Parallel linear system solvers for Runge-Kutta methods

If the nonlinear systems arising in implicit Runge-Kutta methods like the Radau IIA methods are iterated by (modified) Newton, then we have to solve linear systems whose matrix of coefficients is of the form $I-A otimes hJ$ with A the Runge-Kutta matrix and J an approximation to the Jacobian of the righthand side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decompositions. We try to reduce these costs by solving the linear systems by a second (inner) iteration process. This inner iteration process is such that each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems is of the form $I-B otimes hJ$ where B is similar to a diagonal matrix with positive diagonal entries. Hence, after performing a similarity transformation, the linear systems are decoupled into s subsystems, so that the costs of the LU-decomposition are reduced to the costs of s LU-decompositions of dimension d. Since these LU-decompositions can be computed in parallel, the effective LU-costs on a parallel computer system are reduced by a factor $s^3$. It will be shown that matrices B can be constructed such that the inner iterations converge whenever A and J have their eigenvalues in the positive and nonpositive halfplane, respectively. The theoretical results will be illustrated by a few numerical examples. A parallel implementation on the four-processor Cray-C98/4256 shows a speed-up ranging from at least 2.4 until at least 3.1 with respect to RADAU5 applied in one-processor mode

Parallel Störmer-Cowell methods for high-precision orbit computations

Many orbit problems in celestial mechanics are described by (nonstiff) initial-value problems (IVPs) for second-order ordinary differential equations of the form y' = {bf f (y). The most successful integration methods are based on high-order Runge-Kutta-Nyström formulas. However, these methods were designed for sequential is paper, we consider high-order parallel methods that are not based on Runge-Kutta-Nyström formulas, but which fit into the class of general linear methods. In each step, these methods compute blocks of k approximate solution values (or stage values) at k different points using the whole previous block of solution values. The k stage values can be computed in parallel, so that on a k-processor computer system such methods effectively perform as a one-value method. The block methods considered in this paper are such that each equation defining a stage value resembles a linear multistep equation of the familiar Störmer-Cowell type. For k = 4 and k = 5 we constructed explicit PSC methods with stage order q = k and step point order p = k+1 and implicit PSC methods with q = k+1 and p = k+2. For k = 6 we can construct explicit PSC methods with q = k and p = k+2 and implicit PSC methods with q = k+1 and p = k+3. It turns out that for k = 5 the abscissae of the stage values can be chosen such that only k-1 stage values in each block have to be computed, so that the number of computational stages, and hence the number of processors and the number of starting values needed, reduces to k* = k-1. The numerical examples reported in this paper show that the effective number of righthand side evaluations required by a variable stepsize implementation of the 10th-order PSC method is 4 up to 30 times less than required by the Runge-Kutta-Nyström code DOPRIN (which is considered as one of the most efficient sequential codes for second-order ODEs). Furthermore, a comparison with the 12th-order parallel code PIRKN reveals that the PSC code is, in spite of its lower order, at least equally efficient, and in most cases more efficient than PIRKN

Parallel Störmer-Cowell methods for high-precision orbit computations

Many orbit problems in celestial mechanics are described by (nonstiff) initial-value problems (IVPs) for second-order ordinary differential equations of the form y' = {bf f (y). The most successful integration methods are based on high-order Runge-Kutta-Nyström formulas. However, these methods were designed for sequential is paper, we consider high-order parallel methods that are not based on Runge-Kutta-Nyström formulas, but which fit into the class of general linear methods. In each step, these methods compute blocks of k approximate solution values (or stage values) at k different points using the whole previous block of solution values. The k stage values can be computed in parallel, so that on a k-processor computer system such methods effectively perform as a one-value method. The block methods considered in this paper are such that each equation defining a stage value resembles a linear multistep equation of the familiar Störmer-Cowell type. For k = 4 and k = 5 we constructed explicit PSC methods with stage order q = k and step point order p = k+1 and implicit PSC methods with q = k+1 and p = k+2. For k = 6 we can construct explicit PSC methods with q = k and p = k+2 and implicit PSC methods with q = k+1 and p = k+3. It turns out that for k = 5 the abscissae of the stage values can be chosen such that only k-1 stage values in each block have to be computed, so that the number of computational stages, and hence the number of processors and the number of starting values needed, reduces to k* = k-1. The numerical examples reported in this paper show that the effective number of righthand side evaluations required by a variable stepsize implementation of the 10th-order PSC method is 4 up to 30 times less than required by the Runge-Kutta-Nyström code DOPRIN (which is considered as one of the most efficient sequential codes for second-order ODEs). Furthermore, a comparison with the 12th-order parallel code PIRKN reveals that the PSC code is, in spite of its lower order, at least equally efficient, and in most cases more efficient than PIRKN

Palaeozoic oolitic ironstone of the French Armorican Massif: a chemical and structural trap for orogenic base metal-As-Sb-Au mineralization during Hercynian strike-slip deformation.

