Computational Applied Mathematics and Operations Research

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The Conjugate Gradient Method and Trust Regions in Large Scale Optimization
(1981-10) Steihaug, Trond
Local and Superlinear Convergence for Truncated Projections Methods
(1981-10) Steihaug, Trond
Least change secant updates can be obtained as the limit of iterated projections based on other secant updates. We show that these iterated projections can be terminated or truncated after any positive number of iterations and the local and the superlinear rate of convergence are still maintained. The truncated iterated projections method is used to find sparse and symmetric updates that are locally and superlinearly convergent.
Damped Inexact Quasi-Newton Methods
(1981-12) Steihaug, Trond
The inexact quasi-Newton methods are very attractive methods for large scale optimization since they require only an approximate solution of the linear system of equations for each iteration. To achieve global convergence results, we adjust the step using a backtracking strategy. We discuss the backtracking strategy in detail and show that this strategy has similar convergence properties as one obtains by using line searches with the Goldstein-Armijo conditions. The combination of backtracking and inexact quasi-Newton methods is particularly attractive since the conditions for convergence are easily met. We give conditions for Q-linear and Q-superlinear convergence.
PFC: A Program to Convert Fortran to Parallel Form
(1982-03) Allen, John R.; Kennedy, Ken
The recent success of vector computers like the Cray-1 and array processors such as those manufactured by Floating Point Systems has increased interest in making vector operations available to the Fortran programmer. The Fortran standards committee is currently considering extensions to Fortran which will permit the programmer to explicitly specify vector and array operations. The proposed standard is usually referred to as Fortran 8x. This paper describes PFC, a system that translates sequential programs written in Fortran to Fortran 8x, replacing loops by array operations wherever possible. Central to the theory underlying PFC is the concept of dependence. In another work we developed a test for dependence between statements which distinguished dependences that arise due to the iteration of different loops. In this work, we show how that test is incorporated into a powerful program for recognizing parallelism. By using a careful implementation strategy, combined with judicious choice of data abstraction mechanisms, we have been able to implement a flexible and sophisticated software system while maintaining reasonable efficiency. In fact, our implementation outperforms one of our previous efforts by a factor of more than ten. The resulting program is an interesting case study of the application of theoretical techniques to a practical implementation problem.
An Experimental Computer Network to Support Numerical Computation
(1982-03) Cartwright, Robert; Dennis, J.E. Jr.; Jump, J. Robert; Kennedy, Ken
The Computer Science faculty at Rice University proposes to design and implement an experimental distributed computing system to support numerical computation. Although local networks of single user machines have already been proven for many nonnumerical applications, the concept has yet to be tried in the context of numerical program development and execution. The Rice Numerical Network, or R^n, will consist of approximately 24 single-user numerical machines equipped with high-resolution bit-mapped screens, a 32-bit central processor, and vector floating point hardware. It will also include several specialized server nodes supporting a high-performance vector floating point processor and various peripheral devices including a gateway to the SCnet communications network linking the nation's major computer science research centers. The new facility will support a coherent research program in software systems, computer architecture, and quality numerical software, directed at creating a modern reactive environment for numerical computation. Despite stiff competition from industry and other universities, Rice University has recently assembled the nucleus of computer science faculty required to develop an innovative distributed computing system supporting vector numerical computation and to evaluate its utility as a tool for solving important scientific problems.
Local Analysis of Inexact Quasi-Newton Methods
(1982-05) Eisenstat, Stanley C.; Steihaug, Trond
Quasi-Newton methods are well known iterative methods for solving nonlinear problems. At each stage, a system of linear equations has to be solved. However, for large scale problems, solving the linear system of equations can be expensive and may not be justified when the iterate is far from the solution or when the matrix is an approximation to the Jacobian or Hessian matrix. Instead we consider a class of inexact quasi-Newton methods which solves the linear system only approximately. We derive conditions for local and superlinear rate of convergence in terms of a relative residual.
Crossflooding in Steamflood Operation: A Simulation Study
(1983) Potempa,Thom
Crossflooding techniques have the potential for increasing the ultimate oil yield in secondary or tertiary recovery operations. At least one field scale pilot study of crossflooding in a reservoir undergoing a steamflood is being undertaken at this time. The purpose of this investigation is to evaluate using a numerical simulation various crossflooding options on five and nine spot patterns under a continuous steam drive. As the simulation of a crossflooding process is demonstrated to be very sensitive to the grid orientation phenomenon, these studies are undertaken using a steam displacement model that does not exhibit a serious grid orientation effect. The numerical studies indicate that crossflooding significantly improves the ultimate oil yield in steamflooding operations. The simulation studies indicate that an increased yield is achieved without infill drilling by using an expanded pattern crossflood. Crossflood patterns involving infill wells perform only marginally better than the expanded pattern floods. A new crossflooding pattern introduced in an earlier theoretical investigation is shown to improve the ultimate net oil yield in steamflooding operations.
