A constrained pressure-temperature residual (CPTR) method for non-isothermal multiphase flow in porous media

For both isothermal and thermal petroleum reservoir simulation, the Constrained Pressure Residual (CPR) method is the industry-standard preconditioner. This method is a two-stage process involving the solution of a restricted pressure system. While initially designed for the isothermal case, CPR is also the standard for thermal cases. However, its treatment of the energy conservation equation does not incorporate heat diffusion, which is often dominant in thermal cases. In this paper, we present an extension of CPR: the Constrained Pressure-Temperature Residual (CPTR) method, where a restricted pressure-temperature system is solved in the first stage. In previous work, we introduced a block preconditioner with an efficient Schur complement approximation for a pressure-temperature system. Here, we extend this method for multiphase flow as the first stage of CPTR. The algorithmic performance of different two-stage preconditioners is evaluated for reservoir simulation test cases.Comment: 28 pages, 2 figures. Sources/sinks description in arXiv:1902.0009

Regularization-robust preconditioners for time-dependent PDE constrained optimization problems

In this article, we motivate, derive and test �effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems, which arise in PDE constrained optimization problems. We consider the distributed control problem involving the heat equation with two diff�erent functionals, and the Neumann boundary control problem involving Poisson's equation and the heat equation. Crucial to the eff�ectiveness of our preconditioners in each case is an eff�ective approximation of the Schur complement of the matrix system. In each case, we state the problem being solved, propose the preconditioning approach, prove relevant eigenvalue bounds, and provide numerical results which demonstrate that our solvers are eff�ective for a wide range of regularization parameter values, as well as mesh sizes and time-steps

Space-time block preconditioning for incompressible flow

Parallel-in-time methods have become increasingly popular in the simulation of time-dependent numerical PDEs, allowing for the efficient use of additional MPI processes when spatial parallelism saturates. Most methods treat the solution and parallelism in space and time separately. In contrast, all-at-once methods solve the full space-time system directly, largely treating time as simply another spatial dimension. All-at-once methods offer a number of benefits over separate treatment of space and time, most notably significantly increased parallelism and faster time-to-solution (when applicable). However, the development of fast, scalable all-at-once methods has largely been limited to time-dependent (advection-)diffusion problems. This paper introduces the concept of space-time block preconditioning for the all-at-once solution of incompressible flow. By extending well-known concepts of spatial block preconditioning to the space-time setting, we develop a block preconditioner whose application requires the solution of a space-time (advection-)diffusion equation in the velocity block, coupled with a pressure Schur complement approximation consisting of independent spatial solves at each time-step, and a space-time matrix-vector multiplication. The new method is tested on four classical models in incompressible flow. Results indicate perfect scalability in refinement of spatial and temporal mesh spacing, perfect scalability in nonlinear Picard iterations count when applied to a nonlinear Navier-Stokes problem, and minimal overhead in terms of number of preconditioner applications compared with sequential time-stepping.Comment: 28 pages, 7 figures, 4 table

A block preconditioner for non-isothermal flow in porous media

In petroleum reservoir simulation, the industry standard preconditioner, the constrained pressure residual method (CPR), is a two-stage process which involves solving a restricted pressure system with Algebraic Multigrid (AMG). Initially designed for isothermal models, this approach is often used in the thermal case. However, it does not have a specific treatment of the additional energy conservation equation and temperature variable. We seek to develop preconditioners which better capture thermal effects such as heat diffusion. In order to study the effects of both pressure and temperature on fluid and heat flow, we consider a model of non-isothermal single phase flow through porous media. For this model, we develop a block preconditioner with an efficient Schur complement approximation. Both the pressure block and the approximate Schur complement are approximately inverted using an AMG V-cycle. The resulting solver is scalable with respect to problem size and parallelization.Comment: 35 pages, 3 figure

A Predictive Model for Color Pattern Formation in the Butterfly Wing of Papilio dardanus

Previously, we have proposed a mathematical model based on a modified Turing mechanism to account for pigmentation patterning in the butterfly wing of Papilio dardanus, well-known for the spectacular phenotypic polymorphism in the female of the species (Sekimura, et al., Proc. Roy. Soc. Lond. B 267, 851-859 (2000)). In the present paper, we use our model to predict the outcome of a number of different types of cutting experiments and compare our results with those of a model based on different hypotheses.\ud \ud This paper is dedicated to Professor Masayasu Mimura on his sixtieth birthda

Natural preconditioning and iterative methods for saddle point systems

The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or the discrete setting, so saddle point systems arising from the discretization of partial differential equation problems, such as those describing electromagnetic problems or incompressible flow, lead to equations with this structure, as do, for example, interior point methods and the sequential quadratic programming approach to nonlinear optimization. This survey concerns iterative solution methods for these problems and, in particular, shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness---in terms of rapidity of convergence---is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends