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We introduce MPAS-Albany Land Ice (MALI) v6.0, a new variable-resolution land ice model that uses unstructured Voronoi grids on a plane or sphere. MALI is built using the Model for Prediction Across Scales (MPAS) framework for developing variable-resolution Earth system model components and the Albany multi-physics code base for the solution of coupled systems of partial differential equations, which itself makes use of Trilinos solver libraries. MALI includes a three-dimensional first-order momentum balance solver (Blatter–Pattyn) by linking to the Albany-LI ice sheet velocity solver and an explicit shallow ice velocity solver. The evolution of ice geometry and tracers is handled through an explicit first-order horizontal advection scheme with vertical remapping. The evolution of ice temperature is treated using operator splitting of vertical diffusion and horizontal advection and can be configured to use either a temperature or enthalpy formulation. MALI includes a mass-conserving subglacial hydrology model that supports distributed and/or channelized drainage and can optionally be coupled to ice dynamics. Options for calving include “eigencalving”, which assumes that the calving rate is proportional to extensional strain rates. MALI is evaluated against commonly used exact solutions and community benchmark experiments and shows the expected accuracy. Results for the MISMIP3d benchmark experiments with MALI's Blatter–Pattyn solver fall between published results from Stokes and L1L2 models as expected. We use the model to simulate a semi-realistic Antarctic ice sheet problem following the initMIP protocol and using 2 km resolution in marine ice sheet regions. MALI is the glacier component of the Energy Exascale Earth System Model (E3SM) version 1, and we describe current and planned coupling to other E3SM components.

The one-fluid visco-resistive MHD model provides a description of the dynamics of a charged fluid under the influence of an electromagnetic field. This model is strongly coupled, highly nonlinear, and characterized by physical mechanisms that span a wide range of interacting time scales. Solutions of this system can include very fast component time scales to slowly varying dynamical time scales that are long relative to the normal modes of the model equations. Fully implicit time stepping is attractive for simulating this type of wide-ranging physical phenomena. However, it is essential that one has effective preconditioning strategies so that the overall fully implicit methodology is both efficient and scalable. In this paper, we propose and explore the performance of several candidate block preconditioners for this system. One of these preconditioners is based on an operator-split approximation. This method reduces the $3\\times3$ system (momentum, continuity, and magnetics) into two $2\\times2$ operators: a Navier--Stokes operator (momentum and continuity) and a magnetics-velocity operator (momentum and magnetics) which takes into account the critical Lorentz force coupling. Using previously developed preconditioners for Navier--Stokes, and an initial Schur-complement approximation for the magnetics-velocity system, we show that the split preconditioner is scalable and competitive with other preconditioners, including a fully coupled algebraic multigrid method. [PUBLICATION ABSTRACT]

We propose a new variant of smoothed aggregation (SA) suitable for nonsymmetric linear systems. The new algorithm is based on two key generalizations of SA: restriction smoothing and local damping. Restriction smoothing refers to the smoothing of a tentative restriction operator via a damped Jacobi-like iteration. Restriction smoothing is analogous to prolongator smoothing in standard SA and in fact has the same form as the transpose of prolongator smoothing when the matrix is symmetric. Local damping refers to damping parameters used in the Jacobi-like iteration. In standard SA, a single damping parameter is computed via an eigenvalue computation. Here, local damping parameters are computed by considering the minimization of an energy-like quantity for each individual grid transfer basis function. Numerical results are given showing how this method performs on highly nonsymmetric systems.

Algebraic multigrid methods solve sparse linear systems Ax = b by automatic construction of a multilevel hierarchy. This hierarchy is defined by grid transfer operators that must accurately capture algebraically smooth error relative to the relaxation method. We propose a methodology to improve grid transfers through energy minimization. The proposed strategy is applicable to Hermitian, non-Hermitian, definite, and indefinite problems. Each column of the grid transfer operator P is minimized in an energy-based norm while enforcing two types of constraints: a defined sparsity pattern and preservation of specified modes in the range of P. A Krylov-based strategy is used to minimize energy, which is equivalent to solving APj = 0 for each column j of P, with the constraints ensuring a nontrivial solution. For the Hermitian positive definite case, a conjugate gradient (CG-)based method is utilized to construct grid transfers, while methods based on generalized minimum residual (GMRES) and CG on the normal equations (CGNR) are explored for the general case. The approach is flexible, allowing for arbitrary coarsenings, unrestricted sparsity patterns, straightforward long-distance interpolation, and general use of constraints, either user-defined or auto-generated. We conclude with numerical evidence in support of the proposed framework. [PUBLICATION ABSTRACT]

