Contour integration for eigenvector nonlinearities

Solving polynomial eigenvalue problems with eigenvector nonlinearities (PEPv) is an interesting computational challenge, outside the reach of the well-developed methods for nonlinear eigenvalue problems. We present a natural generalization of these methods which leads to a contour integration approach for computing all eigenvalues of a PEPv in a compact region of the complex plane. Our methods can be used to solve any suitably generic system of polynomial or rational function equations

Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems

The final publication is available at Springer via http://dx.doi.org/ 10.1007/s10543-016-0601-5.We investigate how to adapt the Q-Arnoldi method for the case of symmetric quadratic eigenvalue problems, that is, we are interested in computing a few eigenpairs of with M, C, K symmetric matrices. This problem has no particular structure, in the sense that eigenvalues can be complex or even defective. Still, symmetry of the matrices can be exploited to some extent. For this, we perform a symmetric linearization , where A, B are symmetric matrices but the pair (A, B) is indefinite and hence standard Lanczos methods are not applicable. We implement a symmetric-indefinite Lanczos method and enrich it with a thick-restart technique. This method uses pseudo inner products induced by matrix B for the orthogonalization of vectors (indefinite Gram-Schmidt). The projected problem is also an indefinite matrix pair. The next step is to write a specialized, memory-efficient version that exploits the block structure of A and B, referring only to the original problem matrices M, C, K as in the Q-Arnoldi method. This results in what we have called the Q-Lanczos method. Furthermore, we define a stabilized variant analog of the TOAR method. We show results obtained with parallel implementations in SLEPc.This work was supported by the Spanish Ministry of Economy and Competitiveness under Grant TIN2013-41049-P. Carmen Campos was supported by the Spanish Ministry of Education, Culture and Sport through an FPU Grant with reference AP2012-0608.Campos, C.; Román Moltó, JE. (2016). Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems. BIT Numerical Mathematics. 56(4):1213-1236. https://doi.org/10.1007/s10543-016-0601-5S12131236564Bai, Z., Su, Y.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. 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On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.Comment: 20 page

Locking and Restarting Quadratic Eigenvalue Solvers

This paper studies the solution of quadratic eigenvalue problems by the quadratic residual iteration method. The focus is on applications arising from finite-element simulations in acoustics. One approach is the shift-invert Arnoldi method applied to the linearized problem. When more than one eigenvalue is wanted, it is advisable to use locking or deflation of converged eigenvectors (or Schur vectors). In order to avoid unlimited growth of the subspace dimension, one can restart the method by purging unwanted eigenvectors (or Schur vectors). Both locking and restarting use the partial Schur form. The disadvantage of this approach is that the dimension of the linearized problem is twice that of the quadratic problem. The quadratic residual iteration and Jacobi-Davidson methods directly solve the quadratic problem. Unfortunately, the Schur form is not defined, nor are locking and restarting. This paper shows a link between methods for solving quadratic eigenvalue problems and the linearized problem. It aims to combine the benefits of the quadratic and the linearized approaches by employing a locking and restarting scheme based on the Schur form of the linearized problem in quadratic residual iteration and Jacobi-Davidson. Numerical experiments illustrate quadratic residual iteration and Jacobi-Davidson for computing the linear Schur form. It also makes a comparison with the shift-invert Arnoldi method

Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem

Applications such as the modal analysis of structures and acoustic cavities require a number of eigenvalues and eigenvectors of large-scale Hermitian eigenvalue problems. The most popular method is probably the spectral transformation Lanczos method. An important disadvantage of this method is that a change of pole requires a complete restart. In this paper, we investigate the use of the rational Krylov method for this application. This method does not require a complete restart after a change of pole. It is shown that the change of pole can be considered as a change of Lanczos basis. The major conclusion of this paper is that the method is numerically stable when the poles are chosen in between clusters of the approximate eigenvalues. Copyright (C) 2001 John Wiley & Sons, Ltd

Decolorization of textile waste water, with an emphasis on microbial treatment processes

