Rational Krylov for Stieltjes matrix functions: convergence and pole selection

Evaluating the action of a matrix function on a vector, that is x= f(M) v, is an ubiquitous task in applications. When M is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating x when f(z) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent, completely monotonic) and M is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case M= I⊗ A- BT⊗ I, and v obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of x. Pole selection strategies with explicit convergence bounds are given also in this case

Hierarchical adaptive low-rank format with applications to discretized partial differential equations

A novel framework for hierarchical low-rank matrices is proposed that combines an adaptive hierarchical partitioning of the matrix with low-rank approximation. One typical application is the approximation of discretized functions on rectangular domains; the flexibility of the format makes it possible to deal with functions that feature singularities in small, localized regions. To deal with time evolution and relocation of singularities, the partitioning can be dynamically adjusted based on features of the underlying data. Our format can be leveraged to efficiently solve linear systems with Kronecker product structure, as they arise from discretized partial differential equations (PDEs). For this purpose, these linear systems are rephrased as linear matrix equations and a recursive solver is derived from low-rank updates of such equations. We demonstrate the effectiveness of our framework for stationary and time-dependent, linear and nonlinear PDEs, including the Burgers' and Allen-Cahn equations

Compress-and-Restart Block Krylov Subspace Methods for Sylvester Matrix Equations

Block Krylov subspace methods (KSMs) comprise building blocks in many state-of-the-art solvers for large-scale matrix equations as they arise, for example, from the discretization of partial differential equations. While extended and rational block Krylov subspace methods provide a major reduction in iteration counts over polynomial block KSMs, they also require reliable solvers for the coefficient matrices, and these solvers are often iterative methods themselves. It is not hard to devise scenarios in which the available memory, and consequently the dimension of the Krylov subspace, is limited. In such scenarios for linear systems and eigenvalue problems, restarting is a well-explored technique for mitigating memory constraints. In this work, such restarting techniques are applied to polynomial KSMs for matrix equations with a compression step to control the growing rank of the residual. An error analysis is also performed, leading to heuristics for dynamically adjusting the basis size in each restart cycle. A panel of numerical experiments demonstrates the effectiveness of the new method with respect to extended block KSMs

A computational framework for two-dimensional random walks with restarts

The treatment of two-dimensional random walks in the quarter plane leads to Markov processes which involve semi-infinite matrices having Toeplitz or block Toeplitz structure plus a low-rank correction. We propose an extension of the framework introduced in [D. A. Bini, S. Massei, and B. Meini, Math. Comp., 87 (2018), pp. 2811-2830] which allows us to deal with more general situations such as processes involving restart events. This is motivated by the need for modeling processes that can incur in unexpected failures like computer system reboots. We present a theoretical analysis of an enriched Banach algebra that, combined with appropriate algorithms, enables the numerical treatment of these problems. The results are applied to the solution of bidimensional quasi-birth-death processes with infinitely many phases which model random walks in the quarter plane, relying on the matrix analytic approach. The reliability of our approach is confirmed by extensive numerical experimentation on several case studies

Camera traps and activity signs to estimate density and population trends in wild pigs

Massei, G., Cowan, D., Lambert, M., Coats, J., Watola, G., Fox, S., Ward, A., Pietravalle, S

Unusual onset of a case of chronic recurrent multifocal osteomyelitis

Background: Chronic recurrent multifocal osteomyelitis (CRMO) is a rare condition that commonly affects the clavicle and pelvis. Case presentation: We report here a case a 12 years old girl with CRMO arising with recurrent episodes of left supraorbital headache, followed by the appearance of a periorbital dyschromia. Magnetic resonance imaging (MRI) of the skull and orbits revealed an important subacute inflammatory process. Few months after, the child presented a painful swelling of the left clavicle; the histological examination of the related biopsy allowed to establish the diagnosis of CRMO. Conclusion: CRMO presenting as acute headache involving neurocranium is rare; to our knowledge this is the first recognized case in the world literature. This pathological condition is frequently misdiagnosed as infection or neoplasm and needs a deep investigation for the differential diagnosis. The physical, laboratoristic and instrumental diagnostic investigations of the patient and the treatment employed are described in detail

