Evaluation of Directive-Based GPU Programming Models on a Block Eigensolver with Consideration of Large Sparse Matrices

Achieving high performance and performance portability for large-scale scientific applications is a major challenge on heterogeneous computing systems such as many-core CPUs and accelerators like GPUs. In this work, we implement a widely used block eigensolver, Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG), using two popular directive based programming models (OpenMP and OpenACC) for GPU-accelerated systems. Our work differs from existing work in that it adopts a holistic approach that optimizes the full solver performance rather than narrowing the problem into small kernels (e.g., SpMM, SpMV). Our LOPBCG GPU implementation achieves a 2.8$${\times }$$–4.3$${\times }$$ speedup over an optimized CPU implementation when tested with four different input matrices. The evaluated configuration compared one Skylake CPU to one Skylake CPU and one NVIDIA V100 GPU. Our OpenMP and OpenACC LOBPCG GPU implementations gave nearly identical performance. We also consider how to create an efficient LOBPCG solver that can solve problems larger than GPU memory capacity. To this end, we create microbenchmarks representing the two dominant kernels (inner product and SpMM kernel) in LOBPCG and then evaluate performance when using two different programming approaches: tiling the kernels, and using Unified Memory with the original kernels. Our tiled SpMM implementation achieves a 2.9$${\times }$$ and 48.2$${\times }$$ speedup over the Unified Memory implementation on supercomputers with PCIe Gen3 and NVLink 2.0 CPU to GPU interconnects, respectively

Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix

The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace \X, and a Hermitian matrix $A$, the eigenvalues of $X^HAX$ are called Ritz values of $A$ with respect to \X. If the subspace \X is $A$-invariant then the Ritz values are some of the eigenvalues of $A$. If the $A$-invariant subspace \X is perturbed to give rise to another subspace \Y, then the vector of absolute values of changes in Ritz values of $A$ represents the absolute eigenvalue approximation error using \Y. We bound the error in terms of principal angles between \X and \Y. We capitalize on ideas from a recent paper [DOI: 10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute values of differences between Ritz values for subspaces \X and \Y was weakly (sub-)majorized by a constant times the sine of the vector of principal angles between \X and \Y, the constant being the spread of the spectrum of $A$. In that result no assumption was made on either subspace being $A$-invariant. It was conjectured there that if one of the trial subspaces is $A$-invariant then an analogous weak majorization bound should only involve terms of the order of sine squared. Here we confirm this conjecture. Specifically we prove that the absolute eigenvalue error is weakly majorized by a constant times the sine squared of the vector of principal angles between the subspaces \X and \Y, where the constant is proportional to the spread of the spectrum of $A$. For many practical cases we show that the proportionality factor is simply one, and that this bound is sharp. For the general case we can only prove the result with a slightly larger constant, which we believe is artificial.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and Applications (SIMAX

Renormalization of hole-hole interaction at decreasing Drude conductivity

The diffusion contribution of the hole-hole interaction to the conductivity is analyzed in gated GaAs/In$_x$Ga$_{1-x}$As/GaAs heterostructures. We show that the change of the interaction correction to the conductivity with the decreasing Drude conductivity results both from the compensation of the singlet and triplet channels and from the arising prefactor $\alpha_i<1$ in the conventional expression for the interaction correction.Comment: 6 pages, 5 figure

Metal-Insulator Transition in 2D: Experimental Test of the Two-Parameter Scaling

We report a detailed scaling analysis of resistivity \rho(T,n) measured for several high-mobility 2D electron systems in the vicinity of the 2D metal-insulator transition. We analyzed the data using the two parameter scaling approach and general scaling ideas. This enables us to determine the critical electron density, two critical indices, and temperature dependence for the separatrix in the self-consistent manner. In addition, we reconstruct the empirical scaling function describing a two-parameter surface which fits well the \rho(T,n) data.Comment: 4 pages, 4 figures, 1 tabl

Two-kink solution for the Hirota–Ramani equation for degenerated parameters

The Hirota–Ramani equation is considered. Its two-kink solutions are known and may be constructed, for example, by direct methods if the parameters ki, i = 1, 2 of the solution components are not equal to each other. In this paper, such solution is constructed for k1 = k2