Electron petrography of fine-grained matrix in the Karoonda C4 carbonaceous chondrite

The study of matrices of rare Type 4 carbonaceous chondrites can reveal important information on parent body rnetamorp~ic processes and provide a comparison with processes on parent bodies of ordinary chc-idrites. Reflectance spectra (Tholen, 1984) from the two largest asteroids in the asteroid belt, Ceres and Pallas, suggest that they may be metamorphosed carbonaceous chondrites. These two asteroids constitute - onethird of the mass in the asteroid belt implying that type 4-6 carbonaceous chondrites are poorly represented in the meteorite collection and may be of considerable importance. The matrix of the C4 chondrite Karoonda has been investigated using a JEOL 2000FX analytical electron microscope (AEM) with an attached Tracor-Northem TN5500 energy dispersive spectrometer (EDS). In previous studies (Scott and Taylor, 1985; Fitzgerald, 1979; Van Schmus, 1969), the petrography of the Karoonda matrix has been described as consisting largely of coarse-grained (50-200 urn in size) olivine and plagioclase (20-100 um in size), associated with micrometer sized magnetite and rare sulphides. AEM observations on matrix show that in addition to these large grains, there is a significant fraction (10 vol%) of interstitial fine grained phases « 5 urn). The mineralogy of these fine-grained phases differs in some respects from that of the coarser-grained matrix identified by optical and SEM techniques (Scott and Taylor, 1985; Fitzgerald, 1979; Van Schmus, 1969). I~ particular crystals of two compositionally distinct pyroxenes « 2 urn in size) have been identified which have not been previously observed in Karoonda by other analytical techniques. Thin film microanalyses (Mackinnon et al., 1986) of these two pyroxenes indicate compositions consistent with augite and low-Ca pyroxene (- Fs27). Fine-grained anhedral olivine « 2 urn size) is the most abundant phase with composition -Fa29' This composition is essentially indistinguishable from that determined for coarser-grained matrix olivines using an electron microprobe (Scott and Taylor, 1985; Fitzgerald, 1979; Van Schmus, 1969). All olivines are associated with subhedral magnetites « 1 urn size) which contain significant Cr (- 2%) and Al (- 1%) as was also noted for larger sized Karoonda magnetites by Delaney et al. (1985). It has recently been suggested (Burgess et al., 1987) on the basis of sulphur release profiles for S-isotope analyses of Karoonda that CaS04 (anhydrite) may be present. However, no sulphate phase has, as yet, been identified in the matrix of Karoonda. Low magnification contrast images suggest that Karoonda may have a significant porosity within the fine-grained matrix fraction. Most crystals are anhedral and do not show evidence for significant compaction. Individual grains often show single point contact with other grains which result in abundant intergranular voids. These voids frequently contain epoxy which was used as part of the specimen preparation procedure due to the friable nature of the bulk sample.

RcppArmadillo : accelerating R with high-performance C++ linear algebra

The R statistical environment and language has demonstrated particular strengths for interactive development of statistical algorithms, as well as data modelling and visualisation. Its current implementation has an interpreter at its core which may result in a performance penalty in comparison to directly executing user algorithms in the native machine code of the host CPU. In contrast, the C++ language has no built-in visualisation capabilities, handling of linear algebra or even basic statistical algorithms; however, user programs are converted to high-performance machine code, ahead of execution. A new method avoids possible speed penalties in R by using the Rcpp extension package in conjunction with the Armadillo C++ matrix library. In addition to the inherent performance advantages of compiled code, Armadillo provides an easy-to-use template-based meta-programming framework, allowing the automatic pooling of several linear algebra operations into one, which in turn can lead to further speedups. With the aid of Rcpp and Armadillo, conversion of linear algebra centered algorithms from R to C++ becomes straightforward. The algorithms retains the overall structure as well as readability, all while maintaining a bidirectional link with the host R environment. Empirical timing comparisons of R and C++ implementations of a Kalman filtering algorithm indicate a speedup of several orders of magnitude.