Mobility Weighting in Numerical Reservoir Simulation
(1983) Potempa,Thom
The sensitivity of a numerical steamflooding model with respect to mobility weighting is examined in depth. Three numerical discretization procedures are used in this investigation: a new numerical scheme, a five point finite difference method, and a procedure which, under certain assumptions, is equivalent to that introduced by McCracken and Yanosik. Three mobility weighting schemes are investigated. The first approach studied is upstream mobility weighting. The second method investigated uses harmonic total mobility weighting and upstream weighting of fractional flow terms. The approach introduced in this investigation uses the kinematic viscosity in the total mobility and fractional flow terms. Computational results for a simulated steam drive indicate that this new mobility weighting approach is superior to the other two mobility weighting schemes. In particular, the steam displacement model formed from t he combination of this new mobility weighting approach and the McCracken and Yanosik discretization procedure is shown to produce realistic simulation of an inverted seven spot pattern under a continuous steam drive.
A Numerical Model of Two Dimensional, Two Component, Single Phase Miscible Displacement in Porous Media
(1983-03) Potempa,Thom
A new numerical procedure for modeling single phase miscible dispacement in a porous medium is presented. This model is based upon a material balance that is similar to that which is used to derive the differential equations that govern single phase miscible displacement. Since the procedure is defined in terms of a material balance, the data structures arising in a computational implementation are compatible with those that are present in finite difference models of miscible displacement. This procedure obeys a maximum principle due to the upstream weighting of the convective transport terms. This new procedure does not exhibit the grid orientation effect present in the five point finite difference models of this process.
Inaccuracy in Quasi-Newton Methods: Local Improvement Theorems
(1983-03) Dennis, J.E. Jr.; Walker, Homer F.
In this paper, we consider the use of bounded-deterioration quasi-Newton methods implemented in floating-point arithmetic to find solutions to F(x)=0 where only inaccurate F-values are available. Our analysis is for the case where the relative error in F is less than one. We obtain theorems specifying local rates of improvement and limiting accuracies depending on the nearness to Newton's method of the basic algorithm, the accuracy of its implementation, the relative errors in the function values, the accuracy of the solutions of the linear systems for the Newton steps, and the unit-rounding errors in the addition of the Newton steps.
Three Dimensional Simulation of Steam Displacement with Minimal Grid Orientation
(1983-03) Potempa,Thom
Steamflooding is a tertiary oil recovery mechanism with proven economic potential. Reliable numerical simulations of candidate injection schemes can aid in the optimization of process parameters. State-of-the-art numerical models exhibit varying degrees of grid effects which affect their reliability with respect to the modeling of pattern floods. This paper presents a numerical steamflood model which does not exhibit a significant amount of grid orientation. This model utilizes a numerical discretization technique which is an admixture of finite difference and finite element methods. Fluid properties are determined as a function of primary variables in a manner that allows a straightforward implementation of Newton's method for solving the nonlinear system of discretized equations. Fractional flows are defined using mass rather than volumetric mobilities, and are used to upwind the convective flow terms.
The Effect of the Definition of Fractional Flow Upon Grid Effects in a Numerical Model of Thermal Processes
(1983-03) Potempa,Thom
In previous research regarding the numerical simulation of a single phase miscible displacement, a numerical procedure which does not exhibit serious grid effects and is highly compatible with nine point finite difference models has been developed. To determine if this procedure could successfully deal with grid effects in general reservoir simulators, this numerical procedure has been implemented in a simplified thermal recovery model. In the initial implementation of this procedure in a complex setting, multiple phase mass transfer between the computational molecules associated with the discretization procedure utilized the well known concepts of fractional flow and total Darcy velocity. The resulting numerical model exhibits unrealistic phenomena, which is unexpected in light of the realistic simulations earlier obtained for the model problem. By changing the model of multiple phase mass transfer, these physically unrealistic effects are eliminated. The new model uses the total molar flux instead of the total Darcy velocity. The fractional flows are defined in a fashion compatible with the total molar flux.
Toward Direct Sparse Updates of Cholesky Factors
(1983-04) Dennis, J.E. Jr.; Vu, Phuong
A very important problem in numerical optimization is to find a way to update a sparse Hessian approximation so that it will be positive definite under reasonable circumstances. This problem has motivated research, which is yet to show much progress, toward a "sparse BFGS method." In this paper, we suggest a different approach to the problem based on using a sparse Broyden, or Schubert, update directly on the Cholesky factor of the current Hessian approximation to define the next Hessian approximation implicitly in terms of its Cholesky factorization. This approach has the added advantage of being able to cheaply find the Newton step, since no factorization step is required. The difficulty with our approach is in finding a satisfactory secant or quasi-Newton condition to use in the update.
A Convergence Theory for a Class of Quasi-Newton Methods for Constrained Optimization
(1983-05) Fontecilla, Rodrigo; Steihaug, Trond; Tapia, Richard A.
