Numerical Methods for Optimal Control Problems

Dr. Ruchi Sandilya

Abstract: In the first part of the talk, I will present my work on development and analysis of a robust numerical scheme called discontinuous finite volume method (DFVM) for the approximation of Brinkman optimal control problem. The DFVM is a combination of finite volume element method and discontinuous Galerkin method and retain desirable properties of both methods such as mass conservation, local mesh adaptivity, element-wise conservative, allow different degree polynomials in different elements and easily handle the boundary conditions. The Brinkman equations describe the motion of an incompressible viscous fluid within an array of porous particles, and according to the flow regime characterized by the ratio between permeability and viscosity, it can represent both the Darcy and Stokes limits. We have derived optimal order a priori error estimates in suitable natural norms and conducted numerical experiments to illustrate the performance of the proposed scheme and to confirm the predicted accuracy of the theoretical convergence rates. 

The second part of my talk is on numerical stabilization of two-dimensional Navier-Stokes- Boussinesq equations. The stabilization is achieved by boundary feedback controls of finite dimension applied on the velocity and temperature, and localized on parts of domain boundary. We consider a stationary solution of the Boussinesq system which is linearly unstable. The linearized system around this unstable stationary solution is used to construct a feedback law. For that, we determine the projected linearized system onto an invariant subspace (including the unstable subspace), and we determine a feedback control law stabilizing this projected system, by solving a Ricatti equation, of small dimension, associated with this system. The linear feedback law, stabilizing the projected system, is applied to the non-linear model to study its ability to locally stabilize the flow and its temperature.

Speaker’s Bio:  Dr. Ruchi Sandilya is a researcher in numerical analysis and scientific computing. She worked as a postdoctoral researcher at Weierstrass Institute Berlin (2019) and Tata Institute of Fundamental Research Bangalore (Aug 2017-Dec 2018). She received her Ph.D. in applied mathematics from Indian Institute of Space Science and Technology in 2017. Her area of expertise is in developing numerical methods for partial differential equations and optimal control problems. Specific research interests include the development of discontinuous finite volume methods for solving optimal control problems, numerical stabilization of Navier-Stokes-Boussinesq equations with feedback control, and development of adaptive finite element method for generalized Nash equilibrium problem. She is also interested in studying reduced-order models for complex fluid flows to overcome the computational challenges.
Host:  Eirik Endeve, endevee@ornl.gov

About the Seminar: The Computational and Applied Math Seminar features talks by invited speakers, local mathematicians, and domain scientists working on problems of mathematical interest. The seminar is held weekly, every Thursday from 3:00pm-4:00pm. If you are interested in giving a seminar, please contact Eirik Endeve, endevee@ornl.gov. To subscribe to the CAM Seminar mailing list, please contact Kasi Arnold, arnoldkl@ornl.gov. To see the full list of previous and upcoming seminars, go to https://csmd.ornl.gov/events/9/seminars.


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Last Updated: May 6, 2020 - 11:15 am