed and depends on the smoothness of the solution and the type of equation, improving it is a part of future work.

Multi-level Data Fusion Algorithm towards Privacy Protection in Wireless Sensor Networks

Data fusion is one of the key technologies in wireless sensor networks. To promote secure and efficient data fusion for wireless sensor networks, a privacy protection-based multi-level data fusion algorithm is proposed. In order to minimise the network energy consumption, the optimal number of cluster heads is selected and the adaptive clustering is performed according to the node energy and the positional relationship between nodes. Then, the cluster members collect and encrypt data, whereas the cluster head cleans and integrates the encrypted data. Furthermore, by analysing the correlation between data and constructing the BP neural network, cluster heads and the sink node can fuse the data in a cluster and the data between clusters to achieve the optimal data fusion. Results show that the mechanism proposed in this paper can significantly reduce resource overheads, effectively guarantee data security and fusion efficiency.

An enhanced multilevel ML-DFT codebook algorithm for hybrid beamforming of millimetre wave MIMO systems

Millimetre Wave (mmWave) wireless communication is considered an enabling technology to allow 5G cellular achieving high data rates. Large-scale antenna arrays can be adopted to compensate the huge path loss at higher frequencies. MIMO precoding cannot be performed only at baseband due to high power consumption of signal mixers and analogue-to-digital converters. Therefore, hybrid analogue-digital architecture at transceiver is considered as a cost-effective precoding scheme. However, the optimal design of such hybrid precoders and combiners needs to be further investigated. In this paper, a maximum likelihood (ML) beamforming technique is used for estimating signal's direction of departure in the presence of random noise. Moreover, an enhanced Orthogonal Mapping-based Matching Pursuit (OMBMP) algorithm is proposed. Layered orthogonal codebook is used to adjust base stations beamformers. Simulation results demonstrate that the proposed architecture provides acceptable performance gain with almost 90% of the performance of optimal full-digital precoder with great reduced complexity.

An enhanced multilevel ML-DFT codebook algorithm for hybrid beamforming of millimetre wave MIMO systems

An enhanced multilevel ML-DFT codebook algorithm for hybrid beamforming of millimetre wave MIMO systems

Research on Multi-level Inventory Optimization Algorithm of Repairable Spare Parts Based on Two Improved Differential Evolution

Research on Multi-level Inventory Optimization Algorithm of Repairable Spare Parts Based on Two Improved Differential Evolution

Estimating DSGE Models using Multilevel Sequential Monte Carlo in Approximate Bayesian Computation

Dynamic Stochastic General Equilibrium (DSGE) models allow for probabilistic estimations with the aim of formulating macroeconomic policies and monitoring them. In this study, we propose to apply the Sequential Monte Carlo Multilevel algorithm and Approximate Bayesian Computation (MLSMC-ABC) to increase the robustness of DSGE models built for small samples and with irregular data. Our results indicate that MLSMC-ABC improves the estimation of these models in two aspects. Firstly, the accuracy levels of the existing models are increased, and secondly, the cost of the resources used is reduced due to the need for shorter execution time.

Multilevel approximation of Gaussian random fields: Fast simulation

We propose and analyze several multilevel algorithms for the fast simulation of possibly nonstationary Gaussian random fields (GRFs) indexed, for example, by the closure of a bounded domain [Formula: see text] or, more generally, by a compact metric space [Formula: see text] such as a compact [Formula: see text]-manifold [Formula: see text]. A colored GRF [Formula: see text], admissible for our algorithms, solves the stochastic fractional-order equation [Formula: see text] for some [Formula: see text], where [Formula: see text] is a linear, local, second-order elliptic self-adjoint differential operator in divergence form and [Formula: see text] is white noise on [Formula: see text]. We thus consider GRFs on [Formula: see text] with covariance operators of the form [Formula: see text]. The proposed algorithms numerically approximate samples of [Formula: see text] on nested sequences [Formula: see text] of regular, simplicial partitions [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. Work and memory to compute one approximate realization of the GRF [Formula: see text] on the triangulation [Formula: see text] of [Formula: see text] with consistency [Formula: see text], for some consistency order [Formula: see text], scale essentially linearly in [Formula: see text], independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic “coloring” operator [Formula: see text] to white noise [Formula: see text]. For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

