During development, motor axons are guided toward muscle target by various extrinsic cues including extracellular matrix (ECM) proteins whose identities and cellular source remain poorly characterized. Here, using single - cell RNAseq of sorted GFP+ cells from smyhc1:gfp- injected zebrafish embryos, we unravel the slow muscle progenitors (SMP) pseudotemporal trajectory at the single - cell level and show that differentiating SMPs are a major source of ECM proteins. The SMP core- matrisome was characterized and computationally predicted to form a basement membrane-like structure tailored for motor axon guidance, including basement membrane-associated ECM proteins, as collagen XV-B, one of the earliest core- matrisome gene transcribed in differentiating SMPs and the glycoprotein Tenascin C. To investigate how contact- mediated guidance cues are organized along the motor path to exert their function in vivo, we used microscopy - based methods to analyze and quantify motor axon navigation in tnc and col15a1b knock - out fish. We show that motor axon shape and growth rely on the timely expression of the attractive cue Collagen XV-B that locally provides axons with a permissive soft microenvironment and separately organizes the repulsive cue Tenascin C into a unique functional dual topology. Importantly, bioprinted micropatterns that mimic this in vivo ECM topology were sufficient to drive directional motor axon growth. Our study offers evidence that not only the composition of ECM cues but their topology critically influences motor axon navigation in vertebrates with potential applications in regenerative medicine for peripheral nerve injury as regenerating nerves follow their original path.

In the field of sensitivity analysis, Sobol’ indices are widely used to assess the importance of the inputs of a model to its output. Among the methods that estimate these indices, the replication procedure is noteworthy for its efficient cost. A practical problem is how many model evaluations must be performed to guarantee a sufficient precision on the Sobol’ estimates. The present paper tackles this issue by rendering the replication procedure iterative. The idea is to enable the addition of new model evaluations to progressively increase the accuracy of the estimates. These evaluations are done at points located in under-explored regions of the experimental designs, but preserving their characteristics. The key feature of this approach is the construction of nested space-filling designs. For the estimation of first-order indices, a nested Latin hypercube design is used. For the estimation of closed second-order indices, two constructions of a nested orthogonal array design are proposed. Regularity and uniformity properties of the nested designs are studied.

For the characterization of acoustic sources, a common approach is to retro-propagate the sound pressure measured with a microphone array, which is often performed through the resolution of an inverse problem. The ill-posed nature of this problem, as well as the limited number of measurements, are known to reduce the quality of the source reconstruction. A practical solution to these limitations is to increase the number of measurements with new array placements. However, finding the best array positions is not a straightforward process. The present paper tackles this issue by introducing a sequential approach that selects at each iteration the optimal array placement. The proposed approach builds on two features rooted in a Bayesian framework: an inverse method called “Bayesian focusing” and a Bayesian search criterion based on the Kullback-Leibler divergence. Simulations results for the characterization of a directive source are used to illustrate the performance of the approach. It is shown that for a fixed number of iterations, the proposed approach performs better than ones where the successive placements are randomly selected around the source, or others where the placements follow a deterministic spherical grid pattern.

The characterization of acoustic sources typically involves the retro-propagation of the acoustic field measured with a microphone array to a mesh of the surface of interest, which amounts to solve an inverse problem. Such an inverse problem is built on the basis of a forward model prone to uncertainties arising from mismatches with the physics of the experiment. Assessing the effects of these unavoidable uncertainties on the resolution of the inverse problem represents a challenge. The present paper introduces a practical solution to measure these effects by conducting a sensitivity analysis. The latter provides a mean to identify and rank the main sources of uncertainty through the estimation of sensitivity indices. Two inverse methods are investigated through the sensitivity analysis: conventional Beamforming and Bayesian focusing. The propagation of uncertainties is carried on numerically. The consistency between the real experiment and its numerical simulation is assessed by means of a small batch of measurements performed in a semi-anechoic chamber.

Among practitioners, the importance of inputs to a model output is commonly measured via the computation of Sobol' sensitivity indices. Various estimation strategies exist in the literature, most of them requiring a very high number of model evaluations. Designing methods that compete favorably both in terms of computational cost and accuracy is therefore an issue of crucial importance. In this paper, an efficient replication-based strategy is proposed to estimate the full set of first- and second-order Sobol' indices. It relies on a Sobol' pick-freeze estimation scheme and requires only two replicated designs based on randomized orthogonal arrays of strength two. The precision of this procedure is assessed with bootstrap confidence intervals, presented for the first time in the replication framework. Our developments are compared to known approaches and validated on numerical test cases. A way to estimate the full set of first-, second-order but also total-effect Sobol' indices at a very competitive cost is also described, as a combination of our procedure and the one introduced by Saltelli [1].

The reconstruction of sound sources by using inverse methods is known to be prone to estimation errors due to measurement noise, model mismatch, and poor conditioning of the inverse problem. This paper introduces a solution to map the estimation errors together with the reconstructed sound sources. From a Bayesian perspective, it initializes a Gibbs sampler with the Bayesian focusing method. The proposed Gibbs sampler is shown to converge within a few iterations, which makes it realistic for practical purposes. It also turns out to be very flexible in various scenarios. One peculiarity is the capability to directly operate on the cross-spectral matrix. Another one is to easily accommodate sparse priors. Eventually, it can also account for uncertainties in the microphone positions, which reinforces the regularization of the inverse problem.

