Multilevel Monte Carlo covariance estimation for the computation of Sobol' indices

International audienceCrude and quasi Monte Carlo (MC) sampling techniques are common tools dedicated to estimating statistics (expectation, variance, covariance) of a random quantity of interest.We focus here on the uncertainty quantification framework where the quantity of interest is the output of a numerical simulator fed with uncertain input parameters.Then, sampling the output involves running the simulator for different samples of the inputs, which may be computationally time-consuming.To reduce the cost of sampling, a first approach consists in replacing the numerical simulator by a surrogate model that is cheaper to evaluate, thus making it possible to generate more samples of the output and therefore leading to a lower sampling error.However, this approach adds to the sampling error an unavoidable model error.Another approach, which does not introduce any model error, is the so-called multilevel MC (MLMC) method.Given a sequence of levels corresponding to numerical simulators with increasing accuracy and computational cost, MLMC combines samples obtained at different levels to construct an estimator at a reduced cost compared to standard MC sampling.In this paper, we derive and analyze multilevel covariance estimators and adapt the MLMC convergence theorem in terms of the corresponding covariances and fourth order moments. We propose a multilevel algorithm driven by a target cost as an alternative to typical algorithms driven by a target accuracy.These results are used in a sensitivity analysis context in order to derive a multilevel estimation of Sobol' indices, whose building blocks can be written as covariance terms in a pick-and-freeze formulation.These contributions are successfully tested on an initial value problem with random parameters

Pedestrian Spatial Self-organization According to its Nearest Neighbor Position

AbstractThe paper describes relation between positions of individuals in a crowd, namely dependence between position of a pedestrian and position of his/her nearest neighbors. Two main characteristics have been analyzed: nth nearest neighbors’ spatial and angular distributions. At first sight, people in human crowd seem to be located randomly, however, our findings indicate that there are clearly visible patterns in analyzed characteristics. We discover symptoms of strong correlations between position of closely located pedestrians. Simple, local movement rules for pedestrian are proposed to explain observed patterns

FRAP, FLIM, and FRET: Detection and analysis of cellular dynamics on a molecular scale using fluorescence microscopy

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/112297/1/mrd22501.pd

Aligned electrospun nanofibers specify the direction of dorsal root ganglia neurite growth

Nerve injury, a significant cause of disability, may be treated more effectively using nerve guidance channels containing longitudinally aligned fibers. Aligned, electrospun nanofibers direct the neurite growth of immortalized neural stem cells, demonstrating potential for directing regenerating neurites. However, no study of neurite guidance on these fibers has yet been performed with primary neurons. Here, we examined neurites from dorsal root ganglia explants on electrospun poly- L -lactate nanofibers of high, intermediate, and random alignment. On aligned fibers, neurites grew radially outward from the ganglia and turned to follow the fibers upon contact. Neurite guidance was robust, with neurites never leaving the fibers to grow on the surrounding cover slip. To compare the alignment of neurites to that of the nanofiber substrates, Fourier methods were used to quantify the alignment. Neurite alignment, however striking, was inferior to fiber alignment on all but the randomly aligned fibers. Neurites on highly aligned substrates were 20 and 16% longer than neurites on random and intermediate fibers, respectively. Schwann cells on fibers assumed a very narrow morphology compared to those on the surrounding coverslip. The robust neurite guidance demonstrated here is a significant step toward the use of aligned, electrospun nanofibers for nerve regeneration. © 2007 Wiley Periodicals, Inc. J Biomed Mater Res, 2007Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57401/1/31285_ftp.pd

A filtered multilevel Monte Carlo method for estimating the expectation of discretized random fields

We investigate the use of multilevel Monte Carlo (MLMC) methods for estimating the expectation of discretized random fields. Specifically, we consider a setting in which the input and output vectors of the numerical simulators have inconsistent dimensions across the multilevel hierarchy. This requires the introduction of grid transfer operators borrowed from multigrid methods. Starting from a simple 1D illustration, we demonstrate numerically that the resulting MLMC estimator deteriorates the estimation of high-frequency components of the discretized expectation field compared to a Monte Carlo (MC) estimator. By adapting mathematical tools initially developed for multigrid methods, we perform a theoretical spectral analysis of the MLMC estimator of the expectation of discretized random fields, in the specific case of linear, symmetric and circulant simulators. This analysis provides a spectral decomposition of the variance into contributions associated with each scale component of the discretized field. We then propose improved MLMC estimators using a filtering mechanism similar to the smoothing process of multigrid methods. The filtering operators improve the estimation of both the small- and large-scale components of the variance, resulting in a reduction of the total variance of the estimator. These improvements are quantified for the specific class of simulators considered in our spectral analysis. The resulting filtered MLMC (F-MLMC) estimator is applied to the problem of estimating the discretized variance field of a diffusion-based covariance operator, which amounts to estimating the expectation of a discretized random field. The numerical experiments support the conclusions of the theoretical analysis even with non-linear simulators, and demonstrate the improvements brought by the proposed F-MLMC estimator compared to both a crude MC and an unfiltered MLMC estimator

