Petroleum Reservoir Simulation

A fast inversion technique for the interpretation of data from resistivity tomography surveys has been developed for operation on a microcomputer. This technique is based on the smoothness-constrained least-squares method and it produces a two-dimensional subsurface model from the apparent resistivity pseudosection. In the first iteration, a homogeneous earth model is used as the starting model for which the apparent resistivity partial derivative values can be calculated analytically. For subsequent iterations, a quasi-Newton method is used to estimate the partial derivatives which reduces the computer time and memory space required by about eight and twelve times, respectively, compared to the conventional least-squares method. Tests with a variety of computer models and data from field surveys show that this technique is insensitive to random noise and converges rapidly. This technique takes about one minute to invert a single data set on an 80486DX microcomputer.

Rapid least-squared inversion of apparent resisitivity pseudosections by a quasi-Newton method

On etend la methode frontale pour resoudre des systemes lineaires d'equations en permettant a plus d'un front d'apparaitre en meme temps

The Multifrontal Solution of Indefinite Sparse Symmetric Linear

Techniques from numerical analysis and crystallographic refinement have been combined to produce a variant of the Truncated Newton nonlinear optimization procedure. The new algorithm shows particular promise for potential energy minimization of large molecular systems. Usual implementations of Newton's method require storage space proportional to the number of atoms squared (i.e., O(N2)) and computer time of O(N3). Our suggested implementation of the Truncated Newton technique requires storage of less than O(N1.5) and CPU time of less than O(N2) for structures containing several hundred to a few thousand atoms. The algorithm exhibits quadratic convergence near the minimum and is also very tolerant of poor initial structures. A comparison with existing optimization procedures is detailed for cyclohexane, arachidonic acid, and the small protein crambin. In particular, a structure for crambin (662 atoms) has been refined to an RMS gradient of 3.6 × 10−6 kcal/mol/A per atom on the MM2 potential energy surface. Several suggestions are made which may lead to further improvement of the new method.

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An efficient newton‐like method for molecular mechanics energy minimization of large molecules

An approximate minimum degree (AMD) ordering algorithm for preordering a symmetric sparse matrix prior to numerical factorization is presented. We use techniques based on the quotient graph for matrix factorization that allow us to obtain computationally cheap bounds for the minimum degree. We show that these bounds are often equal to the actual degree. The resulting algorithm is typically much faster than previous minimum degree ordering algorithms and produces results that are comparable in quality with the best orderings from other minimum degree algorithms.

An Approximate Minimum Degree Ordering Algorithm

In this paper we present algorithms and data structures that may be used in the efficient implementation of symmetric Gaussian elimination for sparse systems of linear equations with positive definite coefficient matrices. The techniques described here serve as the basis for the symmetric codes in the Yale Sparse Matrix Package.

Algorithms and Data Structures for Sparse Symmetric Gaussian Elimination

: The solution of large sparse systems of linear is an important application of parallel computers. In this paper, we give a unified presentation and compare the performance of three distributed schemes to compute the Cholesky factor of a large sparse symmetric positive definite matrix on a local-memory parallel processor. (kr)

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA228143&Location=U2&doc=GetTRDoc.pdf

A Comparison of Three Column-Based Distributed Sparse Factorization Schemes.

Publisher Summary This chapter discusses the applications of an element model for Gaussian elimination. The system of linear equations is considered A x = b, where A is an N × N sparse symmetric positive definite matrix such as those that arise in finite difference and finite element approximations to elliptic boundary value problems in two and three dimensions. A classic method for solving such systems is Gaussian elimination. The chapter presents a method by which kth equation is used to eliminate the kth variable from the remaining N − k equations for k = 1, 2, …, N − 1 and then back-solve the resulting upper triangular system for the unknown vector x. The chapter reviews the graph-theoretic elimination model and highlights a new element model of the elimination process. The model is used to give simple proofs of inherent lower bounds for the work and storage associated with Gaussian elimination. It also explains the way by which element model, combined with the unusual idea of recomputing rather than saving the factorization, leads to a minimal storage sparse elimination algorithm that requires significantly less storage than regular sparse elimination for the five- or nine-point.

Applications of an Element Model for Gaussian Elimination

This chapter describes the Yale Sparse Matrix Package as a collection of routines for solving the n × n system of linear equations Mx = b when the coefficient matrix M is large and sparse. It explains the release features direct methods based on Gaussian elimination without pivoting. The next release also includes a number of preconditioned conjugate-gradient-like methods. The ordering driver ODRV selects P by applying the minimum degree algorithm to the matrix V+V t , where V denotes the upper triangular part of M . This heuristic algorithm effectively performs a symbolic elimination, at each stage choosing a pivot element from among those uneliminated diagonal entries that require the fewest arithmetic operations to eliminate. Thus, it locally optimizes the elimination process with respect to the number of arithmetic operations performed. If the zero–nonzero structure of M is not symmetric, this can yield a bad ordering. The solution process is broken into three stages, namely, symbolic factorization, numerical factorization, and numerical solution.

The (New) Yale Sparse Matrix Package

Publisher Summary This chapter discusses the application of sparse matrix techniques to reservoir simulation. It presents computing time requirements of sparse Gaussian elimination for some typical problems of reservoir simulation. In many reservoir simulation problems, the solution of a system of nonlinear parabolic partial differential equations describes multiphase flow in two or three space dimensions. The chapter focuses on two-dimensional problems. The most common technique is to approximate the domain by a rectilinear mesh or grid and to approximate the partial differential equations by five point difference equations together with suitable linearizations. In reservoir simulation, systems of the form have usually been solved with iterative rather than elimination methods. This saves both time and storage. However, selecting an efficient iterative method, optimal acceleration parameters, and a good stopping criterion is difficult and expensive. It has been found that in some situations, iterative methods do not converge to an acceptable solution within a reasonable number of iterations because of the increasing complexity of simulation problems.

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Andrew H. Sherman

Papers

Algorithms and Data Structures for Sparse Symmetric Gaussian Elimination

A Comparison of Three Column-Based Distributed Sparse Factorization Schemes.

Applications of an Element Model for Gaussian Elimination

The (New) Yale Sparse Matrix Package

Application of Sparse Matrix Techniques to Reservoir Simulation.