3,185 research outputs found

    Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems

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    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

    Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints

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    This paper presents a stochastic model predictive control approach for nonlinear systems subject to time-invariant probabilistic uncertainties in model parameters and initial conditions. The stochastic optimal control problem entails a cost function in terms of expected values and higher moments of the states, and chance constraints that ensure probabilistic constraint satisfaction. The generalized polynomial chaos framework is used to propagate the time-invariant stochastic uncertainties through the nonlinear system dynamics, and to efficiently sample from the probability densities of the states to approximate the satisfaction probability of the chance constraints. To increase computational efficiency by avoiding excessive sampling, a statistical analysis is proposed to systematically determine a-priori the least conservative constraint tightening required at a given sample size to guarantee a desired feasibility probability of the sample-approximated chance constraint optimization problem. In addition, a method is presented for sample-based approximation of the analytic gradients of the chance constraints, which increases the optimization efficiency significantly. The proposed stochastic nonlinear model predictive control approach is applicable to a broad class of nonlinear systems with the sufficient condition that each term is analytic with respect to the states, and separable with respect to the inputs, states and parameters. The closed-loop performance of the proposed approach is evaluated using the Williams-Otto reactor with seven states, and ten uncertain parameters and initial conditions. The results demonstrate the efficiency of the approach for real-time stochastic model predictive control and its capability to systematically account for probabilistic uncertainties in contrast to a nonlinear model predictive control approaches.Comment: Submitted to Journal of Process Contro

    Completeness of Inertial Modes of an Incompressible Non-Viscous Fluid in a Corotating Ellipsoid

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    Inertial modes are the eigenmodes of contained rotating fluids restored by the Coriolis force. When the fluid is incompressible, inviscid and contained in a rigid container, these modes satisfy Poincar\'e's equation that has the peculiarity of being hyperbolic with boundary conditions. Inertial modes are therefore solutions of an ill-posed boundary-value problem. In this paper we investigate the mathematical side of this problem. We first show that the Poincar\'e problem can be formulated in the Hilbert space of square-integrable functions, with no hypothesis on the continuity or the differentiability of velocity fields. We observe that with this formulation, the Poincar\'e operator is bounded and self-adjoint and as such, its spectrum is the union of the point spectrum (the set of eigenvalues) and the continuous spectrum only. When the fluid volume is an ellipsoid, we show that the inertial modes form a complete base of polynomial velocity fields for the square-integrable velocity fields defined over the ellipsoid and meeting the boundary conditions. If the ellipsoid is axisymmetric then the base can be identified with the set of Poincar\'e modes, first obtained by Bryan (1889), and completed with the geostrophic modes.Comment: 19 pages, 1 figure, to appear in Physical Review

    Graph Signal Processing: Overview, Challenges and Applications

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    Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE
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