Coordinate Transformation and Polynomial Chaos for the Bayesian Inference of a Gaussian Process with Parametrized Prior Covariance Function

This paper addresses model dimensionality reduction for Bayesian inference based on prior Gaussian fields with uncertainty in the covariance function hyper-parameters. The dimensionality reduction is traditionally achieved using the Karhunen-\Loeve expansion of a prior Gaussian process assuming covariance function with fixed hyper-parameters, despite the fact that these are uncertain in nature. The posterior distribution of the Karhunen-Lo\`{e}ve coordinates is then inferred using available observations. The resulting inferred field is therefore dependent on the assumed hyper-parameters. Here, we seek to efficiently estimate both the field and covariance hyper-parameters using Bayesian inference. To this end, a generalized Karhunen-Lo\`{e}ve expansion is derived using a coordinate transformation to account for the dependence with respect to the covariance hyper-parameters. Polynomial Chaos expansions are employed for the acceleration of the Bayesian inference using similar coordinate transformations, enabling us to avoid expanding explicitly the solution dependence on the uncertain hyper-parameters. We demonstrate the feasibility of the proposed method on a transient diffusion equation by inferring spatially-varying log-diffusivity fields from noisy data. The inferred profiles were found closer to the true profiles when including the hyper-parameters' uncertainty in the inference formulation.Comment: 34 pages, 17 figure

Self-Propagating Reactive Fronts in Compacts of Multilayered Particles

Reactive multilayered foils in the form of thin films have gained interest in various applications such as joining, welding, and ignition. Typically, thin film multilayers support self-propagating reaction fronts with speeds ranging from 1 to 20 m/s. In some applications, however, reaction fronts with much smaller velocities are required. This recently motivated Fritz et al. (2011) to fabricate compacts of regular sized/shaped multilayered particles and demonstrate self-sustained reaction fronts having much smaller velocities than thin films with similar layering. In this work, we develop a simplified numerical model to simulate the self-propagation of reactive fronts in an idealized compact, comprising identical Ni/Al multilayered particles in thermal contact. The evolution of the reaction in the compact is simulated using a two-dimensional transient model, based on a reduced description of mixing, heat release, and thermal transport. Computed results reveal that an advancing reaction front can be substantially delayed as it crosses from one particle to a neighboring particle, which results in a reduced mean propagation velocity. A quantitative analysis is thus conducted on the dependence of these phenomena on the contact area between the particles, the thermal contact resistance, and the arrangement of the multilayered particles

A priori testing of sparse adaptive polynomial chaos expansions using an ocean general circulation model database

This work explores the implementation of an adaptive strategy to design sparse ensembles of oceanic simulations suitable for constructing polynomial chaos surrogates. We use a recently developed pseudo-spectral algorithm that is based on a direct application of the Smolyak sparse grid formula and that allows the use of arbitrary admissible sparse grids. The adaptive algorithm is tested using an existing simulation database of the oceanic response to Hurricane Ivan in the Gulf of Mexico. The a priori tests demonstrate that sparse and adaptive pseudo-spectral constructions lead to substantial savings over isotropic sparse sampling in the present setting.United States. Office of Naval Research (award N00014-101-0498)United States. Dept. of Energy. Office of Advanced Scientific Computing Research (award numbers DE-SC0007020, DE-SC0008789, and DE-SC0007099)Gulf of Mexico Research Initiative (contract numbers SA1207GOMRI005 (CARTHE) and SA12GOMRI008 (DEEP-C)

DYNAMIC RAY-TRACING FOR MODELING CELL MANIPULATION USING HIGH-THROUGHPUT OPTICAL STRETCHERS

Optical stretchers have been recently used to identity cell types and to detect disease states. In this method, a cell is trapped within a laser beam then is stretched by optical forces from the interaction of light with the cell surface. Cell stretch is used to calculate cellular elastic properties using an appropriate numerical model. The current dual-beam optical stretcher was successfully used as an efficient and accurate cell biomarker compared to other contact methods such as Atomic Force Microscopy or micropipette aspiration. This is due to the no-direct contact with the cell thus less induced damage. However, in the dual-beam optical stretcher cells are sequentially trapped and stretched within static systems resulting in very low-throughput. In addition, there is a lack of numerical methods that take into account cell deformability, cell-fluid interaction, different cell membrane models, and high-throughput. In the first part of this dissertation, we present simulations of cell deformation by optical forces from a recently developed high-throughput optical stretcher using diode laser bars. The simulations were done using an implemented Dynamic Ray-Tracing method linked with the Immersed Boundary Method to account for both cell-fluid interaction and cell-light interaction. We modeled cell stretching by single and dual laser bar configurations. We showed that single diode laser bar induces cell stretching in addition to translation as opposed to the dual diode laser bar where the cell is fixed at a focus. We also presented a method to calculate cell stiffness from experimental measurements. We used this method to calculate red blood cells' shear modulus. Our calculated cell stiffness was in the range of reported values. In the second part of this dissertation, we investigate the effect of cell deformation on the calculation of optical forces. The implemented Dynamic Ray-Tracing allows calculating these forces on arbitrary cell shapes as opposed to the current Ray-Optics technique that was designed for spherical rigid cells only. We showed that the change in cell shape affects the optical forces calculation. A difference of up to 42% in the net optical force was observed at high laser powers that suggests considering cell deformability in the cell stiffness calculation. Non-spherical cells deformation simulations were also presented. Finally, high-throughput optical stretching simulations were presented by applying an external flow. It was shown that at the experimental average flow rate of 50 μm/sec, cell deformation caused by the external applied flow is insignificant and can be neglected. However, at a flow rate of 500 μm/sec net cell deformation was 10% less compared to the quiescent case. These findings suggest that the external flow rate can be increased to achieve faster cell throughput, however, its effect has to be considered in the calculation of elastic properties. The results of this study coupled with experimental measurements can be used to calculate individual cell's properties at high-throughput. We believe that the developed algorithm can be used to better design high-throughput optical stretchers for different cell models

