On Optimizing Distributed Tucker Decomposition for Dense Tensors

The Tucker decomposition expresses a given tensor as the product of a small core tensor and a set of factor matrices. Apart from providing data compression, the construction is useful in performing analysis such as principal component analysis (PCA)and finds applications in diverse domains such as signal processing, computer vision and text analytics. Our objective is to develop an efficient distributed implementation for the case of dense tensors. The implementation is based on the HOOI (Higher Order Orthogonal Iterator) procedure, wherein the tensor-times-matrix product forms the core routine. Prior work have proposed heuristics for reducing the computational load and communication volume incurred by the routine. We study the two metrics in a formal and systematic manner, and design strategies that are optimal under the two fundamental metrics. Our experimental evaluation on a large benchmark of tensors shows that the optimal strategies provide significant reduction in load and volume compared to prior heuristics, and provide up to 7x speed-up in the overall running time.Comment: Preliminary version of the paper appears in the proceedings of IPDPS'1

Efficient Uncertainty Quantification with the Polynomial Chaos Method for Stiff Systems

The polynomial chaos method has been widely adopted as a computationally feasible approach for uncertainty quantification. Most studies to date have focused on non-stiff systems. When stiff systems are considered, implicit numerical integration requires the solution of a nonlinear system of equations at every time step. Using the Galerkin approach, the size of the system state increases from $n$ to $S \times n$, where $S$ is the number of the polynomial chaos basis functions. Solving such systems with full linear algebra causes the computational cost to increase from $O(n^3)$ to $O(S^3n^3)$. The $S^3$-fold increase can make the computational cost prohibitive. This paper explores computationally efficient uncertainty quantification techniques for stiff systems using the Galerkin, collocation and collocation least-squares formulations of polynomial chaos. In the Galerkin approach, we propose a modification in the implicit time stepping process using an approximation of the Jacobian matrix to reduce the computational cost. The numerical results show a run time reduction with a small impact on accuracy. In the stochastic collocation formulation, we propose a least-squares approach based on collocation at a low-discrepancy set of points. Numerical experiments illustrate that the collocation least-squares approach for uncertainty quantification has similar accuracy with the Galerkin approach, is more efficient, and does not require any modifications of the original code

High-order Discretization of a Gyrokinetic Vlasov Model in Edge Plasma Geometry

We present a high-order spatial discretization of a continuum gyrokinetic Vlasov model in axisymmetric tokamak edge plasma geometries. Such models describe the phase space advection of plasma species distribution functions in the absence of collisions. The gyrokinetic model is posed in a four-dimensional phase space, upon which a grid is imposed when discretized. To mitigate the computational cost associated with high-dimensional grids, we employ a high-order discretization to reduce the grid size needed to achieve a given level of accuracy relative to lower-order methods. Strong anisotropy induced by the magnetic field motivates the use of mapped coordinate grids aligned with magnetic flux surfaces. The natural partitioning of the edge geometry by the separatrix between the closed and open field line regions leads to the consideration of multiple mapped blocks, in what is known as a mapped multiblock (MMB) approach. We describe the specialization of a more general formalism that we have developed for the construction of high-order, finite-volume discretizations on MMB grids, yielding the accurate evaluation of the gyrokinetic Vlasov operator, the metric factors resulting from the MMB coordinate mappings, and the interaction of blocks at adjacent boundaries. Our conservative formulation of the gyrokinetic Vlasov model incorporates the fact that the phase space velocity has zero divergence, which must be preserved discretely to avoid truncation error accumulation. We describe an approach for the discrete evaluation of the gyrokinetic phase space velocity that preserves the divergence-free property to machine precision

New Discretization Methods for the Numerical Approximation of PDEs

The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems. The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years