We propose a new ice sheet model validation framework - the Cryospheric Model Comparison Tool (CmCt) - that takes advantage of ice sheet altimetry and gravimetry observations collected over the past several decades and is applied here to modeling of the Greenland ice sheet. We use realistic simulations performed with the Community Ice Sheet Model (CISM) along with two idealized, non-dynamic models to demonstrate the framework and its use. Dynamic simulations with CISM are forced from 1991 to 2013, using combinations of reanalysis-based surface mass balance and observations of outlet glacier flux change. We propose and demonstrate qualitative and quantitative metrics for use in evaluating the different model simulations against the observations. We find that the altimetry observations used here are largely ambiguous in terms of their ability to distinguish one simulation from another. Based on basin-scale and whole-ice-sheet-scale metrics, we find that simulations using both idealized conceptual models and dynamic, numerical models provide an equally reasonable representation of the ice sheet surface (mean elevation differences of < 1m). This is likely due to their short period of record, biases inherent to digital elevation models used for model initial conditions, and biases resulting from firn dynamics, which are not explicitly accounted for in the models or observations. On the other hand, we find that the gravimetry observations used here are able to unambiguously distinguish between simulations of varying complexity, and along with the CmCt, can provide a quantitative score for assessing a particular model and/or simulation. The new framework demonstrates that our proposed metrics can distinguish relatively better from relatively worse simulations and that dynamic ice sheet models, when appropriately initialized and forced with the right boundary conditions, demonstrate a predictive skill with respect to observed dynamic changes that have occurred on Greenland over the past few decades. An extensible design will allow for continued use of the CmCt as future altimetry, gravimetry, and other remotely sensed data become available for use in ice sheet model validation.

Here, we propose a new algorithm for the fast solution of large, sparse, symmetric positive-definite linear systems, spaND (sparsified Nested Dissection). It is based on nested dissection, sparsification, and low-rank compression. After eliminating all interiors at a given level of the elimination tree, the algorithm sparsifies all separators corresponding to the interiors. This operation reduces the size of the separators by eliminating some degrees of freedom but without introducing any fill-in. This is done at the expense of a small and controllable approximation error. The result is an approximate factorization that can be used as an efficient preconditioner. We then perform several numerical experiments to evaluate this algorithm. We demonstrate that a version using orthogonal factorization and block-diagonal scaling takes fewer CG iterations to converge than previous similar algorithms on various kinds of problems. Furthermore, this algorithm is provably guaranteed to never break down and the matrix stays symmetric positive-definite throughout the process. We evaluate the algorithm on some large problems show it exhibits near-linear scaling. The factorization time is roughly $\\mathcal{O}$(N), and the number of iterations grows slowly with N.

With the rise in popularity of compatible finite element, finite difference, and finite volume discretizations for the time domain eddy current equations, there has been a corresponding need for fast solvers of the resulting linear algebraic systems. However, the traits that make compatible discretizations a preferred choice for the Maxwell's equations also render these linear systems essentially intractable by truly black-box techniques. We propose an algebraic reformulation of the discrete eddy current equations along with a new algebraic multigrid (AMG) technique for this reformulated problem. The reformulation process takes advantage of a discrete Hodge decomposition on cochains to replace the discrete eddy current equations by an equivalent $2\\times2$ block linear system whose diagonal blocks are discrete Hodge-Laplace operators acting on 1-cochains and 0-cochains, respectively. While this new AMG technique requires somewhat specialized treatment on the finest mesh, the coarser meshes can be handled using standard methods for Laplace-type problems. Our new AMG method is applicable to a wide range of compatible methods on structured and unstructured grids, including edge finite elements, mimetic finite differences, covolume methods, and Yee-like schemes. We illustrate the new technique, using edge elements in the context of smoothed aggregation AMG, and present computational results for problems in both two and three dimensions.

We propose a new algebraic multigrid (AMG) method for solving the eddy current approximations to Maxwell's equations. This AMG method has its roots in an algorithm proposed by Reitzinger and Schoberl. The main focus in the Reitzinger and Schoberl method is to maintain null-space properties of the weak $\\nabla \\times \\nabla \\>{\\times}$ operator on coarse grids. While these null-space properties are critical, they are not enough to guarantee $h$-independent convergence rates of the overall multigrid scheme. We present a new strategy for choosing intergrid transfers that not only maintains the important null-space properties on coarse grids but also yields significantly improved multigrid convergence rates. This improvement is related to those we explored in a previous paper, but is fundamentally simpler, easier to compute, and performs better with respect to both multigrid operator complexity and convergence rates. The new strategy builds on ideas in smoothed aggregation to improve the approximation property of an existing interpolation operator. By carefully choosing the smoothing operators, we show how it is sometimes possible to achieve $h$-independent convergence rates with a modest increase in multigrid operator complexity. Though this ideal case is not always possible, the overall algorithm performs significantly better than the original scheme in both iterations and run time. Finally, the Reitzinger and Schoberl method, as well as our previous smoothed method, are shown to be special cases of this new algorithm.