We present a parallel multigrid method for solving variable-coefficient elliptic partial differential equations on arbitrary geometries using highly adapted meshes. Our method is designed for meshes that are built from an unstructured hexahedral macro mesh, in which each macro element is adaptively refined as an octree. This forest-of-octrees approach enables us to generate meshes for complex geometries with arbitrary levels of local refinement. We use geometric multigrid (GMG) for each of the octrees and algebraic multigrid (AMG) as the coarse grid solver. We designed our GMG sweeps to entirely avoid collectives, thus minimizing communication cost.We present weak and strong scaling results for the 3D variablecoefficient Poisson problem that demonstrate high parallel scalability. As a highlight, the largest problem we solve is on a non-uniform mesh with 100 billion unknowns on 262,144 cores of NCCS's Cray XK6 "Jaguar"; in this solve we sustain 272 TFlops/s.

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We present a weighted BFBT approximation (w-BFBT) to the inverse Schur complement of a Stokes system with highly heterogeneous viscosity. When used as part of a Schur complement-based Stokes preconditioner, we observe robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations (Q k × P disc k−1 , order k ≥ 2). For certain difficult problems, we demonstrate numerically that w-BFBT significantly improves Stokes solver convergence over the widely used inverse viscosity-weighted pressure mass matrix approximation of the Schur complement. In addition, we derive theoretical eigenvalue bounds to prove spectral equivalence of w-BFBT. Using detailed numerical experiments, we discuss modifications to w-BFBT at Dirichlet boundaries that decrease the number of iterations. The overall algorithmic performance of the Stokes solver is governed by the efficacy of w-BFBT as a Schur complement approximation and, in addition, by our parallel hybrid spectral-geometric-algebraic multigrid (HMG) method, which we use to approximate the inverses of the viscous block and variable-coefficient pressure Poisson operators within w-BFBT. Building on the scalability of HMG, our Stokes solver achieves a parallel efficiency of 90% while weak scaling over a more than 600-fold increase from 48 to all 30,000 cores of TACC's Lonestar 5 supercomputer. *

Mantle convection is the fundamental physical process within earth's interior responsible for the thermal and geological evolution of the planet, including plate tectonics. The mantle is modeled as a viscous, incompressible, non-Newtonian fluid. The wide range of spatial scales, extreme variability and anisotropy in material properties, and severely nonlinear rheology have made global mantle convection modeling with realistic parameters prohibitive. Here we present a new implicit solver that exhibits optimal algorithmic performance and is capable of extreme scaling for hard PDE problems, such as mantle convection. To maximize accuracy and minimize runtime, the solver incorporates a number of advances, including aggressive multi-octree adaptivity, mixed continuous-discontinuous discretization, arbitrarilyhigh-order accuracy, hybrid spectral/geometric/algebraic multigrid, and novel Schur-complement preconditioning. These features present enormous challenges for extreme scalability. We demonstrate that-contrary to conventional wisdom-algorithmically optimal implicit solvers can be designed that scale out to 1.5 million cores for severely nonlinear, ill-conditioned, heterogeneous, and anisotropic PDEs.

For the analysis of me series data from hydrology, we used a recently developed technique that is by now widely known as the Hilbert-Huang transform (HHT). Specifi cally, it is designed for nonlinear and nonsta onary data. In contrast to data analysis techniques using the short-me, windowed Fourier transform or the con nuous wavelet transform, the new technique is empirically adapted to the data in the following sense. First, an addive decomposi on, called empirical mode decomposi on (EMD), of the data into certain mul scale components is computed. Second, to each of these components, the Hilbert transform is applied. The resul ng Hilbert spectrum of the modes provides a localized me-frequency spectrum and instantaneous ( me-dependent) frequencies. In this study, we applied the HHT to hydrological me series data from the Upper Rur Catchment Area, mostly German territory, taken during a period of 20 yr. Our fi rst observa on was that a coarse approxima on of the data can be derived by trunca ng the EMD representaon. This can be used to be er model pa erns like seasonal structures. Moreover, the corresponding me-frequency energy spectrum applied to the complete EMD revealed seasonal events in a par cular apparent way together with their energy. We compared the Hilbert spectra with Fourier spectrograms and wavelet spectra to demonstrate a be er localiza on of the energy components, which also exhibit strong seasonal components. The Hilbert energy spectrum of the three measurement sta ons appear to be very similar, indica ng li le local variability in drainage.Abbrevia ons: EMD, empirical mode decomposi on; HHT, Hilbert-Huang transform; IMF, intrinsic mode func on.Given empirical data, the detection and parameterization of multiscale patterns and shapes in the measurements is an important task. Specifi cally, to study the eff ect of patterns on water and solute fl uxes, temporal and spatial data have to be analyzed at various stages so that their parameterization can eventually be used in simulation fl ux models.Th is study was part of a SFB/TR32 project (Transregional Collaborative Research Centre 32, www.tr32.de; verifi ed 3 June 2010) for which the overall objective is to improve our knowledge about the mechanisms leading to spatial and temporal patterns in energy and matter fl uxes of the soil-vegetation-atmosphere system. Part of the objectives is the determination, description, and analysis of patterns derived from diff erent sources. For instance, the large spatial and temporal variability of soil moisture patterns is determined by factors like atmospheric forcing, topography, soil properties, and vegetation, which interact in a complex, nonlinear way (see, e.g., Grayson and Blöschl, 2001;Western et al., 2004). Th us, a very large number of continuous soil moisture measurements are necessary to adequately capture this variability. In the framework of the TR32, a dense soil moisture sensor network for monitoring soil water content changes at high spatial and temporal scales has been set up (see Bogena et al., ...

Strain localization and resulting plasticity and failure play an important role in the evolution of the lithosphere. These phenomena are commonly modeled by Stokes flows with viscoplastic rheologies. The nonlinearities of these rheologies make the numerical solution of the resulting systems challenging, and iterative methods often converge slowly or not at all. Yet accurate solutions are critical for representing the physics. Moreover, for some rheology laws, aspects of solvability are still unknown. We study a basic but representative viscoplastic rheology law. The law involves a yield stress that is independent of the dynamic pressure, referred to as von Mises yield criterion. Two commonly used variants, perfect/ideal and composite viscoplasticity, are compared. We derive both variants from energy minimization principles, and we use this perspective to argue when solutions are unique. We propose a new stress‐velocity Newton solution algorithm that treats the stress as an independent variable during the Newton linearization but requires solution only of Stokes systems that are of the usual velocity‐pressure form. To study different solution algorithms, we implement 2‐D and 3‐D finite element discretizations, and we generate Stokes problems with up to 7 orders of magnitude viscosity contrasts, in which compression or tension results in significant nonlinear localization effects. Comparing the performance of the proposed Newton method with the standard Newton method and the Picard fixed‐point method, we observe a significant reduction in the number of iterations and improved stability with respect to problem nonlinearity, mesh refinement, and the polynomial order of the discretization.

We propose to use deep learning to estimate parameters in statistical models when standard likelihood estimation methods are computationally infeasible. We show how to estimate parameters from max-stable processes, where inference is exceptionally challenging even with small datasets but simulation is straightforward. We use data from model simulations as input and train deep neural networks to learn statistical parameters. Our neural-network-based method provides a competitive alternative to current approaches, as demonstrated by considerable accuracy and computational time improvements. It serves as a proof of concept for deep learning in statistical parameter estimation and can be extended to other estimation problems.

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