Hari Sundar

Hari Sundar, Georg Stadler, George Biros

We present a comparison of different multigrid approaches for the solution of systems arising from high-order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the pointwise Jacobi, the Chebyshev-accelerated Jacobi and the symmetric successive over-relaxation (SSOR) smoothers, as well as elementwise block Jacobi smoothing. Three approaches for the multigrid hierarchy are compared: 1) high-order $h$-multigrid, which uses high-order interpolation and restriction between geometrically coarsened meshes; 2) $p$-multigrid, in which the polynomial order is reduced while the mesh remains unchanged, and the interpolation and restriction incorporate the different-order basis functions; and 3), a first-order approximation multigrid preconditioner constructed using the nodes of the high-order discretization. This latter approach is often combined with algebraic multigrid for the low-order operator and is attractive for high-order discretizations on unstructured meshes, where geometric coarsening is difficult. Based on a simple performance model, we compare the computational cost of the different approaches. Using scalar test problems in two and three dimensions with constant and varying coefficients, we compare the performance of the different multigrid approaches for polynomial orders up to 16. Overall, both $h$- and $p$-multigrid work well; the first-order approximation is less efficient. For constant coefficients, all smoothers work well. For variable coefficients, Chebyshev and SSOR smoothing outperforms Jacobi smoothing. While all of the tested methods converge in a mesh-independent number of iterations, none of them behaves completely independent of the polynomial order. When multigrid is used as a preconditioner in a Krylov method, the iteration number decreases significantly compared to using multigrid as a solver.

Jonathan Wittmer, Jacob Badger, Hari Sundar et al.

PDE-constrained inverse problems are some of the most challenging and computationally demanding problems in computational science today. Fine meshes that are required to accurately compute the PDE solution introduce an enormous number of parameters and require large scale computing resources such as more processors and more memory to solve such systems in a reasonable time. For inverse problems constrained by time dependent PDEs, the adjoint method that is often employed to efficiently compute gradients and higher order derivatives requires solving a time-reversed, so-called adjoint PDE that depends on the forward PDE solution at each timestep. This necessitates the storage of a high dimensional forward solution vector at every timestep. Such a procedure quickly exhausts the available memory resources. Several approaches that trade additional computation for reduced memory footprint have been proposed to mitigate the memory bottleneck, including checkpointing and compression strategies. In this work, we propose a close-to-ideal scalable compression approach using autoencoders to eliminate the need for checkpointing and substantial memory storage, thereby reducing both the time-to-solution and memory requirements. We compare our approach with checkpointing and an off-the-shelf compression approach on an earth-scale ill-posed seismic inverse problem. The results verify the expected close-to-ideal speedup for both the gradient and Hessian-vector product using the proposed autoencoder compression approach. To highlight the usefulness of the proposed approach, we combine the autoencoder compression with the data-informed active subspace (DIAS) prior to show how the DIAS method can be affordably extended to large scale problems without the need of checkpointing and large memory.