Approximation of Matrix Functions Arising in Physics and Network Science: Theoretical and Computational Aspects

Many applications in physics and network science require the computation of quantities related to certain matrix functions. In many cases, a straightforward way to proceed is by diagonalization. However, the cost of this method becomes prohibitive for large dimensions. We aim to provide techniques with much lower computational cost that exploit the (approximate) sparse structure of the matrices involved. Spectral projectors associated with Hermitian matrices play a key role in applications such as electronic structure computations. Linear scaling methods in the gapped case are based on the fact that the entries of such projectors decay rapidly with respect to some sparsity patterns. The relation with the sign function, together with an integral representation of the latter, is used to obtain asymptotically optimal decay bounds. The influence of isolated extremal eigenvalues on the decay properties is also investigated. Using similar techniques, we extend our results to related matrix functions. Another problem of interest is the computation of the trace of a matrix function. In certain situations, combining probing methods based on graph colorings with stochastic trace estimation techniques can yield accurate approximations at moderate cost. Until recently, such methods had not been thoroughly analyzed, however, but were rather used as efficient heuristics by practitioners. We perform a detailed analysis of stochastic probing methods and, in particular, expose conditions under which the expected approximation error in the stochastic probing method scales more favorably with the dimension of the matrix than the error in non-stochastic probing. A quantity that often appears in quantum physics and network science is the von Neumann entropy, expressed as the trace of a matrix function. Probing techniques or stochastic trace estimators can be used to obtain approximations of the entropy. The computation of the several quadratic forms and matrix-vector products involved can be carried efficiently by using polynomial and rational Krylov subspace methods. For the probing approach, theoretical bounds and heuristic estimates are provided for the error on the entropy, which can be used to select the number of quadratic forms required to reach a certain accuracy. Moreover, a posteriori error bounds are given for the Krylov approximations. Our results are validated by several numerical experiments on a number of test problems arising in network analysis

Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods

We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix $A$, defined as $\operatorname{tr}(f(A))$ where $f(x)=-x\log x$. After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.Comment: 32 pages, 10 figure

Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices

We consider the problem of estimating the trace of a matrix function $f(A)$. In certain situations, in particular if $f(A)$ cannot be well approximated by a low-rank matrix, combining probing methods based on graph colorings with stochastic trace estimation techniques can yield accurate approximations at moderate cost. So far, such methods have not been thoroughly analyzed, though, but were rather used as efficient heuristics by practitioners. In this manuscript, we perform a detailed analysis of stochastic probing methods and, in particular, expose conditions under which the expected approximation error in the stochastic probing method scales more favorably with the dimension of the matrix than the error in non-stochastic probing. Extending results from [E. Aune, D. P. Simpson, J. Eidsvik, Parameter estimation in high dimensional Gaussian distributions, Stat. Comput., 24, pp. 247--263, 2014], we also characterize situations in which using just one stochastic vector is always -- not only in expectation -- better than the deterministic probing method. Several numerical experiments illustrate our theory and compare with existing methods

Refined decay bounds on the entries of spectral projectors associated with sparse Hermitian matrices

Spectral projectors of Hermitian matrices play a key role in many applications, such as electronic structure computations. Linear scaling methods for gapped systems are based on the fact that these special matrix functions are localized, which means that the entries decay rapidly away from the main diagonal or with respect to more general sparsity patterns. The relation with the sign function together with an integral representation is used to obtain new decay bounds, which turn out to be optimal in an asymptotic sense. The influence of isolated extremal eigenvalues on the decay properties is also investigated and a superexponential behaviour is predicted

Toeplitz-like structures and matrix equations encountered in stochastic modeling

In this thesis, we study methods for computing the invariant probability measure for certain quasi-birth-and-death Markov chains. In particular, we consider the situation of a double QBD, whose transition matrix has the infinite, block-tridiagonal and almost block-Toeplitz form of a QBD, with blocks which are infinite, tridiagonal and almost Toeplitz. This class includes classic situations like the reflected random walk in the quarter plane and the tandem Jackson queue. For QBDs, the problem of finding the invariant measure can be reduced to computing the minimal non-negative solution of quadratic matrix equations. A lot of literature exists about it in the case of finite-dimensional blocks. For the infinite-dimensional case, recently has been introduced an approach based quasi-Toeplitz matrices, which relies on solving matrix equations in a suitable Banach algebra. Other techniques exist. For example, the "truncation and augmentation" method by Taylor and Latouche, and the compensation approach by Adan. The aim of this work is to describe the approach based on quasi-Toeplitz matrices and to improve the existing theory in order to handle problematic cases. A description of the algebra of quasi-Toeplitz matrices is given, with attention to computational issues like the effects of truncation and arithmetic operations. The relation between a suitable factorization of a Laurent matrix polynomial and the solutions of matrix equations is shown. This allows us to prove the quadratic convergence of a classic algorithm, the cyclic reduction. Conditions for the solutions to be quasi-Toeplitz matrices are given, with attention to problematic cases in which this does not happen. To deal with these cases, we introduce a new matrix algebra which extends the class of Quasi-Toeplitz matrices and contains solutions of problematic cases, and we use a shifting technique to solve equations in the extended class

Ectodermal Dysplasia-Syndactyly Syndrome with Toe-Only Minimal Syndactyly Due to a Novel Mutation in NECTIN4: A Case Report and Literature Review