In this paper we develop a general convergence theory for a class of quasi-Newton methods for equality constrained optimization. The theory is set in the framework of the diagonalized multiplier method defined by Tapia and is an extension of the theory developed by Glad. We believe that this framework is flexible and amenable to convergence analysis and generalizations. A key ingredient of a method in this class is a multiplier update. Our theory is tested by showing that a straightforward application gives the best known convergence results for several known multiplier updates. Also a characterization of q-superlinear convergence is presented. It is shown that in the special case when the diagonalized multiplier method is equivalent to the successive quadratic programming approach, our general characterization result gives the Boggs, Tolle and Wang characterization.
An Improved Implementation of the McCracken and Yanosik Nine Point Finite Difference Procedure
(1983-05) Potempa,Thom
The grid orientation phenomenon present in numerical models of fluid flow in a porous medium can give rise to unrealistic predictions when simulating adverse mobility displacements. McCracken and Yanosik proposed a nine point finite difference scheme for approximating the solution of the continuity equations that has the potential of eliminating many of the unrealistic predictions that are observed when using five point finite difference operators. Coats and Ramesh have implemented this scheme in a steamflooding model, and have noted that serious grid effects are present in the simulation of an inverted seven spot pattern. Potempa has described a procedure which exhibits minimal grid effects for the problem described by Coats and Ramesh. This paper describes modifications to the McCracken and Yanosik procedure which allow for realistic simulation of this inverted seven spot pattern under a steam drive. These modifications are based upon an approximation scheme that has been previously reported, and affect the incorporation of upstream weighting in a simulator.
The Lack of Positive Definiteness in the Hessian in Constrained Optimization
(1983-06) Fontecilla, Rodrigo
The use of the DFP or the BFGS secant updates requires the Hessian at the solution to be positive definite. The second order sufficiency conditions insure the positive definiteness only in a subspace of R^n. Conditions are given so we can safely update with either update. A new class of algorithms is proposed which generate a sequence {xk} converging 2-step q-superlinearly. We also propose two specific algorithms: One converges q-superlinearly if the Hessian is positive definite in R^n and converges 2-step q-superlinearly if the Hessian is positive definite only in a subspace; the second one generates a sequence converging 1-step q-superlinearly. While the former costs one extra gradient evaluation, the latter costs one extra gradient evaluation and one extra function evaluation on the constraints.
A New Nonlinear Equations Test Problem
(1983-06) Dennis, J.E. Jr.; Gay, David M.; Vu, Phuong Ahn
This presents a set of test problems for nonlinear equations and nonlinear least-squares algorithms. These problems, sent to us by C.V. Nelson of the Maine Medical Center, come from a dipole model of the heart. They are 6 x 6 or 8 x 8, easy to code, cheap to evaluate, and not easy to solve. In support of the latter contention, we present test results from MINPACK AND NL2SOL.
A User's Guide to Nonlinear Optimization Algorithms
(1983-08) Dennis, J.E. Jr.
The purpose of this paper is to provide a user's introduction to the basic ideas currently favored in nonlinear optimization routines by numerical analysts. The primary focus will be on the unconstrained problem because the main ideas are much more settled. Although this is not a paper about nonlinear least squares, the rich structure of this important practical problem makes it a convenient example to use to illustrate the ideas we will discuss. We will make most use of this example in the first three sections which deal with the helpful concept of a local modeling technique and the attendant local convergence analysis. Stress will be put on ways used to improve a poor initial solution estimate since this is one of the keys to choosing the most suitable routine for a particular application. This material is covered in the rather long Section 4. the discussion of the constrained problem in Section 5 will be a brief outline of the current issues involved in deciding what algorithms to implement. Section 6 is devoted to some concluding remarks including sparse comments on large problems.
On the Successive Projections Approach to Least-Squares Problems
(1983-08) Dennis, J.E. Jr.; Steihaug, Trond
In this paper, we suggest a generalized Gauss-Seidel approach to sparse linear and nonlinear least-squares problems. The algorithm, closely related to one given by Elfving (1980), uses the work of Curtis, Powell, and Reid (1974) as extended by Coleman and Moré (1983) to divide the variables into nondisjoint groups of structurally orthogonal columns and then projects the updated residual into each column subspace of the Jacobian in turn. In the linear case, this procedure can be viewed as an alternate ordering of the variables in the Gauss-Seidel method. Preliminary tests indicate that this leads quickly to cheap solutions of limited accuracy for linear problems, and that this approach is promising for an inexact Gauss-Newton analog of the inexact Newton approach of Dembo, Eisenstat, and Steihaug (1981).
A User's Guide to the Rice Steam Displacement Model
(1983-10) Potempa,Thom
To provide an accurate model of heavy oil steam displacement processes. The primary advantage of this model over other state-of-the-art models is that it does not exhibit grid orientation effects. Serious grid effects have been reported in other steam displacement models.

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