Speaker:

Jehanzeb H. Chaudhry

Title:

A Least-Squares Reduced Order Method, Asymptotically Rigorous Error Bounds, and Eigenvalue Analysis of Coercivity Constants arising in Partial Differential Equations

Abstract:
Many scientific and engineering applications require solutions of partial differential equations for a wide range of parameter values (e.g. in statistical inverse problem, optimal control etc.). Additionally, many applications require inexpensive solution of PDEs, especially in a real-time context. Traditional numerical methods for the solution of PDEs (e.g. finite difference, finite element, etc.) involve a large number of unknowns and hence are unsuitabe for such applications.  Reduced basis methods are a form of model order reduction that offers the potential to decrease the dimension of the problem and hence solutions are constructed with low computational cost. However, in most reduced basis methods the accuracy of a reduced basis solution is typically measured in reference to a full-order finite element solution. Often the accuracy of the full order finite element solution is itself heavily dependent on the value of parameters for certain problems, resulting in an error estimate for the reduced basis solution that is often overly optimistic.

In the first part of this talk, we present a reduced basis method with a sharp error estimate with respect to the exact solution of the PDE. A crucial element in developing such a method is the least-squares finite element method (LSFEM). LSFEMs are widely used for the solution of PDEs arising in many applications in science and engineering. LSFEMs are
based on minimizing the residual of the PDE in an appropriate norm, and have a number of attractive properties. In particular, the property relevant to this work is that  these methods provide a robust and inexpensive a posteriori error estimate with respect to the true solution. This estimate is utilized in developing the Least-Squares Reduced Basis Method presented in this talk.

The second part of the talk concerns a key ingredient in error estimates for variational problems and reduced basis methods: the so-called coercivity or inf-sup constant of the continuous problem. We characterize the coercivity constant as a spectral value of a self-adjoint linear operator; for several differential equations, we show that the coercivity constant is related to the eigenvalue of a compact operator. For these applications, convergence rates are derived and verified with numerical examples.