Recovery of cellular traction in three-dimensional nonlinear hyperelastic matrices

The traction exerted by a cell on the extra-cellular matrix (ECM) is critical to understanding and manipulating important biological processes such as stem cell differentiation, cancer cell metastasis, and embryonic morphogenesis. This traction is typically quantified through traction force microscopy (TFM). In TFM, the displacement of select markers inside the ECM is tracked, and is used in conjunction with an elasticity problem to reconstruct the traction field. Most applications of this technique thus far have assumed that the matrix behaves as a linear elastic solid that undergoes small deformation and infinitesimal strains. In this manuscript, we develop and implement a robust and efficient TFM methodology that overcomes these limitations by accounting for geometric and material nonlinearities in the ECM. We pose the TFM problem as an inverse problem and develop efficient adjoint-based minimization techniques to solve it. We test the effect of measurement noise on the proposed method, and examine the error incurred by not including nonlinear effects when solving the TFM problem. We present these results for in-silico traction fields that are applied to realistic geometric models of microglial and neuronal cells

A few-shot graph Laplacian-based approach for improving the accuracy of low-fidelity data

Low-fidelity data is typically inexpensive to generate but inaccurate. On the other hand, high-fidelity data is accurate but expensive to obtain. Multi-fidelity methods use a small set of high-fidelity data to enhance the accuracy of a large set of low-fidelity data. In the approach described in this paper, this is accomplished by constructing a graph Laplacian using the low-fidelity data and computing its low-lying spectrum. This spectrum is then used to cluster the data and identify points that are closest to the centroids of the clusters. High-fidelity data is then acquired for these key points. Thereafter, a transformation that maps every low-fidelity data point to its bi-fidelity counterpart is determined by minimizing the discrepancy between the bi- and high-fidelity data at the key points, and to preserve the underlying structure of the low-fidelity data distribution. The latter objective is achieved by relying, once again, on the spectral properties of the graph Laplacian. This method is applied to a problem in solid mechanics and another in aerodynamics. In both cases, this methods uses a small fraction of high-fidelity data to significantly improve the accuracy of a large set of low-fidelity data

Final Report on the “New Directions in the Variational Multiscale Formulation of Large Eddy Simulation of Turbulence”

In the report we present a summary of the new models and algorithms developed by the PI and the students supported by this grant. These developments are described in detail in ten peer-reviewed journal articles that acknowledge support from this grant

Final Report on the “ New Directions in the Variational Multiscale Formulation of Large Eddy Simulation of Turbulence”

The work performed as part of this grant may be described in the context of developing numerical methods for solving an abstract, nonlinear variational problem for which the straightforward Galerkin approximation on a space of reasonable dimension is inaccurate. Target applications include flows with turbulence and systems with shocks. With this in mind, an alternate finite dimensional problem is proposed in which a model term (with unknown parameters) is added to the Galerkin approximation to improve its performance. We have developed new formulations for this model term especially for turbulent flows and for systems with shocks. We have also developed a dynamic approach for evaluating the parameters that appear in this model term. We have conducted several numerical studies to confirm the usefulness of the proposed approach

Final Report on the “New Directions in the Variational Multiscale Formulation of Large Eddy Simulation of Turbulence”

In the report we present a summary of the new models and algorithms developed by the PI and the students supported by this grant. These developments are described in detail in ten peer-reviewed journal articles that acknowledge support from this grant

Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priors

Inverse problems are notoriously difficult to solve because they can have no solutions, multiple solutions, or have solutions that vary significantly in response to small perturbations in measurements. Bayesian inference, which poses an inverse problem as a stochastic inference problem, addresses these difficulties and provides quantitative estimates of the inferred field and the associated uncertainty. However, it is difficult to employ when inferring vectors of large dimensions, and/or when prior information is available through previously acquired samples. In this paper, we describe how deep generative adversarial networks can be used to represent the prior distribution in Bayesian inference and overcome these challenges. We apply these ideas to inverse problems that are diverse in terms of the governing physical principles, sources of prior knowledge, type of measurement, and the extent of available information about measurement noise. In each case we apply the proposed approach to infer the most likely solution and quantitative estimates of uncertainty.Comment: Paper: 18 pages, 5 figures. Supplementary: 9 pages, 6 Figures, 2 Table