Newton-type multilevel optimization method

Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. Multilevel methods make more assumptions regarding the structure of the optimization model, and as a result, they outperform single-level methods, especially for large-scale models. The impressive performance of multilevel optimization methods is an empirical observation, and no theoretical explanation has so far been proposed. In order to address this issue, we study the convergence properties of a multilevel method that is motivated by second-order methods. We take the first step toward establishing how the structure of an optimization problem is related to the convergence rate of multilevel algorithms.

Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration

Smolyak’s method, also known as sparse grid method, is a powerful tool to tackle multivariate tensor product problems solely with the help of efficient algorithms for the corresponding univariate problem.In this paper we study the randomized setting, i.e., we randomize Smolyak’s method. We provide upper and lower error bounds for randomized Smolyak algorithms with explicitly given dependence on the number of variables and the number of information evaluations used. The error criteria we consider are the worst case root mean square error (the typical error criterion for randomized algorithms, sometimes referred to as “randomized error”, ) and the root mean square worst case error (sometimes referred to as “worst-case error”).Randomized Smolyak algorithms can be used as building blocks for efficient methods such as multilevel algorithms, multivariate decomposition methods or dimension-wise quadrature methods to tackle successfully high-dimensional or even infinite-dimensional problems. As an example, we provide a very general and sharp result on the convergence rate of Nth minimal errors of infinite-dimensional integration on weighted reproducing kernel Hilbert spaces. Moreover, we are able to characterize the spaces for which randomized algorithms for infinite-dimensional integration are superior to deterministic ones. We illustrate our findings for the special instance of weighted Korobov spaces. We indicate how these results can be extended, e.g., to spaces of functions whose smooth dependence on successive variables increases (“spaces of increasing smoothness”) and to the problem of L2-approximation (function recovery).

A multi-level Demand-Side Management algorithm for offgrid multi-source systems

The use of renewable sources for electricity supply of islands is faced with technical and economic constraints. To ensure that demand is fully met in the event of low generated power, batteries and gensets are often used and these can be expensive due to fuel import and battery costs. To provide more degrees of freedom for these offgrid networks, a multi-level algorithm based on several Demand Side Management strategies is proposed in this paper. The simulated case study concerns an island supplied by a multi-source system including solar, wind, tidal, wave energies and a battery storage solution. To limit the inconvenience for users, a hierarchical application of the proposed strategies is studied, according to a day-ahead forecast. Strategies based on anticipation are firstly applied for electric room heaters and water heaters in order to use the excess of generated power. For the most critical situations, strategies based on load shifting and load shedding are studied. In these cases, the best solution is found using a genetic algorithm. The application of the proposed Demand Side Management algorithm was found to help reduce the unmet load demand rate and adapt load demand according to the power generated.

Quality Control Scheme Selection with a Case of Aviation Equipment Development

Due to the complex outsourcing characteristics of aviation equipment development, one key issue is how to coordinate quality and cost decisions during development. This article focuses on the quality control plan selection from the multilevel perspective and builds a multilevel algorithm. Considering a complex research and development (R&D) project involving an integrator (client, tier 1 in a supply chain) and a contractor (tier 2 in a supply chain), this study builds two models by taking two different kinds of decision sequences into account. In the first case, the client moves first (i.e., the client makes decision first) while the contractor is the first mover in the second case. Two objectives are considered: the quality improvement rate and the resource cost. The results show that it is not necessary for the client to be the first mover in the decision-making process when alternative plans show no obvious difference in both the quality improvement rate and resource cost. In addition, because of the transitivity of quality objectives and resource constraints, the component quality from lower level will take effect on upper-level quality. Moreover, we find that the client, as the first mover, has a marked impact on the total quality improvement rate of the project only when alternative plans show obvious differences in either the quality improvement rate or the resource consumption. Our study offers insights on the importance of exploring the relationship of decision sequences on quality plans for a project manager.

