Gradient-like Method Research Articles

Penalty methods for American options with stochastic volatility

The American early exercise constraint can be viewed as transforming the original linear two dimensional stochastic volatility option pricing PDE into a PDE with a nonlinear source term. Several methods are described for enforcing the early exercise constraint by using a penalty source term in the discrete equations. The resulting nonlinear algebraic equations are solved using an approximate Newton iteration. The solution of the Jacobian is obtained using an incomplete LU (ILU) preconditioned conjugate gradient-like (PCG) method. Some example computations are presented for option pricing problems based on a stochastic volatility model, including an exotic American chooser option written on a put and call with discrete double knockout barriers and discrete dividends.

An implicit numerical scheme for the simulation of internal viscous flow on unstructured grids

The development of a cell-centered unstructured grid solution for the two-dimensional Navier-Stokes equations is described. A finite-volume approach is used to discretize the conservation law form of the compressible flow equations written in terms of primitive variables. Temporal preconditioning is employed so that low Mach number flows can be solved economically. The equations are marched in time using either an implicit Gauss-Seidel iterative procedure or a solver based on a conjugate gradient-like method. A four color scheme is employed to vectorize the block Gauss-Seidel relaxation procedure. This increases the memory requirements minimally and decreases the computer time spent solving the resulting system of equations substantially. A factor of 7.6 speedup in the matrix solver is typical for the viscous equations. The grids are generated based on the method of Delaunay triangulation. Numerical results are obtained for inviscid flow over a bump in a channel at subsonic and transonic conditions for validation with structured solvers. Viscous results are computed for developing flow in a channel, a symmetric sudden expansion, periodic tandem cylinders in a cross-flow, and a fourport valve. Comparisons are made with available results obtained by other investigators.

A note to the gmres method

GMRES method [4] is an effective conjugate gradient-like iterative method for solving linear systems of equations. In each loop of the GMRES we have to solve a special least squares problem (1). A classical way of solving such least squares problems is to factor H into QR using Givens plane rotations. In [4] it is shown that the factorization can be updated progressively as each column of H appears (i.e., at every setup of the Arnoldi process), therefore it enables us to obtain the residual norm of the approximate solution without computing Xj, thus allowing us to decide when to stop the process without wasting needless operations. In this note we give a direct method for the least squares problems, which uses less computational work compared with Givens rotation method and has the nice property mentioned above.

Fast Algorithms for Composite Materials

AbstractWe briefly review recently developed fast algorithms for the evaluation of electrostatic fields in composite materials consisting of a collection of piecewise homogeneous inclusions embedded in a uniform background. These algorithms are based on combining a suitable boundary integral equation with the fast multipole method and a conjugate gradient-like iterative method. The CPU time required grows linearly with the number of points in the discretization of the interface between the inclusions and the background material, bringing large-scale calculations within practical reach.

Comparison of standard and matrix-free implementations of several Newton-Krylov solvers

Fully coupled Newton's method is combined with conjugate gradient-like iterative algorithms to form inexact Newton-Krylov algorithms for solving the steady, incompressible, Navier-Stokes and energy equations in primitive variables. Finite volume differencing is employed using the power law convection-diffusion scheme on a uniform but staggered mesh. The well-known model problem of natural correction in an enclosed cavity is solved. Three conjugate gradient-like algorithms are selected from a clans of algorithms based upon the Lanczos biorthogonalization procedure; namely, the conjugate gradients squared algorithm, the transpose-free quasi-minimal-residual algorithm, and a more smoothly convergent version of the biconjugate gradients algorithm. A fourth algorithm is based upon the Arnoldi process, namely the popular generalized minimal residual algorithm (GMRES). The performance of a standard inexact Newton's method implementation is compared with a matrix-free implementation. Results indicate that the performance of the matrix-free implementation is strongly dependent upon grid size (number of unknowns) and the selection of the conjugate gradient-like method. GMRES appeared to be superior to the Lanczos based algorithms within the context of a matrix-free implementation

Fast Iterative Solution of Stabilised Stokes Systems Part II: Using General Block Preconditioners

