Related papers: Multi-Grid Lanczos
In her seminal 1989 work, Greenbaum demonstrated that the results produced by the finite precision Lanczos algorithm after $k$ iterations can be interpreted as exact Lanczos results applied to a larger matrix, whose eigenvalues lie in small…
The truncated Lanczos method using a variational scheme based on Hilbert space reduction as well as a local basis change is re-examined. The energy is extrapolated as a power law function of the Hamiltonian variance. This systematic…
In this paper we propose and analyze an algorithm for identifying spectral gaps of a real symmetric matrix $A$ by simultaneously approximating the traces of spectral projectors associated with multiple different spectral slices. Our method…
Path-integral techniques are a powerful tool used in open quantum systems to provide an exact solution for the non-Markovian dynamics. However, the exponential scaling of the tensor size with quantum memory length of these techniques limits…
Structured Cartesian grids are a fundamental component in numerical simulations. Although these grids facilitate straightforward discretization schemes, their na\"{i}ve use in sparse domains leads to excessive memory overhead and…
The low rank approximation of matrices is a crucial component in many data mining applications today. A competitive algorithm for this class of problems is the randomized block Lanczos algorithm - an amalgamation of the traditional block…
We motivate a method for transparently identifying ineffectual computations in unmodified Deep Learning models and without affecting accuracy. Specifically, we show that if we decompose multiplications down to the bit level the amount of…
Graphics Processing Units (GPUs) are having a transformational effect on numerical lattice quantum chromodynamics (LQCD) calculations of importance in nuclear and particle physics. The QUDA library provides a package of mixed precision…
This paper discusses the requirements for and performance metrics of the the Grid Computing system used to implement the Locus Algorithm to identify optimum pointings for differential photometry of 61,662,376 stars and 23,779 quasars.…
Quadratic forms of Hermitian matrix resolvents involve the solutions of shifted linear systems. Efficient iterative solutions use the shift-invariance property of Krylov subspaces The Hermitian Lanczos method reduces a given vector and…
We implement the Lanczos algorithm on an Intel Xeon Phi coprocessor and compare its performance to a multi-core Intel Xeon CPU and an NVIDIA graphics processor. The Xeon and the Xeon Phi are parallelized with OpenMP and the graphics…
We consider an autocorrelation function of a quantum mechanical system through the lens of the so-called recursive method, by iteratively evaluating Lanczos coefficients, or solving a system of coupled differential equations in the Mori…
We describe a set of techniques for performing large scale ab initio calculations using multigrid accelerations and a real-space grid as a basis. The multigrid methods provide effective convergence acceleration and preconditioning on all…
We show that any randomized first-order algorithm which minimizes a $d$-dimensional, $1$-Lipschitz convex function over the unit ball must either use $\Omega(d^{2-\delta})$ bits of memory or make $\Omega(d^{1+\delta/6-o(1)})$ queries, for…
Optimizing the performance of stencil algorithms has been the subject of intense research over the last two decades. Since many stencil schemes have low arithmetic intensity, most optimizations focus on increasing the temporal data access…
We present a new structure preserving Lanczos algorithm for approximating the optical absorption spectrum in the context of solving full Bethe--Salpeter equation without Tamm--Dancoff approximation. The new algorithm is based on a structure…
We study the Lanczos algorithm where the initial vector is sampled uniformly from $\mathbb{S}^{n-1}$. Let $A$ be an $n \times n$ Hermitian matrix. We show that when run for few iterations, the output of Lanczos on $A$ is almost…
An application of an effective numerical algorithm for solving eigenvalue problems which arise in modelling electronic properties of quantum disordered systems is considered. We study the electron states at the localization-delocalization…
The past decade has witnessed a dramatic acceleration of lattice quantum chromodynamics calculations in nuclear and particle physics. This has been due to both significant progress in accelerating the iterative linear solvers using…
We propose a thick-restart block Lanczos method, which is an extension of the thick-restart Lanczos method with the block algorithm, as an eigensolver of the large-scale shell-model calculations. This method has two advantages over the…