Textile wastewater is typically intensely colored, containing high concentrations of dyes, dyeing additives and diverse chemicals, some of which are non-biodegradable and/or toxic, mutagenic or carcinogenic. Therefore, it is essential to treat textile wastewater in order to remove these substances before being discharged into the environment. Over the past few decades extensive research has been performed concerning dye removal from different wastewaters using chemical and biological treatment technologies or a combination of both. Nevertheless, only little is known about the microbial ecology and microbial communities in biological wastewater treatment plants (WWTPs) treating textile wastewaters, and about the efficiency of these systems to remove recalcitrant dyes. In this PhD thesis, using reactive azo dyes as model components, several aspects were studied contributing to a better understanding of dye degradation and its removal from textile wastewater. Reactive azo dyes are an important group of toxic, recalcitrant textile dyes and represent the majority of all dyes used in the textile industry, and are therefore highly suited for this study. First, a number of available molecular tools were implemented and evaluated to assess the microbial community composition and some important gene functions in activated sludge from (textile) WWTPs (Chapter 2). More particularly, a molecular-ecological toolbox was developed, consisting of quantitative real-time PCR (qPCR) protocols for monitoring abundance of bacterial and archaeal 16S ribosomal RNA (rRNA) genes as well as a number of functional genes involved in nitrogen removal through nitrification/denitrification. Additionally, a protocol based on 454 pyrosequencing of 16S rRNA gene amplicons was developed to assess the archaeal and bacterial community composition in activated sludge systems. Microbial communities of activated sludge in WWTPs have been profoundly studied over the past decade. However, despite these efforts still little is known about the microbial community composition and their functioning in activated sludge from textile wastewater treatment systems. Therefore, the aim of Chapter 3 was to study the microbial community in activated sludge from well-operating textile WWTPs in comparison with municipal WWTPs over two seasons (winter and summer), and to explain observed differences by environmental variables. In total, 454-pyrosequencing of 16S rRNA gene amplicons generated 160 archaeal and 1645 bacterial Operational Taxonomic Units (OTUs, which are a surrogate for species). Results suggested that activated sludge from textile WWTPs harbors a microbial community which is different from those from municipal WWTPs. Both archaeal and bacterial richness were significantly higher for samples from municipal WWTPs compared to those from textile WWTPs. The bacterial phyla Planctomycetes, Chloroflexi, Chlorobi and Acidobacteria were more abundant in activated sludge samples from textile WWTPs, together with archaeal members of Thaumarchaeota. Additionally, sulfate-reducing bacteria were almost only detected in textile WWTPs, while nitrifying and denitrifying bacteria as well as phosphate-accumulation bacteria were more abundant in municipal WWTPs. It became also clear that microbial communities from textile WWTPs were more dissimilar than those of municipal WWTPs, possibly due to a wider diversity in environmental stresses to which microbial communities in textile WWTPs are subjected. High salinity, high organic loads and a higher water temperature were found as important variables driving the microbial community composition in textile WWTPs. In an attempt to assess how microbial communities in textile WWTPs are established, the response of activated sludge microbial communities when exposed to textile dyes was studied. To this end, we assessed the microbial community composition in activated sludge from municipal WWTPs before and after exposure to azo dyes (Reactive Violet 5 (RV5)) (Chapter 4). Molecular analysis revealed that microbial communities that become exposed to recalcitrant azo dyes shift from diverse communities towards less diverse communities harboring highly adapted taxa with azo dye-degrading activity. Many approaches have been proposed to remove dyes from textile wastewaters, including (physico)chemical and biological methods. A combination of a chemical method to obtain partial dye degradation followed by a biological treatment is believed to be a promising method for cost-effective decolorization of colored wastewater. Therefore, the aim of Chapter 4 was to develop and evaluate a combined method of partial Fenton oxidation and biological treatment using activated sludge for decolorization of azo dyes. Using RV5 as a model dye, color removal was significantly higher when the combined Fenton treatment/activated sludge method was used, as opposed to separate application of these treatment technologies. More specifically, pretreatment with Fenton's reagent removed 52.9, 83.9 and 91.3 % of color from a 500 mg l-1 RV5 aqueous solution within 60 min when H2O2 concentrations of 1.0, 1.5, and 2.0 mM were used, respectively. Subsequent biological treatment significantly enhanced the chemical treatment, with microbial decolorization removing 70.2 % of the remaining RV5 concentration, on average. No apparent lag phase was detected when the chemical and biological method were combined, suggesting that the dye compounds have been partially degraded to compounds readily usable by the sludge microorganisms. Instead of combining a biological with chemical treatment technology to enhance purification of textile wastewater, another alternative is the application of microorganisms with dye-degrading capabilities. Therefore, in