Fast computation of spectral projectors of banded matrices

We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure methods, the presence of small relative spectral gaps challenges existing methods based on approximate sparsity. In this work, we show how a data-sparse approximation based on hierarchical matrices can be used to overcome this problem. We prove a priori bounds on the approximation error and propose a fast algo- rithm based on the QDWH algorithm, along the works by Nakatsukasa et al. Numerical experiments demonstrate that the performance of our algorithm is robust with respect to the spectral gap. A preliminary Matlab implementation becomes faster than eig already for matrix sizes of a few thousand.Comment: 27 pages, 10 figure

Accelerated filtering on graphs using Lanczos method

Signal-processing on graphs has developed into a very active field of research during the last decade. In particular, the number of applications using frames constructed from graphs, like wavelets on graphs, has substantially increased. To attain scalability for large graphs, fast graph-signal filtering techniques are needed. In this contribution, we propose an accelerated algorithm based on the Lanczos method that adapts to the Laplacian spectrum without explicitly computing it. The result is an accurate, robust, scalable and efficient algorithm. Compared to existing methods based on Chebyshev polynomials, our solution achieves higher accuracy without increasing the overall complexity significantly. Furthermore, it is particularly well suited for graphs with large spectral gaps

Effect of Low Selenium Diet on Glutathione Peroxidase 3 Concentration in Male Sprague-Dawley Rats’ Serum

Aim: Determination of antioxidative enzyme glutathione peroxidase 3 (GPx3) serum concentrations after consumption of food which contains different concentrations of selenium (Se). Research subjects and methods: Four-week-old Sprague Dawley rats consumed food containing different concentrations of Se (food Divan) over a period of 10 weeks. The animals were divided into two groups: 1) normal Se (0.363 mg/kg Se) and 2) low Se (0.030 mg/kg Se). Each animal was weighed at the end of protocol, and serum samples were collected for determining GPx3 concentrations. All experimental procedures were in compliance with the European Convention for the Protection of Vertebrate Animals used for Experimental and other Scientific Purposes and approved by the Ethics Committee of the Faculty of Medicine in Osijek and the Ministry of Agriculture of the Republic of Croatia. Results: Different concentrations of Se in food did not cause a change in body weight. Food containing the recommended intake of Se according to the guidelines of the World Health Organization significantly increased GPx3 enzyme concentration (13.96±0.42 mg/ml) when compared to low selective Se (12.04 ± 0.33 mg/ml, p = 0.002). Conclusion: Serum concentration of the antioxidant enzyme GPx3 depends on the concentration of Se in food. It is shown that, in comparison with food with low Se levels, food containing a normal concentration of Se is enriched with the antioxidant GPx3 which, according to numerous studies, has a protective role in the human body

COVID-19 AND THE ENVIRONMENT – THE ROLE OF THE PUBLIC HEALTH INSTITUTE

The Croatian National Health Care Act defines the areas of activities of the public health institute, including the activities of the epidemiology of infectious diseases and chronic non-communicable diseases, public health, health promotion, environmental health, microbiology, school and adolescent medicine, mental health and addiction prevention at Zagreb City level. This paper reviews the highly variable activities in the Andrija Štampar Teaching Institute of Public Health with the aim of promoting a comprehensive approach to the COVID-19 pandemic. Human and analytical resources in the Institute, activities and rapid implementation of innovations testify to the high capacities for adaptation to emerging risks. In the Institute, it is possible to carry out a whole range of tests and to monitor the environmental factors with predominant impact on human health and safety of the Zagreb environment. The supply of safe water for human consumption in the Republic of Croatia during the current COVID-19 crisis has been uninterrupted and in accordance with applicable legislation. Also, our laboratories have been developing and introducing a method for wastewater testing for SARS-CoV-2 presence. The sludge from wastewater treatment plants is used in agriculture, and potential risks associated with the COVID-19 outbreak should be assessed prior to each application on the soil. Increased use of disinfectants during the epidemic may present a higher risk to the aquatic environment. Air quality monitoring indicates a positive impact on air quality as result of isolation measures

