Stochastic models: theory and simulation.

Many problems in applied science and engineering involve physical phenomena that behave randomly in time and/or space. Examples are diverse and include turbulent flow over an aircraft wing, Earth climatology, material microstructure, and the financial markets. Mathematical models for these random phenomena are referred to as stochastic processes and/or random fields, and Monte Carlo simulation is the only general-purpose tool for solving problems of this type. The use of Monte Carlo simulation requires methods and algorithms to generate samples of the appropriate stochastic model; these samples then become inputs and/or boundary conditions to established deterministic simulation codes. While numerous algorithms and tools currently exist to generate samples of simple random variables and vectors, no cohesive simulation tool yet exists for generating samples of stochastic processes and/or random fields. There are two objectives of this report. First, we provide some theoretical background on stochastic processes and random fields that can be used to model phenomena that are random in space and/or time. Second, we provide simple algorithms that can be used to generate independent samples of general stochastic models. The theory and simulation of random variables and vectors is also reviewed for completeness

A decision-theoretic method for surrogate model selection.

The use of surrogate models to approximate computationally expensive simulation models, e.g., large comprehensive finite element models, is widespread. Applications include surrogate models for design, sensitivity analysis, and/or uncertainty quantification. Typically, a surrogate model is defined by a postulated functional form; values for the surrogate model parameters are estimated using results from a limited number of solutions to the comprehensive model. In general, there may be multiple surrogate models, each defined by possibly a different functional form, consistent with the limited data from the comprehensive model. We refer to each as a candidate surrogate model. Methods are developed and applied to select the optimal surrogate model from the collection of candidate surrogate models. The classical approach is to select the surrogate model that best fits the data provided by the comprehensive model; this technique is independent of the model use and, therefore, may be inappropriate for some applications. The proposed approach applies techniques from decision theory, where postulated utility functions are used to quantify the model use. Two applications are presented to illustrate the methods. These include surrogate model selection for the purpose of: (1) estimating the minimum of a deterministic function, and (2) the design under uncertainty of a physical system

Methods for model selection in applied science and engineering.

Mathematical models are developed and used to study the properties of complex systems and/or modify these systems to satisfy some performance requirements in just about every area of applied science and engineering. A particular reason for developing a model, e.g., performance assessment or design, is referred to as the model use. Our objective is the development of a methodology for selecting a model that is sufficiently accurate for an intended use. Information on the system being modeled is, in general, incomplete, so that there may be two or more models consistent with the available information. The collection of these models is called the class of candidate models. Methods are developed for selecting the optimal member from a class of candidate models for the system. The optimal model depends on the available information, the selected class of candidate models, and the model use. Classical methods for model selection, including the method of maximum likelihood and Bayesian methods, as well as a method employing a decision-theoretic approach, are formulated to select the optimal model for numerous applications. There is no requirement that the candidate models be random. Classical methods for model selection ignore model use and require data to be available. Examples are used to show that these methods can be unreliable when data is limited. The decision-theoretic approach to model selection does not have these limitations, and model use is included through an appropriate utility function. This is especially important when modeling high risk systems, where the consequences of using an inappropriate model for the system can be disastrous. The decision-theoretic method for model selection is developed and applied for a series of complex and diverse applications. These include the selection of the: (1) optimal order of the polynomial chaos approximation for non-Gaussian random variables and stationary stochastic processes, (2) optimal pressure load model to be applied to a spacecraft during atmospheric re-entry, and (3) optimal design of a distributed sensor network for the purpose of vehicle tracking and identification

Convergence properties of polynomial chaos approximations for L2 random variables.

Polynomial chaos (PC) representations for non-Gaussian random variables are infinite series of Hermite polynomials of standard Gaussian random variables with deterministic coefficients. For calculations, the PC representations are truncated, creating what are herein referred to as PC approximations. We study some convergence properties of PC approximations for L{sub 2} random variables. The well-known property of mean-square convergence is reviewed. Mathematical proof is then provided to show that higher-order moments (i.e., greater than two) of PC approximations may or may not converge as the number of terms retained in the series, denoted by n, grows large. In particular, it is shown that the third absolute moment of the PC approximation for a lognormal random variable does converge, while moments of order four and higher of PC approximations for uniform random variables do not converge. It has been previously demonstrated through numerical study that this lack of convergence in the higher-order moments can have a profound effect on the rate of convergence of the tails of the distribution of the PC approximation. As a result, reliability estimates based on PC approximations can exhibit large errors, even when n is large. The purpose of this report is not to criticize the use of polynomial chaos for probabilistic analysis but, rather, to motivate the need for further study of the efficacy of the method

Numerical Techniques to Evaluate Moments of Dynamic System Response

Probabilistic uncertainty is a phenomenon that occurs to a certain degree in many engineering applications. The effects that this uncertainty has upon a given system response are a matter of some concern. Techniques which provide insight to these effects will be required as modeling and prediction becomes a more vital tool in the engineering design process. The purpose of this paper is to outline a procedure to evaluate uncertainty in dynamic system response exploiting various numerical methods. Specifically, the goal is to attain the statistics of the response with minimal computational effort. Numerical interpolation and integration techniques are utilized in conjunction with the iterative form of the Advanced Mean Value (AMV+) method to efficiently and accurately estimate statistical moments of the response random process. A numerical example illustrating the use of this analytical tool in a practical framework is presented

An overview of reliability assessment and control for design of civil engineering structures