Research Priorities for Managing Invasive Wild Pigs in North America

With recent increases in distribution and numbers of feral pigs (Sus scrofa; invasive wild pigs) in North America, there has been a concurrent increase in the ecological and economic effects they have had on native and anthropogenic ecosystems. Despite the amplified interest in invasive wild pig research, there remains a significant knowledge gap regarding their basic biology and ecology, the scope of the damage they cause, and the efficacy of many control strategies. Such information is important to support the successful management of invasive wild pigs throughout North America and other areas. In 2016, members of the National Wild Pig Task Force met and developed a set of research priorities to aid in effective management of invasive wild pigs. These research priorities identify 4 topical areas where increased effort and science is most needed to manage invasive wild pigs: biology and ecology, economic and ecological damages, control strategies, and education and human dimensions, with particular emphasis on areas where specific data gaps remain within each topical area. Resolution of such knowledge deficits would advance the understanding of invasive wild pig ecology, enabling more efficient and effective management of this species

Fertility control for managing free-roaming feral cattle in Hong Kong

Human-wildlife conflicts are increasing worldwide. For instance, growing numbers of free-roaming feral cattle in Hong Kong are causing traffic accidents and damaging crops. Public antipathy towards lethal methods to manage wildlife has promoted research into alternative options, such as fertility control. The aims of this study were to assess the potential side effects and effectiveness of the injectable immunocontraceptive vaccine GonaCon on free-roaming feral cattle in Hong Kong. Sixty female cattle were captured and randomly assigned to treatment or control groups. Treatment animals were administered one dose of GonaCon, followed by a booster dose 3–6 months later. Control animals were administered an equivalent dose of a saline solution. The side effects of GonaCon were assessed by monitoring injection site, body condition and body weight at vaccination, at the booster stage and one year after initial vaccination. At the same times, blood samples were collected to quantify antibodies to the vaccine and to assess pregnancy status. GonaCon did not affect the body weight or body condition of cattle and had no adverse side effects such as injection site reactions, limping or abnormal behaviour. GonaCon did not appear to interrupt ongoing pregnancies but reduced fertility significantly: the proportion of pregnant animals in the GonaCon-treated group decreased from 76% at initial vaccination to 6% one year after vaccination, compared to 67% and 57% respectively in the control group. There was no difference between antibody titres at the booster stage or one year post vaccination, suggesting the booster dose maintained antibody levels. This study confirmed that GonaCon is safe and effective in inducing infertility in feral cattle, with a booster dose critical for maintaining infertility

Arctic hydroclimate variability during the last 2000 years : current understanding and research challenges

Reanalysis data show an increasing trend in Arctic precipitation over the 20th century, but changes are not homogenous across seasons or space. The observed hydro-climate changes are expected to continue and possibly accelerate in the coming century, not only affecting pan-Arctic natural ecosystems and human activities, but also lower latitudes through the atmospheric and ocean circulations. However, a lack of spatiotemporal observational data makes reliable quantification of Arctic hydroclimate change difficult, especially in a long-term context. To understand Arctic hydroclimate and its variability prior to the instrumental record, climate proxy records are needed. The purpose of this review is to summarise the current understanding of Arctic hydroclimate during the past 2000 years. First, the paper reviews the main natural archives and proxies used to infer past hydroclimate variations in this remote region and outlines the difficulty of disentangling the moisture from the temperature signal in these records. Second, a comparison of two sets of hydroclimate records covering the Common Era from two data-rich regions, North America and Fennoscandia, reveals inter- and intra-regional differences. Third, building on earlier work, this paper shows the potential for providing a high-resolution hydroclimate reconstruction for the Arctic and a comparison with last-millennium simulations from fully coupled climate models. In general, hydroclimate proxies and simulations indicate that the Medieval Climate Anomaly tends to have been wetter than the Little Ice Age (LIA), but there are large regional differences. However, the regional coverage of the proxy data is inadequate, with distinct data gaps in most of Eurasia and parts of North America, making robust assessments for the whole Arctic impossible at present. To fully assess pan-Arctic hydroclimate variability for the last 2 millennia, additional proxy records are required.Peer reviewe

Changes in the variability and periodicity of precipitation in Scotland

This paper analyses the temporal and spatial changes in the amount and variability of rainfall in Scotland. The sequential Mann–Kendall test reveals that total annual precipitation has increased across Scotland since the 1970s with increasing trends in variability beginning between the mid-1960s and the mid-1970s. Whilst temporally consistent increasing trends in precipitation totals prevail in the West, many weather stations in the East have experienced subsequent trend turning points in the following two decades, explaining the larger magnitude of the trends in western Scotland in recent decades. Trend analyses on six measures of rainfall variability indicate an increase in rainfall variability during the period 1961–2000, as measured by the intra-annual variance, the winter to summer precipitation ratio and the annual cumulative sum range, with decreasing trends observed in the number of dry days. Periodicities associated with the North Atlantic Oscillation and the Atlantic Multidecadal Oscillation could explain the observed temporal variability of rainfall