Fast hierarchical solvers for symmetric eigenvalue problems

In this thesis we address the computation of a spectral decomposition for symmetric banded matrices. In light of dealing with large-scale matrices, where classical dense linear algebra routines are not applicable, it is essential to design alternative techniques that take advantage of data properties. Our approach is based upon exploiting the underlying hierarchical low-rank structure in the intermediate results. Indeed, we study the computation in the hierarchically off-diagonal low rank (HODLR) format of two crucial tools: QR decomposition and spectral projectors, in order to devise a fast spectral divide-and-conquer method. In the first part we propose a new method for computing a QR decomposition of a HODLR matrix, where the factor R is returned in the HODLR format, while Q is given in a compact WY representation. The new algorithm enjoys linear-polylogarithmic complexity and is norm-wise accurate. Moreover, it maintains the orthogonality of the Q factor. The approximate computation of spectral projectors is addressed in the second part. This problem has raised some interest in the context of linear scaling electronic structure methods. There the presence of small spectral gaps brings difficulties to existing algorithms based on approximate sparsity. We propose a fast method based on a variant of the QDWH algorithm, and exploit that QDWH applied to a banded input generates a sequence of matrices that can be efficiently represented in the HODLR format. From the theoretical side, we provide an analysis of the structure preservation in the final outcome. More specifically, we derive a priori decay bounds on the singular values in the off-diagonal blocks of spectral projectors. Consequently, this shows that our method, based on data-sparsity, brings benefits in terms of memory requirements in comparison to approximate sparsity approaches, because of its logarithmic dependence on the spectral gap. Numerical experiments conducted on tridiagonal and banded matrices demonstrate that the proposed algorithm is robust with respect to the spectral gap and exhibits linear-polylogarithmic complexity. Furthermore, it renders very accurate approximations to the spectral projectors even for very large matrices. The last part of this thesis is concerned with developing a fast spectral divide-and-conquer method in the HODLR format. The idea behind this technique is to recursively split the spectrum, using invariant subspaces associated with its subsets. This allows to obtain a complete spectral decomposition by solving the smaller-sized problems. Following Nakatsukasa and Higham, we combine our method for the fast computation of spectral projectors with a novel technique for finding a basis for the range of such a HODLR matrix. The latter strongly relies on properties of spectral projectors, and it is analyzed theoretically. Numerical results confirm that the method is applicable for large-scale matrices, and exhibits linear-polylogarithmic complexity

MATHICSE Technical Report : Fast QR decomposition of HODLR matrices

The efficient and accurate QR decomposition for matrices with hierarchical low-rank structures, such as HODLR and hierarchical matrices, has been challenging. Existing structure-exploiting algorithms are prone to numerical instability as they proceed indi- rectly, via Cholesky decompositions or a block Gram-Schmidt procedure. For a highly ill-conditioned matrix, such approaches either break down in finite-precision arithmetic or result in significant loss of orthogonality. Although these issues can sometimes be addressed by regularization and iterative refinement, it would be more desirable to have an algorithm that avoids these detours and is numerically robust to ill-conditioning. In this work, we propose such an algorithm for HODLR matrices. It achieves accuracy by utilizing House- holder reflectors. It achieves efficiency by utilizing fast operations in the HODLR format in combination with compact WY representations and the recursive QR decomposition by Elmroth and Gustavson. Numerical experiments demonstrate that our newly proposed al- gorithm is robust to ill-conditioning and capable of achieving numerical orthogonality down to the level of roundoff error

MATHICSE Technical Report : A fast spectral divide-and-conquer method for banded matrices

Based on the spectral divide-and-conquer algorithm by Nakatsukasa and Higham [SIAM J. Sci. Comput., 35(3):A1325{A1349, 2013], we propose a new algorithm for computing all the eigenvalues and eigenvectors of a symmetric banded matrix. For this purpose, we combine our previous work on the fast computation of spectral projectors in the so called HODLR format, with a novel technique for extracting a basis for the range of such a HODLR matrix. The numerical experiments demonstrate that our algorithm exhibits quasilinear complexity and allows for conveniently dealing with large-scale matrices

A fast spectral divide-and-conquer method for banded matrices

Based on the spectral divide-and-conquer algorithm by Nakatsukasa and Higham [SIAM J. Sci. Comput., 35(3):A1325-A1349, 2013], we propose a new algorithm for computing all the eigenvalues and eigenvectors of a symmetric banded matrix with small bandwidth, with the eigenvectors given implicitly as a product of orthonormal matrices stored in the so-called hierarchically off-diagonal low-rank (HODLR) format. For this purpose, we combine our previous work on the fast computation of spectral projectors in the HODLR format, with a novel technique for extracting a basis for the range of such a HODLR matrix. Preliminary numerical experiments demonstrate that our algorithm exhibits quasi-linear complexity for matrices that can be efficiently represented in the HODLR format throughout the divide-and-conquer algorithm, and allows for conveniently dealing with such large-scale matrices

MATHICSE Technical Report : Fast computation of spectral projectors of banded matrices

We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure methods, the presence of small relative spectral gaps challenges existing methods based on approximate sparsity. In this work, we show how a data-sparse approximation based on hierarchical matrices can be used to overcome this problem. We prove a priori bounds on the approximation error and propose a fast algo-rithm based on the QDWH algorithm, along the works by Nakatsukasa et al. Numerical experiments demonstrate that the performance of our algorithm is robust with respect to the spectral gap. A preliminary MATLAB implementation becomes faster than eig already for matrix sizes of a few thousand