Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems

Stochastic physical problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zero of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods

Methods for the identification of material parameters in distributed models for flexible structures

Theoretical and numerical results are presented for inverse problems involving estimation of spatially varying parameters such as stiffness and damping in distributed models for elastic structures such as Euler-Bernoulli beams. An outline of algorithms used and a summary of computational experiences are presented

Computational methods for the identification of spatially varying stiffness and damping in beams

A numerical approximation scheme for the estimation of functional parameters in Euler-Bernoulli models for the transverse vibration of flexible beams with tip bodies is developed. The method permits the identification of spatially varying flexural stiffness and Voigt-Kelvin viscoelastic damping coefficients which appear in the hybrid system of ordinary and partial differential equations and boundary conditions describing the dynamics of such structures. An inverse problem is formulated as a least squares fit to data subject to constraints in the form of a vector system of abstract first order evolution equations. Spline-based finite element approximations are used to finite dimensionalize the problem. Theoretical convergence results are given and numerical studies carried out on both conventional (serial) and vector computers are discussed

Resource-Aware Predictive Models in Cyber-Physical Systems

Cyber-Physical Systems (CPS) are composed of computing devices interacting with physical systems. Model-based design is a powerful methodology in CPS design in the implementation of control systems. For instance, Model Predictive Control (MPC) is typically implemented in CPS applications, e.g., in path tracking of autonomous vehicles. MPC deploys a model to estimate the behavior of the physical system at future time instants for a specific time horizon. Ordinary Differential Equations (ODE) are the most commonly used models to emulate the behavior of continuous-time (non-)linear dynamical systems. A complex physical model may comprise thousands of ODEs that pose scalability, performance and power consumption challenges. One approach to address these model complexity challenges are frameworks that automate the development of model-to-model transformation. In this dissertation, a state-based model with tunable parameters is proposed to operate as a reconfigurable predictive model of the physical system. Moreover, we propose a run-time switching algorithm that selects the best model using machine learning. We employed a metric that formulates the trade-off between the error and computational savings due to model reduction. Building statistical models are constrained to having expert knowledge and an actual understanding of the modeled phenomenon or process. Also, statistical models may not produce solutions that are as robust in a real-world context as factors outside the model, like disruptions would not be taken into account. Machine learning models have emerged as a solution to account for the dynamic behavior of the environment and automate intelligence acquisition and refinement. Neural networks are machine learning models, well-known to have the ability to learn linear and nonlinear relations between input and output variables without prior knowledge. However, the ability to efficiently exploit resource-hungry neural networks in embedded resource-bound settings is a major challenge.Here, we proposed Priority Neuron Network (PNN), a resource-aware neural networks model that can be reconfigured into smaller sub-networks at runtime. This approach enables a trade-off between the model's computation time and accuracy based on available resources. The PNN model is memory efficient since it stores only one set of parameters to account for various sub-network sizes. We propose a training algorithm that applies regularization techniques to constrain the activation value of neurons and assigns a priority to each one. We consider the neuron's ordinal number as our priority criteria in that the priority of the neuron is inversely proportional to its ordinal number in the layer. This imposes a relatively sorted order on the activation values. We conduct experiments to employ our PNN as the predictive model in a CPS application. We can see that not only our technique will resolve the memory overhead of DNN architectures but it also reduces the computation overhead for the training process substantially. The training time is a critical matter especially in embedded systems where many NN models are trained on the fly