Uniform Approximation of the Smallest Eigenvalue of a Large Parameter-Dependent Hermitian Matrix

Talk by Emre Mengi, Koc University

We consider the uniform approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix on a compact domain. The problem is  motivated by applications such as the estimation of the coercivity constant for  the numerical solution of parametric PDEs. We present an approach based on approximation of the smallest eigenvalue function by that of a small parameter-dependent matrix obtained by projecting the large matrix onto a suitably constructed subspace. The projection subspaces are formed of eigenvectors of the large matrix at the parameter values where surrogate errors, bounding the actual approximation errors from above, are maximal.  We prove in theory that the projected eigenvalue functions converge uniformly to the actual eigenvalue function in the infinite-dimensional setting as the subspace dimension goes to infinity. This is supported by numerical experiments on real examples arising from parametric PDEs that illustrate that the proposed technique is often able to reduce the size of the large parameter-dependent matrix drastically while ensuring an approximation error below the prescribed tolerance.