Random variations, whether they occur in the input signal or the system parameters, are phenomena that occur in nearly all engineering systems of interest. As a result, nondeterministic modeling techniques must somehow account for these variations to ensure validity of the solution. As might be expected, this is a difficult proposition and the focus of many current research efforts. Controlling seismically excited structures is one pertinent application of nondeterministic analysis and is the subject of the work presented herein. This overview paper is organized into two sections. First, techniques to assess system reliability, in a context familiar to civil engineers, are discussed. Second, and as a consequence of the first, active control methods that ensure good performance in this random environment are presented. It is the hope of the authors that these discussions will ignite further interest in the area of reliability assessment and design of controlled civil engineering structures

Coefficient shifts in geographical ecology: an empirical evaluation of spatial and non-spatial regression

Copyright © 2009 The Authors. Copyright © ECOGRAPHY 2009.A major focus of geographical ecology and macro ecology is to understand the causes of spatially structured ecological patterns. However, achieving this understanding can be complicated when using multiple regressions, because the relative importance of explanatory variables, as measured by regression coefficients, can shift depending on whether spatially explicit or non-spatial modelling is used. However, the extent to which coefficients may shift and why shifts occur are unclear. Here, we analyze the relationship between environmental predictors and the geographical distribution of species richness, body size, range size and abundance in 97 multi-factorial data sets. Our goal was to compare standardized partial regression coefficients of non-spatial ordinary least squares regressions (i.e. models fitted using ordinary least squares without taking autocorrelation into account; “OLS models” hereafter) and eight spatial methods to evaluate the frequency of coefficient shifts and identify characteristics of data that might predict when shifts are likely. We generated three metrics of coefficient shifts and eight characteristics of the data sets as predictors of shifts. Typical of ecological data, spatial autocorrelation in the residuals of OLS models was found in most data sets. The spatial models varied in the extent to which they minimized residual spatial autocorrelation. Patterns of coefficient shifts also varied among methods and datasets, although the magnitudes of shifts tended to be small in all cases. We were unable to identify strong predictors of shifts, including the levels of autocorrelation in either explanatory variables or model residuals. Thus, changes in coefficients between spatial and non-spatial methods depend on the method used and are largely idiosyncratic, making it difficult to predict when or why shifts occur. We conclude that the ecological importance of regression coefficients cannot be evaluated with confidence irrespective of whether spatially explicit modelling is used or not. Researchers may have little choice but to be more explicit about the uncertainty of models and more cautious in their interpretation

Blocking Synthesis of the Variant Surface Glycoprotein Coat in Trypanosoma brucei Leads to an Increase in Macrophage Phagocytosis Due to Reduced Clearance of Surface Coat Antibodies

The extracellular bloodstream form parasite Trypanosoma brucei is supremely adapted to escape the host innate and adaptive immune system. Evasion is mediated through an antigenically variable Variant Surface Glycoprotein (VSG) coat, which is recycled at extraordinarily high rates. Blocking VSG synthesis triggers a precytokinesis arrest where stalled cells persist for days in vitro with superficially intact VSG coats, but are rapidly cleared within hours in mice. We therefore investigated the role of VSG synthesis in trypanosome phagocytosis by activated mouse macrophages. T. brucei normally effectively evades macrophages, and induction of VSG RNAi resulted in little change in phagocytosis of the arrested cells. Halting VSG synthesis resulted in stalled cells which swam directionally rather than tumbling, with a significant increase in swim velocity. This is possibly a consequence of increased rigidity of the cells due to a restricted surface coat in the absence of VSG synthesis. However if VSG RNAi was induced in the presence of anti-VSG221 antibodies, phagocytosis increased significantly. Blocking VSG synthesis resulted in reduced clearance of anti-VSG antibodies from the trypanosome surface, possibly as a consequence of the changed motility. This was particularly marked in cells in the G2/ M cell cycle stage, where the half-life of anti-VSG antibody increased from 39.3 ± 4.2 seconds to 99.2 ± 15.9 seconds after induction of VSG RNAi. The rates of internalisation of bulk surface VSG, or endocytic markers like transferrin, tomato lectin or dextran were not significantly affected by the VSG synthesis block. Efficient elimination of anti-VSG-antibody complexes from the trypanosome cell surface is therefore essential for trypanosome evasion of macrophages. These experiments highlight the essentiality of high rates of VSG recycling for the rapid removal of host opsonins from the parasite surface, and identify this process as a key parasite virulence factor during a chronic infection

Computational thermal, chemical, fluid, and solid mechanics for geosystems management.

This document summarizes research performed under the SNL LDRD entitled - Computational Mechanics for Geosystems Management to Support the Energy and Natural Resources Mission. The main accomplishment was development of a foundational SNL capability for computational thermal, chemical, fluid, and solid mechanics analysis of geosystems. The code was developed within the SNL Sierra software system. This report summarizes the capabilities of the simulation code and the supporting research and development conducted under this LDRD. The main goal of this project was the development of a foundational capability for coupled thermal, hydrological, mechanical, chemical (THMC) simulation of heterogeneous geosystems utilizing massively parallel processing. To solve these complex issues, this project integrated research in numerical mathematics and algorithms for chemically reactive multiphase systems with computer science research in adaptive coupled solution control and framework architecture. This report summarizes and demonstrates the capabilities that were developed together with the supporting research underlying the models. Key accomplishments are: (1) General capability for modeling nonisothermal, multiphase, multicomponent flow in heterogeneous porous geologic materials; (2) General capability to model multiphase reactive transport of species in heterogeneous porous media; (3) Constitutive models for describing real, general geomaterials under multiphase conditions utilizing laboratory data; (4) General capability to couple nonisothermal reactive flow with geomechanics (THMC); (5) Phase behavior thermodynamics for the CO2-H2O-NaCl system. General implementation enables modeling of other fluid mixtures. Adaptive look-up tables enable thermodynamic capability to other simulators; (6) Capability for statistical modeling of heterogeneity in geologic materials; and (7) Simulator utilizes unstructured grids on parallel processing computers