The potential of label-free nonlinear optical molecular microscopy to non-invasively characterize the viability of engineered human tissue constructs

AbstractNonlinear optical molecular imaging and quantitative analytic methods were developed to non-invasively assess the viability of tissue-engineered constructs manufactured from primary human cells. Label-free optical measures of local tissue structure and biochemistry characterized morphologic and functional differences between controls and stressed constructs. Rigorous statistical analysis accounted for variability between human patients. Fluorescence intensity-based spatial assessment and metabolic sensing differentiated controls from thermally-stressed and from metabolically-stressed constructs. Fluorescence lifetime-based sensing differentiated controls from thermally-stressed constructs. Unlike traditional histological (found to be generally reliable, but destructive) and biochemical (non-invasive, but found to be unreliable) tissue analyses, label-free optical assessments had the advantages of being both non-invasive and reliable. Thus, such optical measures could serve as reliable manufacturing release criteria for cell-based tissue-engineered constructs prior to human implantation, thereby addressing a critical regulatory need in regenerative medicine

Min–max optimization of node‐targeted attacks in service networks

peer reviewedThis article considers resilience of service networks that are composed of service and control nodes to node‐targeted attacks. Two complementary problems of selecting attacked nodes and placing control nodes reflect the interaction between the network operator and the network attacker. This interaction can be analyzed within the framework of game theory. Considering the limited performance of the previously introduced iterative solution algorithms based on non‐compact problem models, new compact integer programming formulations of the node attack optimization problem are proposed, which are based on the notion of pseudo‐components and on a bilevel model. The efficiency of the new formulations is illustrated by the numerical study that uses two reference networks (medium‐size and large‐size), and a wide range of the sizes of attacks and controllers placements

Evaluation of the swirl characteristics of a tidal stream turbine wake

© 2015 The Authors. Published by Elsevier Ltd. Tidal stream turbines (TSTs) produce a rotating downstream wake. This paper describes the characteristics of the swirl flow in the wake of a TST with a view of comparing these against classical swirl theory and investigating whether swirl is an important factor in wake recovery prediction. Using computational fluid dynamics the paper describes the characteristics of velocities, pressure drop, viscosity and swirl number of 2, 3 and 4 bladed TSTs. To provide confidence in the results the characteristics are compared to the findings in the literature for a set of generic swirl generators. The swirl numbers for the TSTs in a 3.08 m/s tidal (plug) flow were found to be between 0.14 and 0.28, which describes a weak or very weak swirl flow. Whilst the characteristics are in agreement with theory it also means that the swirl component of the wake is not coupled with the axial component and cannot be used to estimate the wake length. However, peak swirl number for the 4 bladed turbine is close to the threshold of 0.3 at which axial velocity starts to become coupled with tangential velocity and therefore wake recovery may be related to S for some turbine designs

Étude des interactions entre deux hydroliennes à axe horizontal alignées avec l'écoulement

Les fermes d'hydroliennes de seconde génération sont des fermes dans lesquelles les interactions négatives, c'est-à-dire susceptibles de détériorer les performances des hydroliennes en aval, sont inévitables. Ceci est dû au fait que de nouvelles rangées d'hydroliennes doivent être ajoutées à une ferme de première génération afin d'augmenter sa capacité. Ces nouvelles rangées se situent inéluctablement dans le sillage des rangées précédentes et leur performance peut alors être profondément modifiée. Afin de mettre ces interactions en évidence, des essais expérimentaux ont été réalisés dans le bassin de l'IFREMER de Boulogne-Sur-Mer, sur des prototypes d'hydroliennes à l'échelle 1/30ème. Dans cette étude, nous nous concentrons sur l'interaction élémentaire de deux hydroliennes à axe horizontal, alignées l'une derrière l'autre avec l'écoulement. L'évolution des coefficients de puissance et de traînée en fonction de la vitesse de rotation de l'hydrolienne aval sont présentés afin de déterminer son efficacité. Pour cela, nous comparons les performances de l'hydrolienne aval aux performances de référence obtenues sur une hydrolienne seule. Par ailleurs, nous exposons des cartes de vitesse axiale, d'intensité turbulente et de cisaillement dans le sillage des hydroliennes afin d'expliquer leur comportement. Nous considérons une large gamme de distances, allant de deux à douze diamètres, entre les deux hydroliennes. D'autre part, nous étudions deux conditions d'intensité turbulente ambiante, à savoir 5% et 25%. Nous mettons en évidence l'influence de ce paramètre sur le comportement des hydroliennes et ainsi sur les effets d'interaction. En particulier, il apparaît clairement que des taux d'intensité turbulente élevés dans l'écoulement amont favorisent la dissipation des effets de sillage et offrent alors un meilleur compromis entre l'espacement entre les hydroliennes et leur performance individuelle