Accelerating Solution of Volume-Surface Integral Equations With Multiple Right-Hand Sides by Improved Skeletonization Techniques

Skeletonization is an effective method to accelerate the solution of linear systems with multiple right-hand sides (RHSs) by exploiting the rank-deficiency property of the RHS matrix. However, when the size of the RHS matrix is very large, the memory requirement would become a bottleneck of the skeletonization technique. To reduce the memory cost, this letter presents the theoretical basis of how to find out the skeleton basis functions (SBFs) to construct the skeleton RHS matrix more efficiently from a purely algebraic point of view and proposes a multilevel algorithm to figure out the SBFs for solving the integral equations with multiple RHSs. The numerical results show that the proposed method accelerates the solution of the volume–surface integral equation with multiple RHSs and reduces the memory cost significantly.

A GPU-based multi-level algorithm for boundary value problems

A novel geometric single-grid multi-level (GSGML) algorithm is presented for the numerical solution of boundary value problems (BVPs). The algorithm is specifically designed for parallel scalability with high occupancy of streaming processors inside many-core architectures like the Graphics Processing Unit (GPU). The algorithm consists of iterative, superposed operations on a single grid, and is composed of two full-grid routines: a restriction and a coarsened interpolation–relaxation. The restriction is a local average of scalar or vector fields, and the interpolation–relaxation is a simultaneous application of coarsened finite-difference operators and interpolations in complementary subsets of the grid. The routines are scheduled in a saw-like refining cycle. Convergence to machine precision is achieved repeating the full cycle using residuals and superposition. The algorithm is attractive for its computational simplicity and provides a competitive GPU-based fast solver for elliptic partial differential equations like the Poisson–Helmholtz equation. Applications shown in this work include the deformation of two-dimensional grids, the computation of three-dimensional streamlines for a singular trifoil-knot vortex and the calculation of three-dimensional electric potentials in heterogeneous dielectric media.

Multilevel Algorithm Feedback Queue for Restaurant Order System

The purpose of writing this paper is to advance a restaurant that is have started out or even developing, with this business process will be make it easy and integrated to be as simple as possible so that customers who come to a restaurant no longer need to queue. In order to improve the quality of education on an ongoing basis both input, process and output in a study program, applications are needed that can support business processes[1]. The author makes the analysis of the system running using the UML (Unified Model Language), Multilevel Algorithm method used for analyzing the running system. The author designed this application so that the data that will be managed is integrated with the database, can be connected directly to the restaurant website and provide notifications to orders so that it makes will easy for customer to know the order process. The author focuses on combination scheduling order food and multilevel algorithm for queue restaurant order system

A Cumulative Sums Algorithm for Segmentation of Digital X-ray Images

A mathematical model of digital X-ray image of the test object is presented for the case when the main type of image distortion is the noise due to the quantum nature of the radiation. The new multilevel cumulative sums algorithm for automatic image segmentation is proposed. The algorithm is based on the edge detection of the segments homogeneous in brightness along the image rows and columns by repeatedly applying the cumulative sums procedure. The efficiency of the proposed algorithm is compared with the known threshold and Leader algorithms. The comparison was performed on simulated images as well as on the X-ray image of a weld. The mean square errors for the new algorithm were about two and three times less than for the threshold and Leader algorithms, correspondingly.

Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domainD⊂ ℝd. The multilevel algorithmQL*which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuoet al.(Found. Comput. Math.15(2015) 411–449). The random coefficient is assumed to admit a representation withlocally supported coefficient functions, as arise for example in spline- or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math.141(2019) 63–102). It also extends the MLQMC error analysis in Kuoet al.(Math. Comput.86(2017) 2827–2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) inD, and to product weights in QMC integration. In particular, in polytopal domainsD⊂ ℝd,d=2,3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices ofD. This allows fornon-stationaryGRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators inDwith boundary conditions on∂D. In the weighted function spaces inD, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations ofDyield optimal asymptotic convergence rates. Comparison of theε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations,e.g.of Karhunen–Loève type. Our analysis yields general bounds for theε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.