Fast and accurate con-eigenvalue algorithm for optimal rational approximations

The need to compute small con-eigenvalues and the associated con-eigenvectors of positive-definite Cauchy matrices naturally arises when constructing rational approximations with a (near) optimally small $L^{\infty}$ error. Specifically, given a rational function with $n$ poles in the unit disk, a rational approximation with $m\ll n$ poles in the unit disk may be obtained from the $m$th con-eigenvector of an $n\times n$ Cauchy matrix, where the associated con-eigenvalue $\lambda_{m}>0$ gives the approximation error in the $L^{\infty}$ norm. Unfortunately, standard algorithms do not accurately compute small con-eigenvalues (and the associated con-eigenvectors) and, in particular, yield few or no correct digits for con-eigenvalues smaller than the machine roundoff. We develop a fast and accurate algorithm for computing con-eigenvalues and con-eigenvectors of positive-definite Cauchy matrices, yielding even the tiniest con-eigenvalues with high relative accuracy. The algorithm computes the $m$th con-eigenvalue in $\mathcal{O}(m^{2}n)$ operations and, since the con-eigenvalues of positive-definite Cauchy matrices decay exponentially fast, we obtain (near) optimal rational approximations in $\mathcal{O}(n(\log\delta^{-1})^{2})$ operations, where $\delta$ is the approximation error in the $L^{\infty}$ norm. We derive error bounds demonstrating high relative accuracy of the computed con-eigenvalues and the high accuracy of the unit con-eigenvectors. We also provide examples of using the algorithm to compute (near) optimal rational approximations of functions with singularities and sharp transitions, where approximation errors close to machine precision are obtained. Finally, we present numerical tests on random (complex-valued) Cauchy matrices to show that the algorithm computes all the con-eigenvalues and con-eigenvectors with nearly full precision

High-Order Mixed Finite Element Variable Eddington Factor Methods

We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the Discrete Ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory. Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations

Are we responding effectively to bone mineral density loss and fracture risks in people with epilepsy?

© 2020 The Authors. Epilepsia Open published by Wiley Periodicals Inc. on behalf of International League Against Epilepsy. Objective: A 2007 study performed at Montefiore Medical Center (Bronx, NY) identified high prevalence of reduced bone density in an urban population of patients with epilepsy and suggested that bone mineralization screenings should be regularly performed for these patients. We conducted a long-term follow-up study to determine whether bone mineral density (BMD) loss, osteoporosis, and fractures have been successfully treated or prevented. Methods: In the current study, patients from the 2007 study who had two dual-energy absorptiometry (DXA) scans performed at least 5 years apart were analyzed. The World Health Organization (WHO) criteria to diagnose patients with osteopenia or osteoporosis were used, and each patient\u27s probability of developing fractures was calculated with the Fracture Risk Assessment Tool (FRAX). Results: The median time between the first and second DXA scans for the 81 patients analyzed was 9.4 years (range 5-14.7). The median age at the first DXA scan was 41 years (range 22-77). Based on WHO criteria, 79.0% of patients did not have worsening of bone density, while 21.0% had new osteopenia or osteoporosis; many patients were prescribed treatment for bone loss. Older age, increased duration of anti-epileptic drug (AED) usage, and low body mass index (BMI) were risk factors for abnormal BMDs. Based on the first DXA scan, the FRAX calculator estimated that none of the patients in this study had a 10-year risk of more than 20% for developing major osteoporotic fracture (hip, spine, wrist, or humeral fracture). However, in this population, 11 patients (13.6%) sustained a major osteoporotic fracture after their first DXA scan. Significance: Despite being routinely screened and frequently treated for bone mineral density loss and fracture prevention, many patients with epilepsy suffered new major osteoporotic fractures. This observation is especially important as persons with epilepsy are at high risk for falls and traumas