Land Use and Transportation Integrated (LUTI) models have become a norm for representing the interactions between land use and the transportation of goods and people in a territory. Through the use of these models, urban planning policies and development scenarios can be evaluated. The calibration of LUTI models is a heavy task, involving gathering of massive amounts of data and the estimation of an important number of parameters. In this paper, the calibration of the open-source LUTI model Tranus is considered. Classical calibrations of Tranus rely on ad hoc econometric techniques and time-consuming trial and error procedures. Here, a two-step calibration that comprises global sensitivity analysis and optimisation is proposed. The sensitivity analysis presented herein is based on the replication method for the estimation of Sobol' indices and generalised to take into account multivariate outputs. The optimisation step is an iterative process combining stochastic and deterministic procedures. The proposed calibration procedure is applied to a study area in the State of Mississippi. Compared to a previous ad hoc procedure, this new approach results in a significant improvement of the adjustment factors of Tranus while reducing drastically the calibration time.

In the field of sensitivity analysis, Sobol' indices are sensitivity measures widely used to assess the importance of inputs of a model to its output. The estimation of these indices is often performed through Monte Carlo or quasi-Monte Carlo methods. A notable method is the replication procedure that estimates first-order indices at a reduced cost in terms of number of model evaluations. An inherent practical problem of this estimation is how to quantify the number of model evaluations needed to ensure that estimates satisfy a desired error tolerance. This article addresses this challenge by proposing a reliable error bound for first-order and total effect Sobol' indices. Starting from the integral formula of the indices, the error bound is defined in terms of the discrete Walsh coefficients of the different integrands. We propose a sequential estimation procedure of Sobol' indices using the error bound as a stopping criterion. The sequential procedure combines Sobol' sequences with either Saltelli's strategy to estimate both first-order and total effect indices, or the replication procedure to estimate only first-order indices.

In the perspective of estimating main effects of model inputs, two approaches are studied to iteratively construct replicated designs based on Sobol' sequences. Space-filling properties of the resulting designs are studied based on two criteria. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS.

This paper deals with the estimation of grouped Sobol' indices in the case where the dependence inside each group is given by sets of linear ordered constraints. In the framework of independent inputs, the replication method allows to estimate first-order indices with only two designs. Through the use of orthogonal arrays of strength two the replication method can be applied to estimate closed second-order indices. We extend this methodology to estimate first-order and closed second-order grouped Sobol' indices under sets of linear constraints. The construction of the two designs required by the replication method is now based on the simplex geometric structure to handle the constraints within each set. We propose a space filling strategy to construct these designs.

Mathematical models used in various fields are often based on complex computer codes involving poorly-known inputs whose uncertainties can have significant effects on the model outputs. From a model user point of view, to assess the importance of these effects is essential to grasp the model behavior. Global sensitivity analysis methods are useful tools to quantify the influence of the model inputs and detect potential interactions between them. Among the large number of available approaches, the variance based method introduced by Sobol’ [7] allows to calculate sensitivity indices called Sobol’ indices. These indices are scalars between 0 and 1 that summarize the influence of each input or set of inputs. An index close to 1 means that the set is influent. At the opposite, an index equal to 0 means that the set is not correlated to the output. First-order indices estimate the main effect from each input whereas higherorder indices estimate the corresponding order of interactions between inputs. Various estimation procedures of these indices have been proposed in the litterature. Unfortunately, these procedures require a significant number of model evaluations that grows with respect to the input space dimension. A solution to break this dependency lies in the use of replicated designs. A synthesis on the use of replicated designs (referred as permuted column sampling plans) can be found in Morris et al. [6] where the authors use the approach introduced by McKay [5]. Later on, Mara et al. [4] combine replicated designs with “pick-freeze” estimators to estimate first-order Sobol’ indices. This procedure has been further studied (asymptotic properties for first-order indices) and generalized in Tissot et al. [8] to the estimation of closed second-order indices. For closed second-order indices, the procedure relies on the replication of randomized orthogonal arrays. We propose here to present three extensions of this replication procedure. The first one consists in dealing with the presence of dependent inputs. The case of correlated inputs has to be tackled with caution, as the calculation of single input indices does not provide anymore a proper information, that can be easily interpreted. As proposed in Jacques et al. [3], a simple strategy consists to define grouped Sobol’ indices for the groups of correlated inputs. Combining this approach with the replication procedure, grouped Sobol’ indices can be estimated at a reasonable computing cost. The second extension deals with the case of multidimensional outputs. It relies on a generalization of the Sobol’ indices for multivariate outputs as introduced in [2]. These new indices, referred as generalized Sobol’ indices, are constructed based on the multidimensional Hoeffding decomposition of the vectorial output. Using these new indices, a straightforward formulation of the replication procedure can be derived to take into account multivariate outputs. The third extension aims to render the replication procedure recursive. The approach proposed relies on the construction of a recursive estimator of the Sobol’ index. This estimator is evaluated by augmenting the replicated designs with new sets of points. The sampling procedure of these sets insures that the replicated designs keep a space-filling structure at each step. The spacefilling structure is either a Latin Hypercube or an orthogonal array of strength two depending on whether firstor closed second-order Sobol’ indices are estimated. With each set of points added MascotNum Annual Conference, March 23-25 2016, Toulouse, France a new estimation of the Sobol’ indices is performed. The recursive process is carried out until the estimated Sobol’ indices satisfy a stopping criterion. Numerical experimentations are conducted on classical test functions to illustrate each extension of the replication procedure. In particular, for the extension to multivariate outputs a real case application to the land-use and transport integrated model (LUTI) TRANUS [1] is presented. As an example, in Figure 1 boxplots of first-order Sobol’ indices estimated with both the replication method (referred as classical) and its recursive extension are represented. The function test is the Bratley et al. function. The two procedures give overall similar results. Hence, there is no drawback to use the recursive version of the replication method. Figure 1: First-order Sobol’ indices estimation with both recursive and classical replication methods for 100 repetitions.