In the Saint-Aubin-des-Châteaux quarry (Armorican Hercynian belt, western France), an epigenetic hydrothermal alteration affects an oolitic ironstone layer intercalated within the Lower Ordovician Grès armoricain Formation. The hydrothermal overprint produced pervasive and massive sulphidation with stratoid pyritized lenticular bodies within the oolitic ironstone layer. These sulphide lenses are spatially associated with strike-slip faults and extend laterally from them. Following the massive sulphidation stage (Fe-As, stage 1), subsequent fracturing allowed the deposition of base metals (stage 2) and Pb-Sb-Au (stage 3) parageneses in veins. The dominant brittle structures are vertical extension veins, conjugate shear veins and strike-slip faults of various orders. All these structures are filled with the same paragenetic sequence. Deformation analysis allows the identification of structures that developed incrementally via right lateral simple shear compatible with bulk strain affecting the Central Armorican Domain. Each increment corresponds to a fracture set filled with specific parageneses. Successive hydrothermal pulses reflect clockwise rotation of the horizontal shortening direction. Geothermometry on chlorite and arsenopyrite shows an input of hot hydrothermal fluids (maximum of 390-350°C) during the main sulphide stage 1. The subsequent stages present a marked temperature drop (300-275°C). Lead isotopes suggest that the lead source is similar for all hydrothermal stages and corresponds to the underlying Neo-proterozoic basement. Lead isotope data, relative ages of deformation and comparison with neighbouring deposits suggest large-scale fluid pulses occurred during the whole Hercynian orogeny rather than pulses restricted to the late Hercynian period. The vicinity of the Hercynian internal domain appears as a key-control for deformation and fluid flow in the oolitic ironstones which acted as a chemical and structural trap for the hydrothermal fluids. The epigenetic mineralization of Saint-Aubin-des-Châteaux appears to be very similar to epigenetic sulphidation described in BIF-hosted gold deposits

Excimer laser coronary angioplasty in the Netherlands: preamble for a randomized study

The immediate outcome of ELCA by XeCl excimer laser radiation is described in 53 patients who were selected to undergo ELCA from December 1990 to September 1991 in two centers that are currently performing ELCA in the Netherlands. Immediate success rates on the basis of visual assessment of the angiogram were as follows. Laser success (> 20% reduction of diameter stenosis after ELCA alone) was observed in 77% of patients, procedural success (< 50% residual stenosis after ELCA with or without adjunctive balloon dilatation [PTCA]) in 91%, and clinical success (procedural success without clinical complications) in 83% of patients. Quantitative coronary angiography by automated contour detection was performed in 31 patients who underwent ELCA in the Thoraxcenter. The minimal luminal diameter (mean +/- SD) of the treated coronary segments increased from 0.77 +/- 0.41 mm to 1.24 +/- 0.25 mm after ELCA and further to 1.67 +/- 0.29 mm after adjunctive PTCA in 25 patients. The present experience is put in perspective of results initially reported by other centers and compared with data from multicenter registries of ELCA. Finally, a short description is given of the design of a prospective, randomized trial of ELCA versus conventional PTCA (AMRO trial)

Si ion implantation-induced damage in fused silica probed by variable-energy positrons

Samples of synthetic fused silica have been implanted at room temperature with silicon ions of energy 1.5 MeV. Fluences ranged from 1011 to 1013 cm−2. Samples were probed using variable‐energy positron annihilation spectroscopy. The Doppler‐broadening S parameter corresponding to the implanted region decreased with increasing fluence and saturated at a fluence of 1013 cm−2. It is shown that the decrease in the S parameter is due to the suppression of positronium (Ps) which is formed in the preimplanted material, due to the competing process of implantation‐induced trapping of positrons. In order to satisfactorily model the positron data it was necessary to account for positron trapping due to defects created by both electronic and nuclear stopping of the implanted ions. Annealing of the 1013 cm−2 sample resulted in measurable recovery of the preimplanted S parameter spectrum at 350 °C and complete recovery to the preimplanted condition at 600 °C. Volume compaction was also observed afterimplantation. Upon annealing, the compaction was seen to decrease by 75%