Fast Multilevel Methods for Solving Ill-posed Problems

1 Introduetion Many industrial and engineering applieations require numerieally solving ill-posed Problems. Regularization methods are employed to find approximate solutions of these problems.The choice of regularization

The New Application of Multilevel Method of Moments in Analysis of Wire-Antenna Arrays

This paper introduces the principle of the multi-level method of moments (MoM) and its application in the analysis of the wire-antenna arrays. The multi-level MoM broadens the usage of the iterative methods in the MoM. Our numerical results show that when applying it to the wire-antenna array analysis with the consideration of the mutual coupling between elements, it can allow a rapid and accurate evaluation of the current distribution on the antennas, and the computational cost is less, especially when the number of antennas is large.

LOCAL MULTILEVEL METHODS FOR SECOND-ORDER ELLIPTIC PROBLEMS WITH HIGHLY DISCONTINUOUS COEFFICIENTS

In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coeffi- cients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenval- ues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel- preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.

OPTIMALITY OF LOCAL MULTILEVEL METHODS FOR ADAPTIVE NONCONFORMING P1 FINITE ELEMENT METHODS

In this paper, a local multilevel product algorithm and its additive version are con- sidered for linear systems arising from adaptive nonconforming P1 finite element approx- imations of second order elliptic boundary value problems. The abstract Schwarz theory is applied to analyze the multilevel methods with Jaeobi or Gauss-Seidel smoothers per- formed on local nodes on coarse meshes and global nodes on the finest mesh. It is shown that the local multilevel methods are optimal, i.e., the convergence rate of the multilevel methods is independent of the mesh sizes and mesh levels. Numerical experiments are given to confirm the theoretical results.

Multilevel Iteration Methods for Solving Linear Operator Equations of the First Kind

In this paper we develop two multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for linear ill-posed problems. The two algorithms and their convergence analyses are presented in an abstract framework.

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstrate the efficiency and accuracy of the methods. In these examples we use the proposed augmentation method to solve large scale linear systems resulting from the recently developed wavelet Galerkin methods and fast collocation methods applied to integral equations of the secondkind. Our numerical results confirm that this augmentation method is particularly efficient for solving large scale linear systems induced from wavelet compression schemes.

MULTILEVEL ITERATION METHODS FOR SOLVING LINEAR ILL-POSED PROBLEMS

In this paper we develop multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for ill-posed problems. The algorithm and its convergence analysis are presented in an abstract framework.

ALGEBRAIC MULTILEVEL RELAXED PRECONDITIONING METHODS

Some ways of multilevel relaxed preconditioning matrices for the stiffness matrix in the discretization of selfad joint second order elliptic boundary value problems are proposed. For reason-able assumptions of the relaxed factor ω, smaller relative condition numbers are given. The optimal relaxed factor ω is derived, too.

ON ALGEBRAIC MULTILEVEL PRECONDITIONINGMETHODS

In the present paper we extend the method presented by 0. Axelsson and P. Vassilevski called AMLP version (i) of recursively constructing preconditioner for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems. In our extended method the systems to be eliminated on each level containing the major block matrices of the given matrix can be solved approximately, while they must be solved exactly in the original method.

Multilevel Characteristic Basis Function Method with ACA for Accelerated Solution of Electrically Large Scattering Problems

The multilevel characteristic basis function method(MLCBFM)with the adaptive cross approximation(ACA)algorithm for accelerated solution of electrically large scattering problems is studied in this paper.In the conventional MLCBFM based on Foldy-Lax multiple scattering equations,the improvement is only made in the generation of characteristic basis functions(CBFs).However,it does not provide a change in impedance matrix filling and reducing matrix calculation procedure,which is time-consuming.In reality,all the impedance and reduced matrix of each level of the MLCBFM have low-rank property and can be calculated efficiently.Therefore,ACA is used for the efficient generation of two-level CBFs and the fast calculation of reduced matrix in this study.Numerical results are given to demonstrate the accuracy and efficiency of the method.

NONNESTED MULTILEVEL PRECONDITIONING METHODS WITH SIMPLE AND IDENTICAL COARSE MESH CORRECTION SPACES

In this paper, some simple and practical multilevel preconditioners for Hermite conforming and some well known nonconforming finite elements are constructed.

