- July 12, 2018 4:00 PM
Quasi-Newton – A Universal Approach for Coupled Problems and Optimization
Quasi-Newton methods are used in many fields to solve non-linear equations without explicitly known derivatives. This is the case, e.g., in coupled multi-physics applications such as fluid-structure interactions where we combine several independent solvers in a partitioned approach coupled simulation environment. To do so, we have to solve a (in general non-linear) interface equation that contains operator contributions from all involved single-physics solvers.
If we assume that these solvers are black-box, quasi-Newton methods are the best known method to accelerate pure interface fixed point iterations. In PDE-constrained optimization, i.e., inverse solvers that are based on gradient descent, we have to find the root of the (reduced) gradient of the objective function. Though the Hessian can usually be calculated and used in an inner Krylov method, these calculations are typically costly as they involve the solution of forward and adjoint problems. Thus, quasi-Newton methods are an efficient alternative. In both cases, coupled problems and optimization, an additional advantage of quasi-Newton over Newton methods is the fact that we can directly approximate the inverse Jacobian or Hessian such that no inner linear solver is required. We present a comparison of known quasi-Newton methods for multi- physics such as interface quasi-Newton with methods usually used in optimization, in particular the BFGS method that is, e.g., used in PETCs’s TAO package. Results for two applications -- fluid-structure interaction and inverse tumor simulation – demonstrate their potential in terms of robustness, generality, and efficiency.
Date: Thursday, 12 July 2018, 4 pm
Place: MML Multimedia Lab, Pfaffenwaldring 61, 70569 Stuttgart