Jennifer A. Scott

A finite element model of heat transport in the human eye

A mathematical model of the human eye based on the bioheat transfer equation is developed. The intraocular temperature distribution is calculated using the Galerkin finite element method. A difficulty associated with the development of an accurate model of the human eye is the lack of reliable biological data available on the constants and parameters that are used in the model. These parameters include the thermal conductivities of the ocular tissues, the heat loss from the anterior corneal surface to the surroundings by convection and evaporation, and the convective heat loss from the sclera to the body core. The different values for the parameters reported in the ophthalmic literature are employed in the model, and the sensitivity of the temperature distribution to uncertainties in the parameters is investigated. A set of control parameter values is suggested for the normal human eye. The effect of the ambient temperature and the body-core temperature on the temperature distribution in the human eye is considered.

A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations

In recent years a number of solvers for the direct solution of large sparse symmetric linear systems of equations have been developed. These include solvers that are designed for the solution of positive definite systems as well as those that are principally intended for solving indefinite problems. In this study, we use performance profiles as a tool for evaluating and comparing the performance of serial sparse direct solvers on an extensive set of symmetric test problems taken from a range of practical applications.

Algorithm 891: A Fortran virtual memory system

Fortran_Virtual_Memory is a Fortran 95 package that provides facilities for reading from and writing to direct-access files. A buffer is used to avoid actual input/output operations whenever possible. The data may be spread over many files and for very large data sets these may be held on more than one device. We describe the design of Fortran_Virtual_Memory and comment on its use within an out-of-core sparse direct solver.

The computation of temperature rises in the human eye induced by infrared radiation

Long-term industrial exposure to low levels of infrared radiation has for many years been associated with the development of cataracts; the injury mechanism is widely held to be thermal. A finite element model of the human eye is employed to calculate the temperature rises experienced by the intraocular media when exposed to infrared radiation. The model is used to calculate transient and steady-state temperature distributions for various exposure times and a range of incident irradiances. The effect of the eyes natural cooling mechanisms on the heating is investigated. Specific absorption rates in the infrared irradiated eye are presented. For a radiation source of 1500 degrees C, absorption of radiant energy by the iris and the lens combined with conduction of heat from the anterior regions is found to be responsible for increases in the lens temperature of 1-2 degrees C, but under extreme exposure conditions the temperature rises are found to be substantially higher.

Sparse Approximate-Inverse Preconditioners Using Norm-Minimization Techniques

We investigate the use of sparse approximate-inverse preconditioners for the iterative solution of unsymmetric linear systems of equations. We consider the approximations proposed by Cosgrove, Diaz, and Griewank [Internat. J. Comput. Math., 44 (1992), pp. 91--110] and Huckle and Grote [A New Approach to Parallel Preconditioning with Sparse Approximate Inverses, Tech. report SCCM-94-03, Stanford University, 1994] which are based on norm-minimization techniques. Such methods are of particular interest because of the considerable scope for parallelization. We propose a number of enhancements which may improve their performance. When run in a sequential environment, these methods can perform unfavorably when compared with other techniques. However, they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment.

The design of a new frontal code for solving sparse, unsymmetric systems

We describe the design, implementation, and performance of a frontal code for the solution of large, sparse, unsymmetric systems of linear equations. The resulting software package, MA42, is included in Release 11 of the Harwell Subroutine Library and is intended to supersede the earlier MA32 package. We discuss in detail the extensive use of higher-level BLAS kernels within MA42 and illustrate the performance on a range of practical problems on a CRAY Y-MP, an IBM 3090, and an IBM RISC System/6000. We examine extending the frontal solution scheme to use multiple fronts to allow MA42 to be run in parallel. We indicate some directions for future development.

A Multilevel Algorithm for Wavefront Reduction

A multilevel algorithm for reordering sparse symmetric matrices to reduce the wavefront and profile is described. The algorithm is a combinatorial algorithm that uses a maximal independent vertex set for coarsening the adjacency graph of the matrix and an enhanced version of the Sloan algorithm on the coarsest graph. On a range of examples arising from practical applications, the multilevel algorithm is shown to produce orderings that are better than those produced by the Sloan algorithm and are of comparable quality to those obtained using the hybrid Sloan algorithm. Advantages over the hybrid Sloan algorithm are that the multilevel approach requires no spectral information and less CPU time.

Design of a Multicore Sparse Cholesky Factorization Using DAGs

The rapid emergence of multicore machines has led to the need to design new algorithms that are efficient on these architectures. Here, we consider the solution of sparse symmetric positive-definite linear systems by Cholesky factorization. We were motivated by the successful division of the computation in the dense case into tasks on blocks and use of a task manager to exploit all the parallelism that is available between these tasks, whose dependencies may be represented by a directed acyclic graph (DAG). Our sparse algorithm is built on the assembly tree and subdivides the work at each node into tasks on blocks of the Cholesky factor. The dependencies between these tasks may again be represented by a DAG. To limit memory requirements, blocks are updated directly rather than through generated-element matrices. Our algorithm is implemented within a new efficient and portable solver HSL_MA87. It is written in Fortran 95 plus OpenMP and is available as part of the software library HSL. Using problems arising from a range of applications, we present experimental results that support our design choices and demonstrate that HSL_MA87 obtains good serial and parallel times on our 8-core test machines. Comparisons are made with existing modern solvers and show that HSL_MA87 performs well, particularly in the case of very large problems.

An out-of-core sparse Cholesky solver

Direct methods for solving large sparse linear systems of equations are popular because of their generality and robustness. Their main weakness is that the memory they require usually increases rapidly with problem size. We discuss the design and development of the first release of a new symmetric direct solver that aims to circumvent this limitation by allowing the system matrix, intermediate data, and the matrix factors to be stored externally. The code, which is written in Fortran and called HSL_MA77, implements a multifrontal algorithm. The first release is for positive-definite systems and performs a Cholesky factorization. Special attention is paid to the use of efficient dense linear algebra kernel codes that handle the full-matrix operations on the frontal matrix and to the input/output operations. The input/output operations are performed using a separate package that provides a virtual-memory system and allows the data to be spread over many files; for very large problems these may be held on more than one device. Numerical results are presented for a collection of 30 large real-world problems, all of which were solved successfully.

An Arnoldi code for computing selected eigenvalues of sparse, real, unsymmetric matrices

Arnoldi methods can be more effective than subspace iterationmethods for computing the dominant eigenvalues of a large, sparse, real,unsymmetric matrix. A code, <?Pub Fmt bold>EB12<?Pub Fmt /bold>, for thesparse, unsymmetric eigenvalue problem based on a subspace iterationalgorithm, optionally combined with Chebychev acceleration, has recentlybeen described by Duff and Scott and is included in the HarwellSubroutine Library. In this article we consider variants of the methodof Arnoldi and discuss the design and development of a code to implementthese methods. The new code, which is called<?Pub Fmt bold>EB13<?Pub Fmt /bold>, offers the user the choice of abasic Arnoldi algorithm, an Arnoldi algorithm with Chebychevacceleration, and a Chebychev preconditioned Arnoldi algorithm. Eachmethod is available in blocked and unblocked form. The code may be usedto compute either the rightmost eigenvalues, the eigenvalues of largestabsolute value, or the eigenvalues of largest imaginary part. Theperformance of each option in the <?Pub Fmt bold>EB13<?Pub Fmt /bold>package is compared with that of subspace iteration on a range of testproblems, and on the basis of the results, advice is offered to the useron the appropriate choice of method. <abstrbyl>—<?Pub Fmt italic>Authors Abstract<?Pub Fmt /italic></abstrbyl>