多级结构超疏水表面的制备与性能分析

随着橡胶产品应用的广泛拓展,具备超疏水性能的橡胶产品越来越受关注。然而如何便捷、大规模地制备超疏水表面对橡胶的疏水应用具有重要的意义。本工作通过在未硫化的氟硅橡胶表面喷涂一层二氧化硅粒子,构造第一层级微观结构,然后通过模板法在氟硅橡胶表面成功地复制了不锈钢编织网的结构,构造了第二层级结构,并分析了构造的氟硅橡胶表面的官能团存在情况、微观结构、疏水性能以及液滴弹跳和自清洁性能。结果表明:通过模板法和喷涂技术在氟硅橡胶表面制备的多级微观结构为液滴提供了支持力和排斥作用,其静态接触角为(156.7±0.97)°,滚动角为1°。并且构建的多级结构表面具有优异的液滴弹跳性能和良好的自清洁性能。在氟硅橡胶表面构建多级结构可以极大地提升表面的疏水性能,为制备超疏水表面提供一种新的思路。

基于电流注入法的模块化多电平换流器损耗建模

模块化多电平换流器(modular multilevel converter,MMC)的阀体损耗较大,MMC型柔性直流精细化电磁暂态仿真及系统效率研究需要对阀体损耗有更准确的模拟。针对现有柔性直流电磁暂态仿真中的阀体因采用等效模型导致的不存在的虚拟损耗问题,提出了基于电流注入法的柔性直流含阀体损耗电磁暂态建模方法,包括采用有限采样法实现不同运行工况下阀体损耗值的准确自动计算,参考计算结果在建模中采用电流注入法对损耗进行补偿,实现对换流阀损耗的动态跟踪与模拟。最后,通过在PSCAD中搭建柔性直流电磁暂态仿真模型,验证了所提方法的准确性,研究结果可用于精确模拟含阀体损耗的柔性直流电磁暂态建模,且通过仿真可进一步开展降损措施研究。

长江经济带洪涝灾害社会脆弱性评价及时空分析

【目的】系统地分析长江经济带各省洪涝灾害社会脆弱性及时空特征,对其防灾减灾提供决策依据。【方法】以长江经济带11个省作为研究对象,基于暴露度-敏感性-适应性(ESA)脆弱性评估框架构建社会脆弱性评估模型,并采用熵权法和多级模糊综合评价方法,对长江经济带洪涝的社会脆弱性和时空分布特征进行评估和分析。【结果】结果表明:(1)长江经济带洪涝社会脆弱性分布呈现出下游低,中游适中,上游高的特征。上游地区社会脆弱性指数平均值约为下游地区的4倍,中游地区的2倍。(2)2009—2019年长江经济带社会脆弱性的暴露度和适应性小幅上升、敏感性持续降低,整体上社会脆弱性表现为持续降低,经济带的灾害防御能力在逐年增强。(3)长江经济带洪涝社会脆弱性的影响因素表现出显著的空间异质性。高适应性和低敏感性是下游地区低社会脆弱性的原因;低适应性是中上游地区高脆弱度的主要原因;高暴露和高敏感性是湖南省的主要影响因素。【结论】就不同地区的致脆主导因素,提出针对性的防灾减灾措施。对于城市基础设施建设能力不足的地区,如江西省和四川省,其基础设施适应性指数对总适应性指数的贡献率分别为43%和47%,应当兴修其排水基础设施,加强防洪排涝设施的管护,增加绿化面积,建设海绵城市,提高城市韧性等。就水库数量不足导致其灾害抵御能力较弱的地区如贵州省,应当加快病险水利工程加固提标,开展小型水库系统治理评估等。

带散射体的电力系统多级传输线高频耦合模型

高频电磁场会与架空输电线路产生耦合,一旦高频电磁场的波长小于传输线的高度,经典传输线理论就不再适用。传统的全波数值算法虽然可以解决这一问题,但求解效率不高,需要消耗大量的时间与计算资源。此外,电力系统输电线路往往有多个垂直端,线路布局复杂,而以往针对电力系统传输线的高效求解研究都是只有左右两个垂直端的较为简单的模型,与实际电力系统传输线的结构不符,没有考虑中间带有垂直端的情况。为了解决这些问题,文中提出一种理论有效的方法来解决有损耗地面上方带有散射体的电力系统多级传输线的高频电磁场耦合问题。采用计算效率高的渐近法,并在渐近法中引入散射体的概念,以考虑更复杂的电路结构。最后通过数值算例验证了该方法的准确性和有效性。

A Context Sensitive Multilevel Thresholding Using Swarm Based Algorithms

In this paper, a comprehensive energy function is used to formulate the three most popular objective functions:Kapur's, Otsu and Tsalli's functions for performing effective multilevel color image thresholding. These new energy based objective criterions are further combined with the proficient search capability of swarm based algorithms to improve the efficiency and robustness. The proposed multilevel thresholding approach accurately determines the optimal threshold values by using generated energy curve, and acutely distinguishes different objects within the multi-channel complex images. The performance evaluation indices and experiments on different test images illustrate that Kapur's entropy aided with differential evolution and bacterial foraging optimization algorithm generates the most accurate and visually pleasing segmented images.