DYNAMIC RAY-TRACING FOR MODELING CELL MANIPULATION USING HIGH-THROUGHPUT OPTICAL STRETCHERS

Optical stretchers have been recently used to identity cell types and to detect disease states. In this method, a cell is trapped within a laser beam then is stretched by optical forces from the interaction of light with the cell surface. Cell stretch is used to calculate cellular elastic properties using an appropriate numerical model. The current dual-beam optical stretcher was successfully used as an efficient and accurate cell biomarker compared to other contact methods such as Atomic Force Microscopy or micropipette aspiration. This is due to the no-direct contact with the cell thus less induced damage. However, in the dual-beam optical stretcher cells are sequentially trapped and stretched within static systems resulting in very low-throughput. In addition, there is a lack of numerical methods that take into account cell deformability, cell-fluid interaction, different cell membrane models, and high-throughput. In the first part of this dissertation, we present simulations of cell deformation by optical forces from a recently developed high-throughput optical stretcher using diode laser bars. The simulations were done using an implemented Dynamic Ray-Tracing method linked with the Immersed Boundary Method to account for both cell-fluid interaction and cell-light interaction. We modeled cell stretching by single and dual laser bar configurations. We showed that single diode laser bar induces cell stretching in addition to translation as opposed to the dual diode laser bar where the cell is fixed at a focus. We also presented a method to calculate cell stiffness from experimental measurements. We used this method to calculate red blood cells' shear modulus. Our calculated cell stiffness was in the range of reported values. In the second part of this dissertation, we investigate the effect of cell deformation on the calculation of optical forces. The implemented Dynamic Ray-Tracing allows calculating these forces on arbitrary cell shapes as opposed to the current Ray-Optics technique that was designed for spherical rigid cells only. We showed that the change in cell shape affects the optical forces calculation. A difference of up to 42% in the net optical force was observed at high laser powers that suggests considering cell deformability in the cell stiffness calculation. Non-spherical cells deformation simulations were also presented. Finally, high-throughput optical stretching simulations were presented by applying an external flow. It was shown that at the experimental average flow rate of 50 μm/sec, cell deformation caused by the external applied flow is insignificant and can be neglected. However, at a flow rate of 500 μm/sec net cell deformation was 10% less compared to the quiescent case. These findings suggest that the external flow rate can be increased to achieve faster cell throughput, however, its effect has to be considered in the calculation of elastic properties. The results of this study coupled with experimental measurements can be used to calculate individual cell's properties at high-throughput. We believe that the developed algorithm can be used to better design high-throughput optical stretchers for different cell models

Drag Parameter Estimation Using Gradients and Hessian from a Polynomial Chaos Model Surrogate

Abstract A variational inverse problem is solved using polynomial chaos expansions to infer several critical variables in the Hybrid Coordinate Ocean Model’s (HYCOM’s) wind drag parameterization. This alternative to the Bayesian inference approach in Sraj et al. avoids the complications of constructing the full posterior with Markov chain Monte Carlo sampling. It focuses instead on identifying the center and spread of the posterior distribution. The present approach leverages the polynomial chaos series to estimate, at very little extra cost, the gradients and Hessian of the cost function during minimization. The Hessian’s inverse yields an estimate of the uncertainty in the solution when the latter’s probability density is approximately Gaussian. The main computational burden is an ensemble of realizations to build the polynomial chaos expansion; no adjoint code or additional forward model runs are needed once the series is available. The ensuing optimal parameters are compared to those obtained in Sraj et al. where the full posterior distribution was constructed. The similarities and differences between the new methodology and a traditional adjoint-based calculation are discussed

Global sensitivity analysis in an ocean general circulation model: a sparse spectral projection approach

Polynomial chaos (PC) expansions are used to propagate parametric uncertainties in ocean global circulation model. The computations focus on short-time, high-resolution simulations of the Gulf of Mexico, using the hybrid coordinate ocean model, with wind stresses corresponding to hurricane Ivan. A sparse quadrature approach is used to determine the PC coefficients which provides a detailed representation of the stochastic model response. The quality of the PC representation is first examined through a systematic refinement of the number of resolution levels. The PC representation of the stochastic model response is then utilized to compute distributions of quantities of interest (QoIs) and to analyze the local and global sensitivity of these QoIs to uncertain parameters. Conclusions are finally drawn regarding limitations of local perturbations and variance-based assessment and concerning potential application of the present methodology to inverse problems and to uncertainty management