Ectodermal dysplasia-syndactyly syndrome 1 (EDSS1) is characterized by cutaneous syndactyly of the toes and fingers and abnormalities of the hair and teeth, variably associated with nail dystrophy and palmoplantar keratoderma (PPK). EDSS1 is caused by biallelic mutations in the NECTIN4 gene, encoding the adherens junction component nectin-4. Nine EDSS1 cases have been described to date. We report a 5.5-year-old female child affected with EDSS1 due to the novel homozygous frameshift mutation c.1150delC (p.Gln384ArgfsTer7) in the NECTIN4 gene. The patient presents brittle scalp hair, sparse eyebrows and eyelashes, widely spaced conical teeth and dental agenesis, as well as toenail dystrophy and mild PPK. She has minimal proximal syndactyly limited to toes 2–3, which makes the phenotype of our patient peculiar as the overt involvement of both fingers and toes is typical of EDSS1. All previously described mutations are located in the nectin-4 extracellular portion, whereas p.Gln384ArgfsTer7 occurs within the cytoplasmic domain of the protein. This mutation is predicted to affect the interaction with afadin, suggesting that impaired afadin activation is sufficient to determine EDSS1. Our case, which represents the first report of a NECTIN4 mutation with toe-only minimal syndactyly, expands the phenotypic and molecular spectrum of EDSS1

Double-outlet left ventricle: single-center experience and literature review

Double-outlet left ventricle (DOLV) is an abnormal ventriculo-arterial connection characterized by origin of both great arteries, or more than 50% of each arterial root, from the morphological left ventricle. The aim of our paper is to describe the anatomic, echocardiographic, and multi-modality imaging characteristics of DOLV and associated malformations, and to assess its surgical outcomes. Methods: From 2011 to 2022, we retrospectively reviewed case records, intra-operatory reports and follow-up data of patients diagnosed with DOLV at Bambino Gesu Children’s Hospital. A systematic search was developed in MEDLINE, EMBASE and Web of Science databases, to identify original reports between January 1, 1975 and May 30, 2022, assessing the morphology and surgical outcomes of DOLV. Retrospective cohort studies, cross-sectional and case series were included in the analysis. Single case reports and reviews were excluded. Results: At our center, four cases of DOLV were identified. Patient 1 was diagnosed with (S,D,D) DOLV and hypoplastic right ventricle. The aorta overrode a large, doubly-committed VSD with absence of infundibular septum. A tenuous mitro-aortic discontinuity and a well-developed subpulmonary conus were present. Associated abnormalities included crossed pulmonary arteries and two adjacent, side-by-side coronary ostia, located in the anterior facing sinus, which gave origin to the left anterior descending (LAD) and the right coronary artery (RCA). Left circumflex artery (LCx) had a retro-aortic course and originated from the RCA. After pulmonary artery banding, Damus-Kaye-Stansel and Glenn intervention were proposed as first-stage of univentricular palliation. Patient 2 and 3 were diagnosed with (S,D,D) DOLV, subaortic VSD and pulmonary stenosis. Patient 2 underwent Rastelli operation and no anatomic detail were available. Patient 3 showed absence of the infundibular septum and mitro-pulmonary continuity, whereas subaortic conus was well developed. Anomalous origin of the LCx, originating from the posterior facing sinus with retro-aortic course was present. Rastelli procedure was performed to reconstruct right ventricular outflow tract. LCA and RCA were respectively caudal to subvalvular and supravalvular segments of the RV-to-PA conduit. After a 6-years follow-up, severe stenosis of the RV-to-PA conduit was present, nevertheless percutaneous conduit dilatation was contraindicated, due to coronary abnormality, and an aortic homograft was implanted Patient 4 was diagnosed with (S,D,L) DOLV with subaortic VSD and mitro-pulmonary fibrous continuity. A large subaortic conus was present. Reparation à l’etage ventriculaire was performed to reconstruct RVOT. Follow-up MRI at 8 years showed severe pulmonary artery regurgitation with mild RV dilatation (indexed volume 99mL/m2) and normal RV ejection fraction (54 %) Systematic review:Through our systematic research strategy we scrutinized 96 records for inclusion criteria (Figure 4). After systematic evaluation, a total of 9 reports fulfilled eligibility criteria and were included in our study. Morphological findings and surgical outcomes are summarized in Table 1. Among 191 cases of DOLV included, the most common subtypes of VSD were subaortic (128/191), subpulmonary (23/191) or doubly committed (14/191) (Figure 5). d-transposition of the aorta was present in 117/191 (61%) cases, whereas l-transposition was reported in 63/191 (32%) (Figure 6

Impact of hard lockdown on interventional cardiology procedures in congenital heart disease: a survey on behalf of the Italian Society of Congenital Heart Disease

The Coronavirus disease 2019 (COVID-19) pandemic has thoroughly and deeply affected the provision of healthcare services worldwide. In order to limit the in-hospital infections and to redistribute the healthcare professionals, cardiac percutaneous intervention in Pediatric and Adult Congenital Heart Disease (ACHD) patients were limited to urgent or emergency ones. The aim of this article is to describe the impact of the COVID-19 pandemic on Pediatric and ACHD cath laboratory activity during the so-called 'hard lockdown' in Italy. Eleven out of 12 Italian institutions with a dedicated Invasive Cardiology Unit in Congenital Heart Disease actively participated in the survey. The interventional cardiology activity was reduced by more than 50% in 6 out of 11 centers. Adolescent and ACHD patients suffered the highest rate of reduction. There was an evident discrepancy in the management of the hard lockdown, irrespective of the number of COVID-19 positive cases registered, with a higher reduction in Southern Italy compared with the most affected regions (Lombardy, Piedmont, Veneto and Emilia Romagna). Although the pandemic was brilliantly addressed in most cases, we recognize the necessity for planning new, and hopefully homogeneous, strategies in order to be prepared for an upcoming new outbreak