Tim Wildey

Tim Wildey, Sriramkrishnan Muralikrishnan, Tan Bui-Thanh

We consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. The proposed unified approach is applicable to any locally conservative hybridized finite element method including multinumerics with different hybridized methods in different parts of the domain. For these methods, the linear system involves only the unknowns residing on the mesh skeleton, and constructing intergrid transfer operators is therefore not trivial. The key to our geometric multigrid algorithm is the physics-based energy-preserving intergrid transfer operators which depend only on the fine scale DtN maps. Thanks to these operators, we completely avoid upscaling of parameters and no information regarding subgrid physics is explicitly required on coarse meshes. Moreover, our algorithm is agglomeration-based and can straightforwardly handle unstructured meshes. We perform extensive numerical studies with hybridized mixed methods, hybridized discontinuous Galerkin method, weak Galerkin method, and a hybridized version of interior penalty discontinuous Galerkin methods on a range of elliptic problems including subsurface flow through highly heterogeneous porous media. We compare the performance of different smoothers and analyze the effect of stabilization parameters on the scalability of the multigrid algorithm.

Anh Tran, Mike Eldred, Tim Wildey et al.

High-fidelity complex engineering simulations are highly predictive, but also computationally expensive and often require substantial computational efforts. The mitigation of computational burden is usually enabled through parallelism in high-performance cluster (HPC) architecture. In this paper, an asynchronous constrained batch-parallel Bayesian optimization method is proposed to efficiently solve the computationally-expensive simulation-based optimization problems on the HPC platform, with a budgeted computational resource, where the maximum number of simulations is a constant. The advantages of this method are three-fold. First, the efficiency of the Bayesian optimization is improved, where multiple input locations are evaluated massively parallel in an asynchronous manner to accelerate the optimization convergence with respect to physical runtime. This efficiency feature is further improved so that when each of the inputs is finished, another input is queried without waiting for the whole batch to complete. Second, the method can handle both known and unknown constraints. Third, the proposed method considers several acquisition functions at the same time and sample based on an evolving probability mass distribution function using a modified GP-Hedge scheme, where parameters are corresponding to the performance of each acquisition function. The proposed framework is termed aphBO-2GP-3B, which corresponds to asynchronous parallel hedge Bayesian optimization with two Gaussian processes and three batches. The aphBO-2GP-3B framework is demonstrated using two high-fidelity expensive industrial applications, where the first one is based on finite element analysis (FEA) and the second one is based on computational fluid dynamics (CFD) simulations.