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

Genetic variability of some Italian and Polish duck breeds

This study is aimed to estimate and compare the inter- and within-breed variability of duck populations under genetic conservation programmes. The following four duck breeds were analysed: Germanata Veneta (AGV) and Mignon (AMG) from Italy, Pekin Krajowy (33P) and Pomniejszona (2K) from Poland. The characterisation of the four populations was carried out through a panel of 23 microsatellite markers. The analysis involved 180 individuals: 39 for AGV, 41 for AMG, 50 for 33P and 50 for 2K. An average of 11.36 alleles per locus was identified. Twenty-two loci showed high values of polymorphism information content from 0.575 to 0.912, while CAUD136 was monomorphic for the Italian breeds. The breeds showed relatively high heterozygosity: higher for the Polish populations (0.6920 for 33P and 0.6521 for 2K), and lower for the Italian (0.4497 and 0.3718 for AGV and AMG, respectively). The inbreeding coefficient was higher for the Italian breeds, AMG in particular (0.133, 0.097 and 0.121), as well as the differentiation index (0.253). The Nei’s minimum distances (DM) and Reynolds distances (DR) were low between the Polish populations (0.131 and 0.088, respectively); these were associated to AGV (DM = 0.191 and DR = 0.259 for 33P; DM = 0.174 and DR = 0.226 for 2K). Finally, AGV was distant from AMG (DM = 0.259 and DR = 0.317). The molecular coancestry, or mean kinship was higher for the Italian breeds compared to Polish populations. The Italian populations showed intermediate values. The obtained results can be perceived as an important tool for the applied genetic conservation programmes

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