Sparse bayesian polynomial chaos approximations of elasto-plastic material models

In this paper we studied the uncertainty quantification in a functional approximation form of elastoplastic models parameterised by material uncertainties. The problem of estimating the polynomial chaos coefficients is recast in a linear regression form by taking into consideration the possible sparsity of the solution. Departing from the classical optimisation point of view, we take a slightly different path by solving the problem in a Bayesian manner with the help of new spectral based sparse Kalman filter algorithms

Parameter Estimation via Conditional Expectation --- A Bayesian Inversion

When a mathematical or computational model is used to analyse some system, it is usual that some parameters resp.\ functions or fields in the model are not known, and hence uncertain. These parametric quantities are then identified by actual observations of the response of the real system. In a probabilistic setting, Bayes's theory is the proper mathematical background for this identification process. The possibility of being able to compute a conditional expectation turns out to be crucial for this purpose. We show how this theoretical background can be used in an actual numerical procedure, and shortly discuss various numerical approximations

Bayesian parameter identification in plasticity

To evaluate the cyclic behaviour under different loading conditions using the kinematic and isotropic hardening theory of steel a Chaboche visco-plastic material model is employed. The parameters of a constitutive model are usually identified by minimization of the distance between model response and experimental data. However, measurement errors and differences in the specimens lead to deviations in the determined parameters. In this article the Choboche model is used and a stochastic simulation technique is applied to generate artificial data which exhibit the same stochastic behaviour as experimental data. Then the model parameters are identified by applying a variaty of Bayes’s theorem. Identified parameters are compared with the true parameters in the simulation and the efficiency of the identification method is discussed

Plasticity described by uncertain parameters - a variational inequality approach -

In this paper we consider the mixed variational formulation of the quasi-static stochastic plasticity with combined isotropic and kinematic hardening. By applying standard results in convex analysis we show that criteria for the existence, uniqueness, and convergence can be easily derived. In addition, we demonstrate the mathematical similarity with the corresponding deterministic formulation which further may be extended to a stochastic variational inequality of the first kind. The aim of this work is to consider the numerical approximation of variational inequalities by a “white noise analysis”. By introducing the random fields/processes used to model the displacements, stress and plastic strain and by approximating them by a combination of Karhunen-Lo`eve and polynomial chaos expansion, we are able to establish stochastic Galerkin and collocation methods. In the first approach, this is followed by a stochastic closest point projection algorithm in order to numerically solve the problem, giving an intrusive method relying on the introduction of the polynomial chaos algebra. As it does not rely on sampling, the method is shown to be very robust and accurate. However, the same procedure may be applied in another way, i.e. by calculating the residuum via high-dimensional integration methods (the second approach) giving a non-intrusive Galerkin techniques based on random sampling—Monte Carlo and related techniques—or deterministic sampling such as collocation methods. The third approach we present is in pure stochastic collocation manner. By highlighting the dependence of the random solution on the uncertain parameters, we try to investigate the influence of individual uncertain characteristics on the structure response by testing several numerical problems in plain strain or plane stress conditions

A deterministic filter for estimation of parameters describing inelastic heterogeneous media

We present a new, fully deterministic method to compute the updates for parameter estimates of quasi-static plasticity with combined kinematic and isotropic hardening from noisy measurements. The materials describing the elastic (reversible) and/or inelastic (irreversible) behaviour have an uncertain structure which further influences the uncertainty in the parameters such as bulk and shear modulus, hardening characteristics, etc. Due to this we formulate the problem as one of stochastic plasticity and try to identify parameters with the help of measurement data. However, in this setup the inverse problem is regarded as ill-posed and one has to apply some of regularisation techniques in order to ensure the existence, uniqueness and stability of the solution. Providing the apriori information next to the measurement data, we regularize the problem in a Bayesian setting which further allow us to identify the unknown parameters in a pure deterministic, algebraic manner via minimum variance estimator. The new approach has shown to be effective and reliable in comparison to most methods which take the form of integrals over the posterior and compute them by sampling, e.g. Markov chain Monte Carlo (MCMC)

Stochastic Plasticity - A Variational Inequality Formulation and Functional Approximation Approach. I: The Linear Case

In this paper we formulate and study the existence and uniqueness of the solution for a class of stochastic mixed variational inequalities arising in problems of infinitesimal elastoplasticity described by uncertain parameters. As a particular example we consider the quasi-static von Mises elastoplastic rate-independent evolution problem with linear elastic behaviour and hardening. For such a problem under the neccessary assumptions we show the equivalency between the variational inequality and a quadratic minimization problem described by a strictly convex, continuous, G^{a}teaux differentiable, and coercive functional on a Hilbert space. In order to find the unique minimiser we propose the stochastic closest point projection method, obtained by extension of the well known classical return alogorithms to the more general stochastic case. The method is, similarly to its deterministic counterpart, described by non-dissipative and dissipative operators

