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In this work, we present a novel balancing domain decomposition by constraints preconditioner that is robust for multi-material problems. We start with a well-balanced subdomain partition, and based on an aggregation of elements according to their physical coecients, we end up with a ner physics-based (PB) subdomain partition. Next, we dene geometrical objects (corners, edges, and faces) for this PB partition, and select some of them to enforce subdomain continuity (primal objects). When the physical coecient in each PB subdomain is constant and the set of selected primal objects satisfy a mild condition on the existence of acceptable paths, we can show both theoretically and numerically that the condition number does not depend on the contrast of the coecient. An extensive set of numerical experiments for 2D and 3D Poisson's and linear elasticity problems is provided to support our ndings. In particular, we show robustness and weak scalability of the new preconditioner up to 8232 cores when applied to 3D multi-material problems with the contrast of the physical coecient up to 10 8 and more than half a billion degrees of freedom. For the scalability analysis, we have exploited a highly scalable advanced interlevel overlapped implementation of the preconditioner that deals very eciently with the coarse problem computation. The proposed preconditioner is compared against a state-of-the-art implementation of an adaptive BDDC method in PETSc for thermal and mechanical multi-material problems.

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Soil erosion induced by rainfall is a critical problem in many regions in the world, particularly in tropical areas where the annual rainfall amount often exceeds 2000 mm. Predicting soil erosion is a challenging task, subjecting to variation of soil characteristics, slope, vegetation cover, land management, and weather condition. Conventional models based on the mechanism of soil erosion processes generally provide good results but are time-consuming due to calibration and validation. The goal of this study is to develop a machine learning model based on support vector machine (SVM) for soil erosion prediction. The SVM serves as the main prediction machinery establishing a nonlinear function that maps considered influencing factors to accurate predictions. In addition, in order to improve the accuracy of the model, the history-based adaptive differential evolution with linear population size reduction and population-wide inertia term (L-SHADE-PWI) is employed to find an optimal set of parameters for SVM. Thus, the proposed method, named L-SHADE-PWI-SVM, is an integration of machine learning and metaheuristic optimization. For the purpose of training and testing the method, a dataset consisting of 236 samples of soil erosion in Northwest Vietnam is collected with 10 influencing factors. The training set includes 90% of the original dataset; the rest of the dataset is reserved for assessing the generalization capability of the model. The experimental results indicate that the newly developed L-SHADE-PWI-SVM method is a competitive soil erosion predictor with superior performance statistics. Most importantly, L-SHADE-PWI-SVM can achieve a high classification accuracy rate of 92%, which is much better than that of backpropagation artificial neural network (87%) and radial basis function artificial neural network (78%).

Abstract. In this paper, we formulate and analyze a perturbed formulation of the balancing domain decomposition by constraints (BDDC) method. We prove that the perturbed BDDC has the same polylogarithmic bound for the condition number as the standard formulation. Two types of properly scaled zero-order perturbations are considered: one uses a mass matrix, and the other uses a Robin-type boundary condition, i.e, a mass matrix on the interface. With perturbation, the wellposedness of the local Neumann problems and the global coarse problem is automatically guaranteed, and coarse degrees of freedom can be defined only for convergence purposes but not well-posedness. This allows a much simpler implementation as no complicated corner selection algorithm is needed. Minimal coarse spaces using only face or edge constraints can also be considered. They are very useful in extreme scale calculations where the coarse problem is usually the bottleneck that can jeopardize scalability. The perturbation also adds extra robustness as the perturbed formulation works even when the constraints fail to eliminate a small number of subdomain rigid body modes from the standard BDDC space. This is extremely important when solving problems on unstructured meshes partitioned by automatic graph partitioners since arbitrary disconnected subdomains are possible. Numerical results are provided to support the theoretical findings.Key words. BDDC, preconditioner, coarse space, parallel solver, scalability AMS subject classifications. 65N55, 65N22, 65F08 DOI. 10.1137/15M10456481. Introduction. The development of highly scalable linear solvers for the solution of large scale linear systems arising from the finite element (FE) discretization of second-order elliptic problems on distributed-memory machines is of great importance in many applications. In this work, we consider nonoverlapping domain decomposition (DD) methods [49], which take advantage of the partition of the FE mesh into submeshes to define effective preconditioners that can exploit large levels of concurrency. In particular, we focus on a variant of the balancing DD by constraints (BDDC) method. The BDDC method was first introduced in 2003 by Dohrmann [18]. It can be regarded as an improved version of the balancing domain decomposition (BDD) method by Mandel [43]. It also has a very close connection with the dual primal finite element tearing and interconnecting (FETI-DP) method [25,24]. In fact, the eigenvalues of the preconditioned systems in the two approaches are almost identical [44,41,15]. The BDDC method is particularly well suited for extreme scale simulations, since it allows for a very aggressive coarsening, the computations at different levels can be computed in parallel, the subdomain problems can be solved inexactly