Mixed finite element approximation of the classical Stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables. Iterative solution of such indefinite systems is feasible and is an attractive approach for large problems. Part I of this work described a conjugate gradient-like method (the method of preconditioned conjugate residuals) which is applicable to symmetric indefinite problems [A. J. Wathen and D. J. Silvester, SIAM J. Numer. Anal., 30 (1993), pp. 630–649]. Using simple arguments, estimates for the eigenvalue distribution of the discrete Stokes operator on which the convergence rate of the iteration depends are easily derived. Part I discussed the important case of diagonal preconditioning (scaling). This paper considers the more general class of block preconditioners, where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables. It is shown that, provided the appropriate scaling is used for the blocks corresponding to the pressure variables, the preconditioning of the Laplacian (viscous) terms determines the complete eigenvalue spectrum of the preconditioned Stokes operator. All conventional preconditioners for the Laplacian including (modified) incomplete factorisation, hierarchical basis, multigrid and domain decomposition methods are covered by the analysis. It is shown that if a spectrally equivalent preconditioner is used for the Laplacian terms then the convergence rate of iterative solution algorithms is independent of the mesh-size. The results apply to both locally and globally stabilised mixed approximations as well as to mixed methods that are inherently stable.

Fast Iterative Solution of Stabilised Stokes Systems. Part I: Using Simple Diagonal Preconditioners

Mixed finite element approximation of the classical Stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables. Iterative solution of such indefinite systems is feasible and is an attractive approach for large problems. The use of stabilisation methods for convenient (but unstable) mixed elements introduces stabilisation parameters. The authors show how these can be chosen to obtain rapid iterative convergence. The authors propose a conjugate gradient-like method (the method of preconditioned conjugate residuals) that is applicable to symmetric indefinite problems, describe the effects of stabilisation on the algebraic structure of the discrete Stokes operator, and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends. The simple case of diagonal preconditioning is discussed. The results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable. It is demonstrated that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete Laplacian are approachable for the Stokes problem.

Domain decomposition and multigrid solvers for flow simulation in porous media on distributed memory parallel processors

Realistic simulations of fluid flow in oil reservoirs have been proven to be computationally intensive. In this work, techniques for solving large sparse systems of linear equations that arise in simulating these processes are developed for parallel computers such as INTEL hypercubes iPSC/2 and iPSC/860. This solver is based on a combined multigrid and domain decomposition approach. The Algorithm uses line corrections solved using a multigrid method, line Jacobi and block incomplete domain decomposition as an overall preconditioner for a conjugate gradient-like acceleration method, ORTHOMIN (k). This is shown to be a factor of ten times faster on a 32-processor hypercube compared to widely used sequential solvers. Three test problems are used to validate the results which include implicit wells and faults: The first is based on highly heterogeneous two-phase flow, the second on the SPE Third Comparative Solution and the third on real production compositional data.

Tracing post-limit-point paths with incomplete or without factorization of the stiffness matrix

In this work nonlinear versions of preconditioned conjugate gradient-like methods have been employed in conjunction with a work control constraint in order to trace the complete loadisplacement path beyond limit points. The so calledconjugate and secant-Newton methods combine the covergence properties of Newton-like methods and the low storage requirements of vector iteration methods by varying the storage demands for the preconditioning matrix according to the available computer storage facilities. Special care to be taken in the line search routine which is indispensable for any nonlinear version of a conjuagte gradient-like method. The constraint formulation of the methods is properly modifies to incorporate the line search, while a special implementation is proposed for the untable branches of the load-displacement curves where the corresponding stationary point of the total potential energy is a saddle point.

A Hybrid Chebyshev Krylov Subspace Algorithm for Solving Nonsymmetric Systems of Linear Equations

We present an iterative method for solving large sparse nonsymmetric linear systems of equations that enhances Manteuflel’s adaptive Chebyshev method with a conjugate gradient-like method. The new method replaces the modified power method for computing needed eigenvalue estimates with Arnoldi’s method, which can be used to simultaneously compute eigenvalues and to improve the approximate solution. Convergence analysis and numerical experiments suggest that the method is more efficient than the original adaptive Chebyshev algorithm.