In this paper, we describe block matrix algorithms for the iterative solution of large scale linear-quadratic optimal control problems arising from the optimal control of parabolic partial differential equations over a finite control horizon. We describe three iterative algorithms. The first algorithm employs a CG method for solving a symmetric positive definite reduced linear system involving only the unknown control variables. This system can be solved using the CG method, but requires double iteration. The second algorithm is designed to avoid double iteration by introducing an auxiliary variable. It yields a symmetric indefinite system and a positive definite block preconditioner. The third algorithm uses a symmetric positive definite block diagonal preconditioner for the saddle point system and is based on the parareal algorithm. Theoretical results show that the preconditioned algorithm has optimal convergence properties and parallel scalability. Numerical experiments are provided to confirm the theoretical results.

Background: Changes in DNA topology during replication are still poorly understood. Results: Classical genetics and two-dimensional agarose gel electrophoresis showed that RIs tensioned in the absence of Topo IV. Conclusion:The results indicated that replication forks swivel in vivo leading to the formation of precatenanes as replication progresses. Significance: This conclusion ends a long lasting debate on the formation of precatenanes during replication.

We systematically varied conditions of two-dimensional (2D) agarose gel electrophoresis to optimize separation of DNA topoisomers that differ either by the extent of knotting, the extent of catenation or the extent of supercoiling. To this aim we compared electrophoretic behavior of three different families of DNA topoisomers: (i) supercoiled DNA molecules, where supercoiling covered the range extending from covalently closed relaxed up to naturally supercoiled DNA molecules; (ii) postreplicative catenanes with catenation number increasing from 1 to ∼15, where both catenated rings were nicked; (iii) knotted but nicked DNA molecules with a naturally arising spectrum of knots. For better comparison, we studied topoisomer families where each member had the same total molecular mass. For knotted and supercoiled molecules, we analyzed dimeric plasmids whereas catenanes were composed of monomeric forms of the same plasmid. We observed that catenated, knotted and supercoiled families of topoisomers showed different reactions to changes of agarose concentration and voltage during electrophoresis. These differences permitted us to optimize conditions for their separation and shed light on physical characteristics of these different types of DNA topoisomers during electrophoresis.

SUMMARYIn this paper, we describe and analyse several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linear-quadratic elliptic control problem with Neumann boundary conditions. To ensure that the problem is well posed, a regularization term with a parameter is included. The first algorithm reduces the saddle point system to a symmetric positive definite Schur complement system for the control variable and employs conjugate gradient (CG) acceleration, however, double iteration is required (except in special cases). A preconditioner yielding a rate of convergence independent of the mesh size h is described for ⊂ R 2 or R 3 , and a preconditioner independent of h and when ⊂ R 2 . Next, two algorithms avoiding double iteration are described using an augmented Lagrangian formulation. One of these algorithms solves the augmented saddle point system employing MINRES acceleration, while the other solves a symmetric positive definite reformulation of the augmented saddle point system employing CG acceleration. For both algorithms, a symmetric positive definite preconditioner is described yielding a rate of convergence independent of h. In addition to the above algorithms, two heuristic algorithms are described, one a projected CG algorithm, and the other an indefinite block matrix preconditioner employing GMRES acceleration. Rigorous convergence results, however, are not known for the heuristic algorithms.

Abstract. Hysteresis effects in two-phase flow in porous media are important in applications such as waterflooding or gas storage in sand aquifers. In this paper, we develop a numerical scheme for such a flow where the permeability hysteresis is modeled by a family of reversible scanning curves enclosed by irreversible imbibition and drainage permeability curves. The scheme is based on associated local Riemann solutions and can be viewed as a modification of the classical Godunov method. The Riemann solutions necessary for the scheme are presented, as well as the criteria that guarantee the well-posedness of the Riemann problem with respect to perturbations of left and right states. The numerical and analytical results show strong influence of the permeability hysteresis on the flow. In addition, the numerical scheme accurately reproduces the available experimental data once hysteresis is taken into account in the model.

No abstract

No abstract

No abstract