An algebraic multigrid method for Q2−Q1Q_2-Q_1 mixed discretizations of the Navier-Stokes equations

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Specifically, we investigate a $Q_2-Q_1$ mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches leveraging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity dof relationships of the $Q_2-Q_1$ discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the finest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa

A first-in-human study of AMG 208, an oral MET inhibitor, in adult patients with advanced solid tumors.

BackgroundThis first-in-human study evaluated AMG 208, a small-molecule MET inhibitor, in patients with advanced solid tumors.MethodsThree to nine patients were enrolled into one of seven AMG 208 dose cohorts (25, 50, 100, 150, 200, 300, and 400 mg). Patients received AMG 208 orally on days 1 and days 4-28 once daily. The primary objectives were to evaluate the safety, tolerability, pharmacokinetics, and maximum tolerated dose (MTD) of AMG 208.ResultsFifty-four patients were enrolled. Six dose-limiting toxicities were observed: grade 3 increased aspartate aminotransferase (200 mg), grade 3 thrombocytopenia (200 mg), grade 4 acute myocardial infarction (300 mg), grade 3 prolonged QT (300 mg), and two cases of grade 3 hypertension (400 mg). The MTD was not reached. The most frequent grade ≥3 treatment-related adverse event was anemia (n = 3) followed by hypertension, prolonged QT, and thrombocytopenia (two patients each). AMG 208 exposure increased linearly with dose; mean plasma half-life estimates were 21.4-68.7 hours. One complete response (prostate cancer) and three partial responses (two in prostate cancer, one in kidney cancer) were observed.ConclusionsIn this study, AMG 208 had manageable toxicities and showed evidence of antitumor activity, particularly in prostate cancer

Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methods

Large linear systems with sparse, non-symmetric matrices arise in the modeling of Markov chains or in the discretization of convection-diffusion problems. Due to their potential to solve sparse linear systems with an effort that is linear in the number of unknowns, algebraic multigrid (AMG) methods are of fundamental interest for such systems. For symmetric positive definite matrices, fundamental theoretical convergence results are established, and efficient AMG solvers have been developed. In contrast, for non-symmetric matrices, theoretical convergence results have been provided only recently. A property that is sufficient for convergence is that the matrix be an M-matrix. In this paper, we present how the simulation of incompressible fluid flows with particle methods leads to large linear systems with sparse, non-symmetric matrices. In each time step, the Poisson equation is approximated by meshfree finite differences. While traditional least squares approaches do not guarantee an M-matrix structure, an approach based on linear optimization yields optimally sparse M-matrices. For both types of discretization approaches, we investigate the performance of a classical AMG method, as well as an AMLI type method. While in the considered test problems, the M-matrix structure turns out not to be necessary for the convergence of AMG, problems can occur when it is violated. In addition, the matrices obtained by the linear optimization approach result in fast solution times due to their optimal sparsity.Comment: 16 pages, 7 figure

The effect of calcitriol, paricalcitol, and a calcimimetic on extraosseous calcifications in uremic rats

Vitamin D derivatives and calcimimetics are used to treat secondary hyperparathyroidism in patients with chronic renal failure. We investigated the effect of calcitriol, paricalcitol, and the calcimimetic AMG 641 on soft-tissue calcification in uremic rats with secondary hyperparathyroidism. Control and uremic rats were treated with vehicle, calcitriol, paricalcitol, AMG 641, or a combination of AMG 641 plus calcitriol or paricalcitol. Parathyroid hormone levels were reduced by all treatments but were better controlled by the combination of paricalcitol and AMG 641. The calcimimetic alone did not induce extraosseous calcification but co-administration of AMG 641 reduced soft-tissue calcification and aortic mineralization in both calcitriol- and paricalcitol-treated rats. Survival was significantly reduced in rats treated with calcitriol and this mortality was attenuated by co-treatment with AMG 641. Our study shows that extraskeletal calcification was present in animals treated with calcitriol and paricalcitol but not with AMG 641. When used in combination with paricalcitol, AMG 641 provided excellent control of secondary hyperparathyroidism and prevented mortality associated with the use of vitamin D derivatives without causing tissue calcification

From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation

We propose a new method for the approximate solution of the Lyapunov equation with rank-$1$ right-hand side, which is based on extended rational Krylov subspace approximation with adaptively computed shifts. The shift selection is obtained from the connection between the Lyapunov equation, solution of systems of linear ODEs and alternating least squares method for low-rank approximation. The numerical experiments confirm the effectiveness of our approach.Comment: 17 pages, 1 figure

A robust adaptive algebraic multigrid linear solver for structural mechanics

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM