We present a survey of the fundamentals and the applications of sparse grids, with a focus on the solution of partial differential equations (PDEs). The sparse grid approach, introduced in Zenger (1991), is based on a higherdimensional multiscale basis, which is derived from a one-dimensional multiscale basis by a tensor product construction. Discretizations on sparse grids involve O(N ·(log N ) d−1 ) degrees of freedom only, where d denotes the underlying problem's dimensionality and where N is the number of grid points in one coordinate direction at the boundary. The accuracy obtained with piecewise linear basis functions, for example, is O(N −2 · (log N ) d−1 ) with respect to the L 2 -and L ∞ -norm, if the solution has bounded second mixed derivatives. This way, the curse of dimensionality, i.e., the exponential dependence O(N d ) of conventional approaches, is overcome to some extent. For the energy norm, only O(N ) degrees of freedom are needed to give an accuracy of O(N −1 ). That is why sparse grids are especially well-suited for problems of very high dimensionality.The sparse grid approach can be extended to nonsmooth solutions by adaptive refinement methods. Furthermore, it can be generalized from piecewise linear to higher-order polynomials. Also, more sophisticated basis functions like interpolets, prewavelets, or wavelets can be used in a straightforward way.We describe the basic features of sparse grids and report the results of various numerical experiments for the solution of elliptic PDEs as well as for other selected problems such as numerical quadrature and data mining. CONTENTS1 Introduction 148 2 Breaking the curse of dimensionality 151 3 Piecewise linear interpolation on sparse grids 154 4 Generalizations, related concepts, applications 188 5 Numerical experiments 219 6 Concluding remarks 255 References 256 Breaking the curse of dimensionalityClassical approximation schemes exhibit the curse of dimensionality (Bellmann 1961) mentioned above. We havewhere r and d denote the isotropic smoothness of the function f and the problem's dimensionality, respectively. This is one of the main obstacles in the treatment of high-dimensional problems. Therefore, the question is whether we can find situations, i.e., either function spaces or error norms, for which the curse of dimensionality can be broken. At first glance, there is an easy way out: if we make a stronger assumption on the smoothness of the function f such thatOf course, such an assumption is completely unrealistic. However, about ten years ago, Barron (1993) found an interesting result. Denote by FL 1 the class of functions with Fourier transforms in L 1 . Then, consider the class of functions of R d with ∇f ∈ FL 1 .We expect an approximation rateMeanwhile, other function classes are known with such properties. These comprise certain radial basis schemes, stochastic sampling techniques, and approaches that work with spaces of functions with bounded mixed derivatives.A better understanding of these results is possible with the help of harmon...

No abstract

Abstract. Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N 3 ) with the size of the investigated problem, N (e.g., the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the ELPA library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the ScaLAPACK library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g., Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPIOpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem sizes arising in the electronic structure theory is demonstrated for current high-performance computer architectures such as Cray or Intel/Infiniband. For a ELPA -Eigenvalue Solutions for Electronic Structure Theory 2 matrix of dimension 260,000, scalability up to 295,000 CPU cores has been shown on BlueGene/P.

a b s t r a c tSparse grids allow one to employ grid-based discretization methods in data-driven problems. We present an extension of the classical sparse grid approach that allows us to tackle highdimensional problems by spatially adaptive refinement, modified ansatz functions, and efficient regularization techniques. The competitiveness of this method is shown for typical benchmark problems with up to 166 dimensions for classification in data mining, pointing out properties of sparse grids in this context. To gain insight into the adaptive refinement and to examine the scope for further improvements, the approximation of non-smooth indicator functions with adaptive sparse grids has been studied as a model problem. As an example for an improved adaptive grid refinement, we present results for an edge-detection strategy.

This paper presents a new approach to construct more efficient reduced-order models for nonlinear partial differential equations with proper orthogonal decomposition and the discrete empirical interpolation method (DEIM). Whereas DEIM projects the nonlinear term onto one global subspace, our localized discrete empirical interpolation method (LDEIM) computes several local subspaces, each tailored to a particular region of characteristic system behavior. Then, depending on the current state of the system, LDEIM selects an appropriate local subspace for the approximation of the nonlinear term. In this way, the dimensions of the local DEIM subspaces, and thus the computational costs, remain low even though the system might exhibit a wide range of behaviors as it passes through different regimes. LDEIM uses machine learning methods in the offline computational phase to discover these regions via clustering. Local DEIM approximations are then computed for each cluster. In the online computational phase, machine-learning-based classification procedures select one of these local subspaces adaptively as the computation proceeds. The classification can be achieved using either the system parameters or a low-dimensional representation of the current state of the system obtained via feature extraction. The LDEIM approach is demonstrated for a reacting flow example of an H 2-Air flame. In this example, where the system state has a strong nonlinear dependence on the parameters, the LDEIM provides speedups of two orders of magnitude over standard DEIM.

No abstract

A novel and versatile numerical form-finding procedure that requires only a minimal knowledge of the structure is presented. The procedure only needs the type of each member, i.e. either compression or tension, and the connectivity of the nodes to be known. Both equilibrium geometry and force densities are iteratively calculated. A condition of a maximal rank of the force density matrix and minimal member length, were included in the form-finding procedure to guide the search of a state of self-stress with minimal elastic potential energy. It is indeed able to calculate novel configurations, with no assumptions on cable lengths or cable-to-strut ratios. Moreover, the proposed approach compares favourably with all the leading techniques in the field. This is clearly exemplified through a series of examples.

The molecular dynamics simulation code ls1 mardyn is presented. It is a highly scalable code, optimized for massively parallel execution on supercomputing architectures, and currently holds the world record for the largest molecular simulation with over four trillion particles. It enables the application of pair potentials to length and time scales which were previously out of scope for molecular dynamics simulation. With an efficient dynamic load balancing scheme, it delivers high scalability even for challenging heterogeneous configurations. * To whom correspondence should be addressed † High Performance Computing Center Stuttgart (HLRS), Germany ‡ Laboratory of Engineering Thermodynamics (LTD), Univ. of Kaiserslautern, Germany ¶ Scientific Computing in Computer Science (SCCS), TU München, Germany § Thermodynamics and Energy Technology (ThEt), Univ. of Paderborn, Germany 1 Presently, multi-center rigid potential models based on Lennard-Jones sites, point charges and higher-order polarities are supported. Due to its modular design, ls1 mardyn can be extended to new physical models, methods, and algorithms, allowing future users to tailor it to suit their respective needs. Possible applications include scenarios with complex geometries, e.g.for fluids at interfaces, as well as non-equilibrium molecular dynamics simulation of heat and mass transfer.