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.
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.
Regularni politopi Berljafa, Mario; Šušnjara, Ana
Math.e,
12/2010, Letnik:
18
Paper
Odprti dostop
U ovom članku promatramo dvodimenzionalne, trodimenzionalne i višedimenzionalne regularne politope. Pozabavili smo se ponajprije njihovom egzistencijom, konstrukcijom te svojstvima koja zadovoljavaju ...njihovi elementi. Najvažniji teorem članka, poznat pod nazivom Euler–Poincaréova formula, nalazi se u šestom poglavlju te povezuje broj svih elemenata n–dimenzionalnog politopa.
http://e.math.hr/math_e_article/br18/berljafa_susnjara
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 indirectly, 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 Householder 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 algorithm is robust to ill-conditioning and capable of achieving numerical orthogonality down to the level of roundoff error.
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.
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.
Recent relevant safety guidelines IEEE-Std C95.1- 2019 and ICNIRP-RF Guidelines 2020 have converged towards 6 GHz as a transition frequency from specific absorption rate (SAR), as basic restriction ...quantity, to absorbed power density (APD). Namely, the penetration of electromagnetic waves into the human tissue rapidly decreases as frequency increases, therefore, tissue heating can be considered as superficial above 6 GHz. However, besides the APD, an alternative internal dosimetric quantity transmitted power density or TPD is sometimes computed since its relation to SAR is more obvious and is easier to obtain. This paper deals with an analytical/numerical approach to determine TPD in planar multi-layered model of the human tissue exposed to the dipole antenna radiation. Analytical approach deals with assumed sinusoidal current distribution, while numerical approach pertains to the determination of current by solving the corresponding Pocklington integro-differential equation via Galerkin-Bubnov Indirect Boundary Element Method. The novelty presented in this paper with respect to previous work is a multilayer geometry whose effects are considered via the corresponding Fresnel plane wave reflection/transmission approximation. Some illustrative results for current distribution, transmitted field, volume power density (VPD) and TPD at various frequencies and distances of the antenna from the interface are given.
This study aimed to investigate the specific role of nitric oxide (NO) in micro- and macrovascular response to a 7-day high-salt (HS) diet, specifically by measuring skin microvascular local thermal ...hyperemia and the flow-mediated dilation of the brachial artery, as well as serum NO and three NO synthase enzyme (NOS) isoform concentrations in healthy individuals. It also aimed to examine the concept of non-osmotic sodium storage in the skin following the HS diet by measuring body fluid status and systemic hemodynamic responses, as well as serum vascular endothelial growth factor C (VEGF-C) concentration. Forty-six young, healthy individuals completed a 7-day low-salt diet, followed by a 7-day HS diet protocol. The 7-day HS diet resulted in impaired NO-mediated endothelial vasodilation in peripheral microcirculation and conduit arteries, in increased eNOS, decreased nNOS, and unchanged iNOS concentration and NO serum level. The HS diet did not change the volume of interstitial fluid, the systemic vascular resistance or the VEGF-C serum level. These results indicate that the 7-day HS-diet induces systemic impairment of NO-mediated endothelial vasodilation, while dissociation in the eNOS and nNOS response indicates complex adaptation of main NO-generating enzyme isoforms to HS intake in healthy individuals. Our results failed to support the concept of non-osmotic sodium storage.
The effects of consumption of n-3 polyunsaturated fatty acids (n-3 PUFAs) enriched hen eggs on endothelium-dependent and endothelium-independent vasodilation in microcirculation, and on endothelial ...activation and inflammation were determined in young healthy individuals. Control group (
= 21) ate three regular hen eggs/daily (249 mg n-3 PUFAs/day), and n-3 PUFAs group (
= 19) ate three n-3 PUFAs enriched hen eggs/daily (1053 g n-3 PUFAs/day) for 3 weeks. Skin microvascular blood flow in response to iontophoresis of acetylcholine (AChID; endothelium-dependent) and sodium nitroprusside (SNPID; endothelium-independent) was assessed by laser Doppler flowmetry. Blood pressure (BP), body composition, body fluid status, serum lipid and free fatty acids profile, and inflammatory and endothelial activation markers were measured before and after respective dietary protocol. Results: Serum n-3 PUFAs concentration significantly increased, AChID significantly improved, and SNPID remained unchanged in n-3 PUFAs group, while none was changed in Control group. Interferon-γ (pro-inflammatory) significantly decreased and interleukin-10 (anti-inflammatory) significantly increased in n-3 PUFAs. BP, fat free mass, and total body water significantly decreased, while fat mass, interleukin-17A (pro-inflammatory), interleukin-10 and vascular endothelial growth factor A significantly increased in the Control group. Other measured parameters remained unchanged in both groups. Favorable anti-inflammatory properties of n-3 PUFAs consumption potentially contribute to the improvement of microvascular endothelium-dependent vasodilation in healthy individuals.
This study aimed to investigate the impact of dietary salt intake during normal pregnancy on maternal microvascular and macrovascular endothelium-dependent reactivity and oxidative stress level.
In ...this cross-sectional study, based on their 24-h urinary sodium excretion, pregnant women (37-40 weeks of gestation) were divided into three groups: normal salt (<5.75 g/day, N = 12), high salt (5.75-10.25 g/day, N = 36), and very high salt (VHS;>10.25 g/day, N = 17). Forearm skin microvascular reactivity in response to vascular occlusion, local heating (LTH) and iontophoresis of acetylcholine (AChID), as well as brachial artery flow mediated dilation (FMD) were measured. Serum nitric oxide, endocan, 8-iso-prostaglandin F2α (8-iso-PGF2α), thiobarbituric acid reactive substances (TBARS), and ferric-reducing ability of plasma assay were measured as biomarkers of endothelial function/activation and oxidative stress.
Brachial artery FMD, microvascular AChID, and LTH were significantly decreased in VHS compared with NS group, while LTH was also decreased in normal salt compared with high salt group. Nitric oxide was significantly decreased in both high salt and VHS groups compared with normal salt. Endocan, 8-iso-PGF2α, and TBARS were significantly increased in VHS compared with the normal salt group.
High dietary salt intake is associated with decreased nitric oxide mediated endothelium-dependent vasodilation in peripheral microcirculation and macrocirculation of healthy pregnant women due to increased oxidative stress.
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