Edoardo Di Napoli

Efficient estimation of eigenvalue counts in an interval

Summary Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications, and it is a prerequisite of eigensolvers based on a divide-and-conquer paradigm. Often, an exact count is not necessary, and methods based on stochastic estimates can be utilized to yield rough approximations. This paper examines a number of techniques tailored to this specific task. It reviews standard approaches and explores new ones based on polynomial and rational approximation filtering combined with a stochastic procedure. We also discuss how the latter method is particularly well-suited for the FEAST eigensolver. Copyright

Dissecting the FEAST algorithm for generalized eigenproblems

We analyze the FEAST method for computing selected eigenvalues and eigenvectors of large sparse matrix pencils. After establishing the close connection between FEAST and the well-known Rayleigh-Ritz method, we identify several critical issues that influence convergence and accuracy of the solver: the choice of the starting vector space, the stopping criterion, how the inner linear systems impact the quality of the solution, and the use of FEAST for computing eigenpairs from multiple intervals. We complement the study with numerical examples, and hint at possible improvements to overcome the existing problems.

Towards an efficient use of the BLAS library for multilinear tensor contractions

Abstract Mathematical operators whose transformation rules constitute the building blocks of a multi-linear algebra are widely used in physics and engineering applications where they are very often represented as tensors. In the last century, thanks to the advances in tensor calculus, it was possible to uncover new research fields and make remarkable progress in the existing ones, from electromagnetism to the dynamics of fluids and from the mechanics of rigid bodies to quantum mechanics of many atoms. By now, the formal mathematical and geometrical properties of tensors are well defined and understood; conversely, in the context of scientific and high-performance computing, many tensor-related problems are still open. In this paper, we address the problem of efficiently computing contractions among two tensors of arbitrary dimension by using kernels from the highly optimized BLAS library. In particular, we establish precise conditions to determine if and when GEMM, the kernel for matrix products, can be used. Such conditions take into consideration both the nature of the operation and the storage scheme of the tensors, and induce a classification of the contractions into three groups. For each group, we provide a recipe to guide the users towards the most effective use of BLAS.

Chiral rings of deconstructive [SU(nc)]N quivers

Dimensional deconstruction of 5D SQCD with general nc, nf and kCS gives rise to 4D N = 1 gauge theories with large quivers of SU(nc) gauge factors. We construct the chiral rings of such [SU(nc)] N theories, off-shell and on-shell. Our results are broadly similar to the chiral rings of single U(nc) theories with both adjoint and fundamental matter, but there are also some noteworthy differences such as nonlocal meson-like operators where the quark and antiquark fields belong to different nodes of the quiver. And because our gauge groups are SU(nc) rather than U(nc), our chiral rings also contain a whole zoo of baryonic and antibaryonic operators.

Correlations in sequences of generalized eigenproblems arising in Density Functional Theory

Abstract Density Functional Theory (DFT) is one of the most used ab initio theoretical frameworks in materials science. It derives the ground state properties of a multi-atomic ensemble directly from the computation of its one-particle density n ( r ) . In DFT-based simulations the solution is calculated through a chain of successive self-consistent cycles; in each cycle a series of coupled equations (Kohn–Sham) translates to a large number of generalized eigenvalue problems whose eigenpairs are the principal means for expressing n ( r ) . A simulation ends when n ( r ) has converged to the solution within the required numerical accuracy. This usually happens after several cycles, resulting in a process calling for the solution of many sequences of eigenproblems. In this paper, the authors report evidence showing unexpected correlations between adjacent eigenproblems within each sequence. By investigating the numerical properties of the sequences of generalized eigenproblems it is shown that the eigenvectors undergo an “evolution” process. At the same time it is shown that the Hamiltonian matrices exhibit a similar evolution and manifest a specific pattern in the information they carry. Correlation between eigenproblems within a sequence is of capital importance: information extracted from the simulation at one step of the sequence could be used to compute the solution at the next step. Although they are not explored in this work, the implications could be manifold: from increasing the performance of material simulations, to the development of an improved iterative solver, to modifying the mathematical foundations of the DFT computational paradigm in use, thus opening the way to the investigation of new materials.

Solving Dense Generalized Eigenproblems on Multi-threaded Architectures

We compare two approaches to compute a fraction of the spectrum of dense symmetric definite generalized eigenproblems: one is based on the reduction to tridiagonal form, and the other on the Krylov-subspace iteration. Two large-scale applications, arising in molecular dynamics and material science, are employed to investigate the contributions of the application, architecture, and parallelism of the method to the performance of the solvers. The experimental results on a state-of-the-art 8-core platform, equipped with a graphics processing unit (GPU), reveal that in realistic applications, iterative Krylov-subspace methods can be a competitive approach also for the solution of dense problems.

An optimized and scalable eigensolver for sequences of eigenvalue problems

In many scientific applications, the solution of nonlinear differential equations are obtained through the setup and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution of one problem fosters the initialization of the next. In addition, in some eigenproblem sequences, there is a connection between the solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a connection, the eigenproblem sequence is said to be correlated. When facing with a sequence of correlated eigenproblems, the current strategy amounts to solving each eigenproblem in isolation. We propose an alternative approach that exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (Chebyshev filtered subspace iteration (ChFSI)). The resulting eigensolver is optimized by minimizing the number of matrix–vector multiplications and parallelized using the Elemental library framework. Numerical results show that ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel eigensolvers. Copyright

Quantum deconstruction of 5D SQCD

We deconstruct the fifth dimension of 5D SCQD with general numbers of colors and flavors and general 5D Chern-Simons level; the latter is adjusted by adding extra quarks to the 4D quiver. We use deconstruction as a non-stringy UV completion of the quantum 5D theory; to prove its usefulness, we compute quantum corrections to the SQCD5s prepotential. We also explore the moduli/parameter space of the deconstructed SQCD5 and show that for |kcs| < nc−½nf it continues to negative values of 1/g25d. In many cases there are flop transitions connecting SQCD5 to exotic 5D theories such as E0, and we present several examples of such transitions. We compare deconstruction to brane-web engineering of the same SQCD5 and show that the phase diagram is the same in both cases; indeed, the two UV completions are in the same universality class, although they are not dual to each other. Hence, the phase structure of an SQCD5 (and presumably any other 5D gauge theory) is inherently five-dimensional and does not depends on a UV completion.

Can quantum de Sitter space have finite entropy

If one tries to view de Sitter as a true (as opposed to a meta-stable) vacuum, there is a tension between the finiteness of its entropy and the infinite dimensionality of its Hilbert space. We investigate the viability of one proposal to reconcile this tension using q-deformation. After defining a differential geometry on the quantum de Sitter space, we try to constrain the value of the deformation parameter by imposing the condition that in the undeformed limit, we want the real form of the (inherently complex) quantum group to reduce to the usual SO(4, 1) of de Sitter. We find that this forces q to be a real number. Since it is known that quantum groups have finite-dimensional representations only for q= root of unity, this suggests that standard q-deformations cannot give rise to finite-dimensional Hilbert spaces, ruling out finite entropy for q-deformed de Sitter.

Anomaly cancellation and conformality in quiver gauge theories

Abelian quiver gauge theories provide non-supersymmetric candidates for the conformality approach to physics beyond the standard model. Written as N=0, U(N)n gauge theories, however, they have mixed U(1)pU(1)q2 and U(1)pSU(N)q2 triangle anomalies. It is shown how to construct explicitly a compensatory term ΔLcomp which restores gauge invariance of Leff=L+ΔLcomp under U(N)n. It can lead to a negative contribution to the U(1) β-function and hence to one-loop conformality at high energy for all dimensionless couplings.