Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

An Energy Formulation of Surface Tension or Willmore Force For Two-Phase Flow

The motion of a biological cell in liquid is a rich subject for modeling. In the early 1970’s, it was realized by Canham that biological vesicles with lipid bilayer membranes reach a steady state shape that minimizes bending. Helfrich soon after mathematically quantified the related bending energy and showed that the shapes from minimizing this bending energy match the types of shapes observed in nature. The resulting Canham-Helfrich energy, consisting of bending energy and a constant surface area and volume constraint, is a major component of any model of cellular motility. To this end, we consider the cellular vesicle to be a closed interface between two fluids and we present a finite element model for a two-phase flow coupling the minimization of some given energy defined on the interface to the incompressible flow of the two fluids, which is then advected according to the resulting velocity field. We provide a general framework for incorporating the energies on the interface and then focus on three applications of energy on the interface: the first is surface tension minimizing the surface area energy, the second minimizes the bending energy without explicit surface area or volume constraints, the third minimizes the Canham-Helfrich energy including the constraints. We present a semi-implicit model for bending energy which uses an implicit levelset formulation for the interface and couples the forces from the interface to the two phase incompressible Navier-Stokes system through the use of an approximate Dirac delta function defined on a band around the interface. By using energies to describe the motion, our model is immediately provided with a sense of energy stability. We provide various numerical simulations and validations of flow under these three energies in two and three dimensions. Our simulations confirm that enforcing the volume constraint in the incompressible flow is vital to achieve the desired steady state shapes

An Energy Formulation of Surface Tension or Willmore Force For Two-Phase Flow

The motion of a biological cell in liquid is a rich subject for modeling. In the early 1970’s, it was realized by Canham that biological vesicles with lipid bilayer membranes reach a steady state shape that minimizes bending. Helfrich soon after mathematically quantified the related bending energy and showed that the shapes from minimizing this bending energy match the types of shapes observed in nature. The resulting Canham-Helfrich energy, consisting of bending energy and a constant surface area and volume constraint, is a major component of any model of cellular motility. To this end, we consider the cellular vesicle to be a closed interface between two fluids and we present a finite element model for a two-phase flow coupling the minimization of some given energy defined on the interface to the incompressible flow of the two fluids, which is then advected according to the resulting velocity field. We provide a general framework for incorporating the energies on the interface and then focus on three applications of energy on the interface: the first is surface tension minimizing the surface area energy, the second minimizes the bending energy without explicit surface area or volume constraints, the third minimizes the Canham-Helfrich energy including the constraints. We present a semi-implicit model for bending energy which uses an implicit levelset formulation for the interface and couples the forces from the interface to the two phase incompressible Navier-Stokes system through the use of an approximate Dirac delta function defined on a band around the interface. By using energies to describe the motion, our model is immediately provided with a sense of energy stability. We provide various numerical simulations and validations of flow under these three energies in two and three dimensions. Our simulations confirm that enforcing the volume constraint in the incompressible flow is vital to achieve the desired steady state shapes