Linear iterative solvers for implicit ODE methods

The numerical solution of stiff initial value problems, which lead to the problem of solving large systems of mildly nonlinear equations are considered. For many problems derived from engineering and science, a solution is possible only with methods derived from iterative linear equation solvers. A common approach to solving the nonlinear equations is to employ an approximate solution obtained from an explicit method. The error is examined to determine how it is distributed among the stiff and non-stiff components, which bears on the choice of an iterative method. The conclusion is that error is (roughly) uniformly distributed, a fact that suggests the Chebyshev method (and the accompanying Manteuffel adaptive parameter algorithm). This method is described, also commenting on Richardson's method and its advantages for large problems. Richardson's method and the Chebyshev method with the Mantueffel algorithm are applied to the solution of the nonlinear equations by Newton's method

Leapfrog variants of iterative methods for linear algebra equations

Two iterative methods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given

Orthogonal Hessenberg reduction and orthogonal Krylov subspace bases

We study necessary and sufficient conditions that a nonsingular matrix A can be B-orthogonally reduced to upper Hessenberg form with small bandwidth. By this we mean the existence of a decomposition AV=VH, where H is upper Hessenberg with few nonzero bands, and the columns of V are orthogonal in an inner product generated by a hermitian positive definite matrix B. The classical example for such a decomposition is the matrix tridiagonalization performed by the hermitian Lanczos algorithm, also called the orthogonal reduction to tridiagonal form. Does there exist such a decomposition when A is nonhermitian? In this paper we completely answer this question. The related (but not equivalent) question of necessary and sufficient conditions on A for the existence of short-term recurrences for computing B-orthogonal Krylov subspace bases was completely answered by the fundamental theorem of Faber and Manteuffel [SIAM J. Numer. Anal.}, 21 (1984), pp. 352--362]. We give a detailed analysis of B-normality, the central condition in both the Faber--Manteuffel theorem and our main theorem, and show how the two theorems are related. Our approach uses only elementary linear algebra tools. We thereby provide new insights into the principles behind Krylov subspace methods, that are not provided when more sophisticated tools are employed

Analysis of liquid-waste injection wells in Illinois by mathematical models

This report contains the results of a preliminary theoretical study of the fate of liquid industrial wastes injected into deep geologic formations. The Jones and Laughlin Corporation well was used as a model and the geology of the area was idealized into a 15-layered homogeneous and an isotropic mathematical model. The finite element method was tested and proved to be an effective mathematical tool in the solution of the equation of flow. The flow and pressure build-up show that the rocks are capable of receiving greater volumes of waste than are now being injected without endangering the integrity of the aquifer or the confining layer.The mass-transport equation for large and complex ground-water reservoir systems was investigated, and it was concluded that the dispersion and diffusion parts of the equation are relatively insignificant, and under extreme conditions the dispersed zone will not be more than a few feet wide. Therefore, it was concluded that a more practical approach to the problem would be the solution of a system with a moving interface boundary in which mass transport results mainly from convection.To overcome difficulties encountered with computer time and memory in the solution of the mass-transport equation for large complex systems, an iterative method is proposed for the solution of the equations, which substantially reduces these difficulties.U.S. Department of the InteriorU.S. Geological Surve

IFNβ Protects Neurons from Damage in a Murine Model of HIV-1 Associated Brain Injury.

Infection with human immunodeficiency virus-1 (HIV-1) causes brain injury. Type I interferons (IFNα/β) are critical mediators of any anti-viral immune response and IFNβ has been implicated in the temporary control of lentiviral infection in the brain. Here we show that transgenic mice expressing HIV-1 envelope glycoprotein 120 in their central nervous system (HIVgp120tg) mount a transient IFNβ response and provide evidence that IFNβ confers neuronal protection against HIVgp120 toxicity. In cerebrocortical cell cultures, neuroprotection by IFNβ against gp120 toxicity is dependent on IFNα receptor 1 (IFNAR1) and the β-chemokine CCL4, as IFNAR1 deficiency and neutralizing antibodies against CCL4, respectively, abolish the neuroprotective effects. We find in vivo that IFNβ mRNA is significantly increased in HIVgp120tg brains at 1.5, but not 3 or 6 months of age. However, a four-week intranasal IFNβ treatment of HIVgp120tg mice starting at 3.5 months of age increases expression of CCL4 and concomitantly protects neuronal dendrites and pre-synaptic terminals in cortex and hippocampus from gp120-induced damage. Moreover, in vivo and in vitro data suggests astrocytes are a major source of IFNβ-induced CCL4. Altogether, our results suggest exogenous IFNβ as a neuroprotective factor that has potential to ameliorate in vivo HIVgp120-induced brain injury

Tumour genomic and microenvironmental heterogeneity as integrated predictors for prostate cancer recurrence: a retrospective study

Clinical prognostic groupings for localised prostate cancers are imprecise, with 30–50% of patients recurring after image-guided radiotherapy or radical prostatectomy. We aimed to test combined genomic and microenvironmental indices in prostate cancer to improve risk stratification and complement clinical prognostic factors

Implementation of an adaptive algorithm for Richardson's method

AbstractWe discuss the implementation of an adaptive algorithm proposed by one of us. The algorithm is a hybrid of the gmres method and Richardson's method. Richardson's method (RM) depends on a set of parameters that are computed by minimizing the L2 norm of a polynomial over the convex hull of eigenvalues. Execution of GMRES yields not only an approximate solution but also the approximate convex hull. RM is used to avoid storing and working with a large number of vectors as GMRES often requires. This method is also advantageous for the solution of large problems. We consider several test problems and compare our algorithm primarily with the conjugate-gradient-squared algorithm, but also with GMRES and to CG (applied to the normal equations). For many (test) problems our algorithm takes roughly 50 percent more work than the conjugate-gradient-squared algorithm, although if the matrix is either preconditioned or indefinite, our algorithm is more efficient. However, our algorithm currently imposes an undesirable burden on the user, who is invited to consider a variety of numerical parameters to manipulate, such as the number of steps of rm, in order to enhance performance: the values we suggest are only empirical