The Trilinos Project is an effort to facilitate the design, development, integration and ongoing support of mathematical software libraries within an object-oriented framework for the solution of large-scale, complex multi-physics engineering and scientific problems. Trilinos addresses two fundamental issues of developing software for these problems: (i) Providing a streamlined process and set of tools for development of new algorithmic implementations and (ii) promoting interoperability of independently developed software. Trilinos uses a two-level software structure designed around collections of packages. A Trilinos package is an integral unit usually developed by a small team of experts in a particular algorithms area such as algebraic preconditioners, nonlinear solvers, etc. Packages exist underneath the Trilinos top level, which provides a common look-and-feel, including configuration, documentation, licensing, and bug-tracking.Here we present the overall Trilinos design, describing our use of abstract interfaces and default concrete implementations. We discuss the services that Trilinos provides to a prospective package and how these services are used by various packages. We also illustrate how packages can be combined to rapidly develop new algorithms. Finally, we discuss how Trilinos facilitates highquality software engineering practices that are increasingly required from simulation software.

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This paper surveys the techniques that are necessary for constructing computationally efficient parallel multigrid solvers. Both geometric and algebraic methods are considered. We first cover the sources of parallelism, including traditional spatial partitioning and more novel additive multilevel methods. We then cover the parallelism issues that must be addressed: parallel smoothing and coarsening, operator complexity, and parallelization of the coarsest grid solve.

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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 Schöberl. The main focus in the Reitzinger and Schöberl method is to maintain null-space properties of the weak ∇ × ∇× 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 Schöberl method, as well as our previous smoothed method, are shown to be special cases of this new algorithm.

Abstract. We develop stochastic mixed finite element methods for spatially adaptive simulations of fluid-structure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuation-dissipation balance condition to develop compatible stochastic driving fields for our discretization. We perform analysis that shows our condition is sufficient to ensure results consistent with statistical mechanics. We show the Gibbs-Boltzmann distribution is invariant under the stochastic dynamics of the semi-discretization. To generate efficiently the required stochastic driving fields, we develop a Gibbs sampler based on iterative methods and multigrid to generate fields with O(N ) computational complexity. Our stochastic methods provide an alternative to uniform discretizations on periodic domains that rely on Fast Fourier Transforms. To demonstrate in practice our stochastic computational methods, we investigate within channel geometries having internal obstacles and no-slip walls how the mobility/diffusivity of particles depends on location. Our methods extend the applicability of fluctuating hydrodynamic approaches by allowing for spatially adaptive resolution of the mechanics and for domains that have complex geometries relevant in many applications. Introduction. We develop general computational methods for applications involving the microscopic mechanics of spatially extended elastic bodies within a fluid that are subjected to thermal fluctuations. Motivating applications include the study of the microstructures of complex fluids [17], lipid bilayer membranes [28,32,48], and micro-mechanical devices [37,29]. Even in the deterministic setting, the mechanics of fluid-structure interactions pose a number of difficult and long-standing challenges owing to the rich behaviors that can arise from the interplay of the fluid flow and elastic stresses of the microstructures [19,42]. To obtain descriptions tractable for analysis and simulations, approximations are often introduced into the fluid-structure coupling. For deterministic systems, many spatially adaptive numerical methods have been developed for approximate fluid-structure interactions [25,35,26,39,2,30]. In the presence of thermal fluctuations, additional challenges arise from the need to capture in computational methods the appropriate propagation of fluctuations throughout the discretized system to obtain results consistent with statistical mechanics. In practice, challenges arise from the very different dissipative properties of the discrete operators relative to their continuum differential counterparts. These issues have important implications for how stochastic fluctuations should be handled in the discrete setting. Even when it is possible to formulate stochastic driving fields in a well-founded manner consistent with statistical mechanics, these Gaussian random

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Quantifying simulation uncertainties is a critical component of rigorous predictive simulation. A key component of this is forward propagation of uncertainties in simulation input data to output quantities of interest. Typical approaches involve repeated sampling of the simulation over the uncertain input data, and can require numerous samples when accurately propagating uncertainties from large numbers of sources. Often simulation processes from sample to sample are similar and much of the data generated from each sample evaluation could be reused. We explore a new method for implementing sampling methods that simultaneously propagates groups of samples together in an embedded fashion, which we call embedded ensemble propagation. We show how this approach takes advantage of properties of modern computer architectures to improve performance by enabling reuse between samples, reducing memory bandwidth requirements, improving memory access patterns, improving opportunities for fine-grained parallelization, and reducing communication costs. We describe a software technique for implementing embedded * ensemble propagation based on the use of C++ templates and describe its integration with various scientific computing libraries within Trilinos. We demonstrate improved performance, portability and scalability for the approach applied to the simulation of partial differential equations on a variety of CPU, GPU, and accelerator architectures, including up to 131,072 cores on a Cray XK7 (Titan).