Ordered mixed-layer structures in the Mighei carbonaceous chondrite matrix

High resolution transmission electron microscopy of the Mighei carbonaceous chondrite matrix has revealed the presence of a new mixed layer structure material. This mixed-layer material consists of an ordered arrangement of serpentine-type (S) and brucite-type (B) layers in the sequence ... SBBSBB. ... Electron diffraction and imaging techniques show that the basal periodicity is ~ 17 Å. Discrete crystals of SBB-type material are typically curved, of small size (<1 μm) and show structural variations similar to the serpentine group minerals. Mixed-layer material also occurs in association with planar serpentine. Characteristics of SBB-type material are not consistent with known terrestrial mixed-layer clay minerals. Evidence for formation by a condensation event or by subsequent alteration of preexisting material is not yet apparent. © 1982.

New phyllosilicate types in a carbonaceous chondrite matrix

CARBONACEOUS chondrites provide valuable information as they are the least altered examples of early Solar System material1. The matrix constitutes a major proportion of carbonaceous chondrites. Despite many past attempts, unambiguous identification of the minerals in the matrix has not been totally successful2. This is mainly due to the extremely fine-grained nature of the matrix phases. Recently, progress in the characterisation of these phases has been made by electron diffraction studies3,4. We present here the direct observation, by high resolution imaging, of phases in carbonaceous chondrite matrices. We used ion-thinned sections from the Murchison C2(M) meteorite for transmission electron microscopy. The Murchison matrix contains both ordered and disordered inter-growths of serpentine-like and brucite-like layers. Such mixed-layer structures are new types of layer silicates. © 1979 Nature Publishing Group.

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focused recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the "best" choice for an initial stepsize, as well as developing effective strategies for stepsize control-the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy.

Numerical simulation of stochastic ordinary differential equations in biomathematical modelling

In this work we discuss the effects of white and coloured noise perturbations on the parameters of a mathematical model of bacteriophage infection introduced by Beretta and Kuang in [Math. Biosc. 149 (1998) 57]. We numerically simulate the strong solutions of the resulting systems of stochastic ordinary differential equations (SDEs), with respect to the global error, by means of numerical methods of both Euler-Taylor expansion and stochastic Runge-Kutta type.

Numerical methods for strong solutions of stochastic differential equations : an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations

The pioneering work of Runge and Kutta a hundred years ago has ultimately led to suites of sophisticated numerical methods suitable for solving complex systems of deterministic ordinary differential equations. However, in many modelling situations, the appropriate representation is a stochastic differential equation and here numerical methods are much less sophisticated. In this paper a very general class of stochastic Runge-Kutta methods is presented and much more efficient classes of explicit methods than previous extant methods are constructed. In particular, a method of strong order 2 with a deterministic component based on the classical Runge-Kutta method is constructed and some numerical results are presented to demonstrate the efficacy of this approach.

Order conditions of stochastic Runge-Kutta methods by B-series

In this paper, general order conditions and a global convergence proof are given for stochastic Runge Kutta methods applied to stochastic ordinary differential equations ( SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.

Numerical solutions of stochastic differential equations – implementation and stability issues

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively.

A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively. (C) 2000 Elsevier Science B.V. All rights reserved.

General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems

In many modeling situations in which parameter values can only be estimated or are subject to noise, the appropriate mathematical representation is a stochastic ordinary differential equation (SODE). However, unlike the deterministic case in which there are suites of sophisticated numerical methods, numerical methods for SODEs are much less sophisticated. Until a recent paper by K. Burrage and P.M. Burrage (1996), the highest strong order of a stochastic Runge-Kutta method was one. But K. Burrage and P.M. Burrage (1996) showed that by including additional random variable terms representing approximations to the higher order Stratonovich (or Ito) integrals, higher order methods could be constructed. However, this analysis applied only to the one Wiener process case. In this paper, it will be shown that in the multiple Wiener process case all known stochastic Runge-Kutta methods can suffer a severe order reduction if there is non-commutativity between the functions associated with the Wiener processes. Importantly, however, it is also suggested how this order can be repaired if certain commutator operators are included in the Runge-Kutta formulation. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if an s-stage SRK contains Stratonovich integrals up to order p then the strong order of the SRK cannot exceed min{(p + 1)/2, (s - 1)/2), p greater than or equal to 2, s greater than or equal to 3 or 1 if p = 1.