Georgia Institute of Technology College of Sciences

We consider a spectrally unstable steady state $(\rho_0,v_0)$ of the incompressible stratified Euler equations on a class of $d$-dimensional domains. Assuming that the linearized equation admits an exponential dichotomy with a reasonably large spectral gap relative to the maximal Lyapunov exponent of the background steady flow $v_0$, we construct the local stable and unstable manifold of $(\rho_0,v_0)$. The proof is based on the Lyapunov--Perron method after reformulating the Euler equation as an ODE on the infinite-dimensional manifold of volume-preserving Lagrangian maps, with the density treated as a frozen Lagrangian parameter as well as the weight in the $L^2$ metric. We also discuss some applications to two-dimensional steady flows. This is a joint work with Zhiwu Lin and Chongchun Zeng.

TBA

In many applications of mathematical optimization, one may wish to optimize an objective function without access to its derivatives (gradients). These situations call for derivative-free optimization (DFO) methods. Among the most successful approaches in practice are model-based trust-region methods, such as those pioneered by M.J.D Powell. These methods rely on function approximations via low degree polynomials and carefully adapt the local geometry of interpolation points to balance exploration and exploitation.  While relatively complex to implement, these methods are now available in standard scientific computing platforms, including MATLAB and SciPy. However, theoretical analysis of their computational complexity lagged behind practice. In particular, it is important to bound the number of function evaluations required to achieve a desired level of accuracy. Using a mix of handy concepts from approximation, sample set geometry, Lagrangian interpolation and linear algebra we systematically derive complexity bounds for classical model-based trust-region methods and their modern variations. We establish, for the first time, that these methods can have the same worst case complexity than any other known DFO method.