Optimal multilevel thresholding based on molecular kinetic theory optimization algorithm and line intercept histogram

Among all segmentation techniques, Otsu thresholding method is widely used. Line intercept histogram based Otsu thresholding method(LIH Otsu method) can be more resistant to Gaussian noise, highly efficient in computing time, and can be easily extended to multilevel thresholding. But when images contain salt-and-pepper noise, LIH Otsu method performs poorly. An improved LIH Otsu method(ILIH Otsu method) is presented, which can be more resistant to Gaussian noise and salt-and-pepper noise. Moreover, it can be easily extended to multilevel thresholding. In order to improve the efficiency, the optimization algorithm based on the kinetic-molecular theory(KMTOA) is used to determine the optimal thresholds. The experimental results show that ILIH Otsu method has stronger anti-noise ability than two-dimensional Otsu thresholding method(2-D Otsu method), LIH Otsu method, K-means clustering algorithm and fuzzy clustering algorithm.

Fast Multilevel CVT-Based Adaptive Data Visualization Algorithm

Efficient data visualization techniques are critical for many scientific applications. Centroidal Voronoi tessellation(CVT) based algorithms offer a convenient vehicle for performing image analysis,segmentation and compression while allowing to optimize retained image quality with respect to a given metric.In experimental science with data counts following Poisson distributions,several CVT-based data tessellation algorithms have been recently developed.Although they surpass their predecessors in robustness and quality of reconstructed data,time consumption remains to be an issue due to heavy utilization of the slowly converging Lloyd iteration.This paper discusses one possible approach to accelerating data visualization algorithms.It relies on a multidimensional generalization of the optimization based multilevel algorithm for the numerical computation of the CVTs introduced in[1],where a rigorous proof of its uniform convergence has been presented in 1-dimensional setting.The multidimensional implementation employs barycentric coordinate based interpolation and maximal independent set coarsening procedures.It is shown that when coupled with bin accretion algorithm accounting for the discrete nature of the data,the algorithm outperforms Lloyd-based schemes and preserves uniform convergence with respect to the problem size.Although numerical demonstrations provided are limited to spectroscopy data analysis,the method has a context-independent setup and can potentially deliver significant speedup to other scientific and engineering applications.

Comparison of Alternative Strategies for Multilevel Optimization of Hierarchical Systems

The augmented Lagrangian penalty formulation and four different coordination strategies are used to examine the nu- merical behavior of Analytical Target Cascading (ATC) for multilevel optimization of hierarchical systems. The coordination strategies considered include augmented Lagrangian using the method of multipliers and alternating direction method of multipliers, diagonal quadratic approximation, and truncated diagonal quadratic approximation. Properties examined include computational cost and solution accuracy based on the selected values for the different parameters that appear in each formulation. The different strategies are implemented using two- and three-level decomposed example problems. While the results show the interaction between the selected ATC formulation and the values of associated parameters, they clearly highlight the impact they could have on both the solution accuracy and computational cost.

Analysis of Reduced Switch Topology Multilevel Inverter with Different Pulse Width Modulation Technique and Its Application with DSTATCOM

Multilevel inverter has played a vital role in medium and high power applications in the recent years. In this paper, Reduced Switch Count Multi Level Inverter structure (RSCMLI) topology is presented with different pulse width modulation techniques. The harmonic level analysis is carried out for the reduced switch count multilevel inverter with the different PWM technique such as with Alternate Phase Opposition Disposition (APOD) method, In Phase Disposition (IPD) method and multi reference pulse width modulation method for five level, seven level , nine level and eleven level inverter. The simulation results are compared with the cascaded H Bridge Multi Level Inverter (CHBMLI). The nine level RSCMLI inverter with APOD method is used for the Distribution Static Synchronous Compensator (DSTATCOM) application in the nonlinear load connected system for power factor improvement. The result shows that the harmonic level and the number of switches required for RSCMLI is reduced compared to CHBMLI. RSCMLI employed in DSTATCOM improves the power factor and harmonic level of the system when it is connected to the nonlinear load.