Bayesian statistical inverse problems are often solved with Markov chain Monte Carlo (MCMC)-type schemes. When the problems are governed by large-scale discrete nonlinear partial differential equations (PDEs), they are computationally challenging because one would then need to solve the forward problem at every sample point. In this talk, the use of reduced-order models (ROMs), as well as deep neural network techniques, is considered for the forward solves within an MCMC routine. In particular, a preconditioning strategy for the ROMs is also proposed to accelerate the forward solves. Numerical experiments are provided to demonstrate the efficiency of the approach for solving forward problems and the associated statistical inverse problems.
We talk about preconditioned Nesterov's accelerated gradient descent methods (PAGD) for approximating the minimizer of locally Lipschitz smooth, strongly convex objective functionals. We introduce a second order ordinary differential equation (ODE) as the limiting case of PAGD as the step size tends to zero. Using a simple energy argument, we will show an exponential convergence of the ODE solution to its steady state. The PAGD method may be viewed as an explicit-type time-discretization scheme of the ODE system, which requires a natural time step restriction for energy stability. Assuming this restriction, an exponential rate of convergence of the PAGD sequence is demonstrated by mimicking the convergence of the solution to the ODE via energy methods. Applications of the PAGD method are made in the context of solving certain nonlinear elliptic PDE using Fourier collocation methods, and several numerical experiments are conducted. The results confirm the global geometric and h-independent convergence of the PAGD method, with an accelerated rate that is improved over the preconditioned gradient descent (PGD) method.
We prove quasi-optimal L∞ norm error estimates (up to logarithmic factors) for the solution of Poisson's problem in two dimensional space by the standard Hybridizable Discontinuous Galerkin (HDG) method. Although such estimates are available for conforming and mixed finite element methods, this is the first proof for HDG. The method of proof is motivated by known L∞ norm estimates for mixed finite elements. We show two applications: the first is to prove optimal convergence rates for boundary flux estimates, and the second is to prove that numerically observed convergence rates for the solution of a Dirichlet boundary control problem are to be expected theoretically. Numerical examples show that the predicted rates are seen in practice.