A Deterministic Filter for Non-Gaussian Bayesian Estimation

We present a fully deterministic method to compute sequential updates for stochastic state estimates of dynamic models from noisy measurements. It does not need any assumptions about the type of distribution for either data or measurement — in particular it does not have to assume any of them as Gaussian. It is based on a polynomial chaos expansion (PCE) of the stochastic variables of the model. We use a minimum variance estimator that combines an a priori state estimate and noisy measurements in a Bayesian way. For computational purposes, the update equation is projected onto a finite-dimensional PCE-subspace. The resulting Kalman-type update formula for the PCE coefficients can be efficiently computed solely within the PCE. As it does not rely on sampling, the method is deterministic, robust, and fast. In this paper we discuss the theory and practical implementation of the method. The original Kalman filter is shown to be a low-order special case. In a first experiment, we perform a bi-modal identification using noisy measurements. Additionally, we provide numerical experiments by applying it to the well known Lorenz-84 model and compare it to a related method, the ensemble Kalman filter

Direct Bayesian Update of Polynomial Chaos Representations

We present a fully deterministic approach to a probabilistic interpretation of inverse problems in which unknown quantities are represented by random fields or processes, described by a non-Gaussian prior distribution. The description of the introduced random fields is given in a ``white noise'' framework, which enables us to solve the stochastic forward problem through Galerkin projection onto polynomial chaos. With the help of such representation, the probabilistic identification problem is cast in a polynomial chaos expansion setting and the linear Bayesian form of updating. This representation leads to a corresponding new formulation of the minimum squared error estimator, obtained by its additional projection onto the polynomial chaos basis. By introducing the Hermite algebra this becomes a direct, purely algebraic way of computing the posterior, which is inexpensive to evaluate. In addition, we show that the well-known Kalman filter method is the low order part of this update. The proposed method has been tested on a stationary diffusion equation with prescribed source terms, characterised by an uncertain conductivity parameter which is then identified from limited and noisy data obtained by a measurement of the diffusing quantity

Fear of falling in obese women under 50 years of age: a cross-sectional study with exploration of the relationship with physical activity

An understanding of capacity for physical activity in obese populations should help guide interventions to promote physical activity. Fear of falling is a phenomenon reported in the elderly, which is associated with reduced mobility and lower physical activity levels. However, although falls are reportedly common in obese adults, fear of falling and its relationship with activity has not been investigated in younger obese populations. In a cross-sectional study, fear of falling was measured in 63 women aged 18 to 49 years, with mean BMI 42.1 kg/m (SD 10.3) using the Modified Falls Efficacy (MFES), the Consequences of Falling (COF) and the Modified Survey of Activities and Fear of Falling in the Elderly (MSAFFE) scales. The choice of scales was informed by prior qualitative interviews with obese younger women. Physical activity levels were measured at the same time using the International Physical Activity Questionnaire. The mean score for fear of falling scales, with 95% confidence intervals, were estimated. Chi-square tests and t-tests were used to explore differences in age, body mass index and fear of falling scores between fallers and non-fallers. For each fear of falling scale, binomial logistic regression was used to explore its relationship with physical activity. Mean scores suggested high levels of fear of falling: MFES [mean 7.7 (SD 2.7); median 8.5]; COF [mean 31.3 (SD 9.4)]; MSAFFE [mean 25.9 (SD 8.7); median 23]. Scores were significantly worse in fallers (  = 42) compared to non-fallers (  = 21). MFES and MSAFFE were independently associated with lower levels of physical activity [odds ratio = 0.65, 95% Cl 0.44 to 0.96 and odds ratio = 1.14, 95% CI 1.01 to 1.28 respectively], when adjusted for age, BMI and depression. This study confirms that fear of falling is present in obese women under 50 years of age. It suggests that it is associated with low levels of physical activity. These novel findings warrant further research to understand capacity for physical and incidental activity in obese adults in both genders and suggest innovative interventions to promote lifestyle changes and/or consideration of falls prevention in this population

The genome of the crustacean Parhyale hawaiensis, a model for animal development, regeneration, immunity and lignocellulose digestion

The amphipod crustacean Parhyale hawaiensis is a blossoming model system for studies of developmental mechanisms and more recently regeneration. We have sequenced the genome allowing annotation of all key signaling pathways, transcription factors, and non-coding RNAs that will enhance ongoing functional studies. Parhyale is a member of the Malacostraca clade, which includes crustacean food crop species. We analysed the immunity related genes of Parhyale as an important comparative system for these species, where immunity related aquaculture problems have increased as farming has intensified. We also find that Parhyale and other species within Multicrustacea contain the enzyme sets necessary to perform lignocellulose digestion ('wood eating'), suggesting this ability may predate the diversification of this lineage. Our data provide an essential resource for further development of Parhyale as an experimental model. The first malacostracan genome will underpin ongoing comparative work in food crop species and research investigating lignocellulose as an energy source. DOI: http://dx.doi.org/10.7554/eLife.20062.00