Chapter 5 bacterial strains capable of decolorizing and/or degrading azo dyes commonly applied in textile production (monoazo dye Reactive Orange 16 and diazo dye Reactive Green 19) were isolated and characterized from activated sludge systems used in the treatment of (textile) wastewater. Following a prescreening of 125 isolates for their decolorization potential, five strains were retained for further evaluation of decolorization rate and effects of physicochemical parameters using a microtiter plate method. Of those five strains, one strain belonging to the genus Acinetobacter (ST16.16/164) and another belonging to Klebsiella (ST16.16/034) outperformed the other tested strains. Interestingly, it was suggested that this Acinetobacter strain represents a novel species, which is closely related to Acinetobacter johnsonii. Both strains exhibited strong decolorization ability (> 80 %) within a wide temperature range (20 °C to 40 °C) and retained good decolorization activity at temperatures as low as 10 °C (especially strain ST16.16/034), offering promising perspectives on a practical level, which requires a stable enzymatic performance of the isolates during the different phases of the purification cycle (thermotolerant). Among the different pH values tested (4, 7 and 10), highest dye removal for both strains occurred at pH 7, with decolorization efficiency remaining relatively high under alkaline conditions (pH 10), and neither isolates decolorization efficiency was negatively impacted by high salt or high dye concentration. Furthermore, both strains displayed the highest rate of decolorization and were able to completely (ST16.16/034) or partly (ST16.16/164) degrade the azo dyes. Altogether, this PhD thesis clearly increased our knowledge on the microbial ecology and microbial communities in textile WWTPs as well as the treatment of textile wastewaters. Eventually, this study should contribute to a more effective, feasible and sustainable treatment of dye contaminated wastewater.Preface – Voorwoord I List of abbreviations III Samenvatting V Summary IX Table of content XIII CHAPTER 1: DECOLORIZATION OF TEXTILE WASTEWATER 1 1.1 Textile wastewater 2 1.2 Textile wastewater treatment methods 8 1.2.1 Physical and physicochemical methods 11 1.2.1.1 Adsorption 11 1.2.1.2 Ion exchange 12 1.2.1.3 Membrane technology 12 1.2.1.4 Irradiation 13 1.2.1.5 Coagulation-Flocculation 14 1.2.1.6 Electrocoagulation 15 1.2.2 Chemical methods 15 1.2.2.1 Chemical oxidation 15 1.2.2.2 Advanced Oxidation Processes 17 1.2.3 Biological methods 22 1.2.3.1 Aerobic, anaerobic and combined anaerobic/aerobic decolorization systems 23 1.2.3.1.1 Aerobic treatment 23 1.2.3.1.2 Anaerobic treatment 24 1.2.3.1.3 Combined anaerobic/aerobic treatment 25 1.2.3.2 Microorganisms involved in the biological degradation of dyes 27 1.2.4 Combination of biological and chemical methods 31 1.3 Goals and objectives of the study 33 CHAPTER 2: IMPLEMENTATION OF METHODS AND PROTOCOLS TO STUDY THE COMPOSITION AND FUNCTION OF MICROBIAL COMMUNITIES IN TEXTILE WASTEWATER TREATMENT PLANTS 37 2.1 Introduction 38 2.2 Materials and methods 40 2.2.1 Study samples and DNA extraction 40 2.2.2 Microbial community characterization 40 2.2.3 Quantification of microbial groups and functional genes 43 2.3 Results and discussion 44 2.3.1 Characterization of activated sludge microbial communities using 454 pyrosequencing 44 2.3.2 Quantification of microbial groups and functional genes in activated sludge 53 CHAPTER 3: ASSESSING THE COMPOSITION OF MICROBIAL COMMUNITIES IN TEXTILE WASTEWATER TREATMENT PLANTS IN COMPARISON WITH MUNICIPAL WASTEWATER TREATMENT PLANTS* 59 3.1 Introduction 60 3.2 Materials and methods 61 3.2.1 Study samples 61 3.2.2 Microbial community characterization 62 3.2.3 Real-time quantitative PCR 63 3.2.4 Chemical analyses 63 3.2.5 Data analysis 64 3.3 Results and discussion 65 3.3.1 Archaeal and bacterial community composition 65 3.3.2 Environmental factors explaining differences in microbial communities 79 CHAPTER 4: DECOLORIZATION OF REACTIVE AZO DYES USING A SEQUENTIAL CHEMICAL AND ACTIVATED SLUDGE TREATMENT* 83 4.1 Introduction 84 4.2 Materials and methods 85 4.2.1 Reagents 85 4.2.2 Fenton oxidation 86 4.2.3 Microbial decolorization 87 4.2.4 Sequential Fenton oxidation and microbial decolorization 88 4.2.5 Microbial community characterization 88 4.3 Results and discussion 89 4.3.1 Chemical decolorization of Reactive Violet 5 89 4.3.2 Microbial decolorization of Reactive Violet 5 using activated sludge 91 4.3.3 A combination of Fenton’s reagent and activated sludge performs better than a single chemical and single activated sludge treatment for decolorization of Reactive Violet 5 92 4.3.4 Microbial community composition of activated sludge changes when exposed to Reactive Violet 5 95 CHAPTER 5: ISOLATION AND SCREENING OF BACTERIAL ISOLATES FROM WASTEWATER TREATMENT PLANTS TO DECOLORIZE AZO DYES* 101 5.1 Introduction 102 5.2 Materials and methods 104 5.2.1 Sampling, bacterial isolation and identification 104 5.2.2 Screening azo dye decolorizing bacterial isolates 105 5.2.3 Impact of physicochemical parameters 107 5.2.4 Analysis of dye decolorization and degradation 108 5.3 Results and discussion 108 5.3.1 Isolation, identification and screening of potential azo dye decolorizing bacteria 108 5.3.2 Effects of physicochemical parameters 116 5.3.2.1 Carbon and nitrogen sources 116 5.3.2.2 Temperature and pH 119 5.3.2.3 Dye concentration and increased salt concentration 121 5.3.3 UV-VIS analysis 122 CHAPTER 6: GENERAL CONCLUSION AND PERSPECTIVES 127 6.1 Main results of this study 128 6.2 Research perspectives 135 6.3 Implementation of research results 139 References 141 Supplementary files 163 Publication list 223nrpages: 242status: publishe