arXiv subjects

Ruipeng Li

Publications and source records attributed to Ruipeng Li.

32 records · Page 2Linked to original sources

arXiv2021

A non-perturbative approach to computing seismic normal modes in rotating planets

A Continuous Galerkin method-based approach is presented to compute the seismic normal modes of rotating planets. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core using a polynomial filtering eigensolver. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method. Our discretization utilizes fully unstructured tetrahedral meshes for both solid and fluid regions. The relevant eigenvalue problem is solved by a combination of several highly parallel and computationally efficient methods. We validate our three-dimensional results in the non-rotating case using analytical results for constant elastic balls, as well as numerical results for an isotropic Earth model from standard ``radial" algorithms. We also validate the computations in the rotating case, but only in the slowly-rotating regime where perturbation theory applies, because no other independent algorithms are available in the general case. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Earth and Mars models studying the effects of heterogeneity on a large range of scales.

physics.comp-ph↗

arXiv2021

The computation of seismic normal modes with rotation as a quadratic eigenvalue problem

A new approach is presented to compute the seismic normal modes of a fully heterogeneous, rotating planet. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method (FMM). To solve the resulting quadratic eigenvalue problem (QEP), the approach utilizes extended Lanczos vectors forming a subspace computed from a non-rotating planet -- with the shape of boundaries of a rotating planet and accounting for the centrifugal potential -- to reduce the dimension of the original problem significantly. The subspace is guaranteed to be contained in the space of functions to which the seismic normal modes belong. The reduced system can further be solved with a standard eigensolver. The computational accuracy is illustrated using all the modes with relative small meshes and also tested against standard perturbation calculations relative to a standard Earth model. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Mars models studying the effects of heterogeneity on a large range of scales.

physics.comp-ph↗

arXiv2021

Learning optimal multigrid smoothers via neural networks

Multigrid methods are one of the most efficient techniques for solving linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). The CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories, and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.

math.NA↗

arXiv2020

Liquid crystal phases with unusual structures and physical properties formed by acute-angle bent-core molecules

Liquid crystals formed by acute-angle bent-core (ABC) molecules with a 1,7 naphthalene central core show an intriguing phase behavior with the nematic phase accompanied by poorly understood additional phases. In this work, we characterize the physical properties of an ABC material, such as birefringence, dielectric permittivities, elastic constants, and surface alignment, and present X-ray diffraction and transmission electron microscopy studies of their ordering. The ABC molecular shape resembling the letter $λ$ yields a very small splay elastic constant in the uniaxial nematic phase and results in the formation of a tetragonal positionally ordered columnar phase consisting of molecular columns with a uniform uniaxial director that can be bent but not splayed.

cond-mat.soft↗

arXiv2020

Shear-induced polydomain structures of nematic lyotropic chromonic liquid crystal disodium cromoglycate

Lyotropic chromonic liquid crystals (LCLCs) represent aqueous dispersions of organic disk-like molecules that form cylindrical aggregates. Despite the growing interest in these materials, their flow behavior is poorly understood. Here, we explore the effect of shear on dynamic structures of the nematic LCLC, formed by 14wt ${\%}$ water dispersion of disodium cromoglycate (DSCG). We employ in-situ polarizing optical microscopy (POM) and small-angle and wide-angle X-ray scattering (SAXS/WAXS) to obtain independent and complementary information on the director structures over a wide range of shear rates. The DSCG nematic shows a shear-thinning behavior with two shear-thinning regions (Region I at $\dotγ<1\,s^{-1}$ and Region III at $\dotγ>10 s^{-1}$) separated by a pseudo-Newtonian Region II ($1 s^{-1}<\dotγ<10 s^{-1}$). The material is of a tumbling type. In Region I, $\dotγ<1 s^{-1}$, the director realigns along the vorticity axis. An increase of $\dotγ$ above $1 s^{-1}$ triggers nucleation of disclination loops. The disclinations introduce patches of the director that deviates from the vorticity direction and form a polydomain texture. Extension of the domains along the flow and along the vorticity direction decreases with the increase of the shear rate to $10 s^{-1}$. Above $10 s^{-1}$, the domains begin to elongate along the flow. At $\dotγ>100 s^{-1}$, the texture evolves into periodic stripes in which the director is predominantly along the flow with left and right tilts. The period of stripes decreases with an increase of $\dotγ$. The shear-induced transformations are explained by the balance of the elastic and viscous energies. In particular, nucleation of disclinations is associated with an increase of the elastic energy at the walls separating nonsingular domains with different director tilts.

cond-mat.soft↗

arXiv2020

Autonomous Materials Discovery Driven by Gaussian Process Regression with Inhomogeneous Measurement Noise and Anisotropic Kernels

A majority of experimental disciplines face the challenge of exploring large and high-dimensional parameter spaces in search of new scientific discoveries. Materials science is no exception; the wide variety of synthesis, processing, and environmental conditions that influence material properties gives rise to particularly vast parameter spaces. Recent advances have led to an increase in efficiency of materials discovery by increasingly automating the exploration processes. Methods for autonomous experimentation have become more sophisticated recently, allowing for multi-dimensional parameter spaces to be explored efficiently and with minimal human intervention, thereby liberating the scientists to focus on interpretations and big-picture decisions. Gaussian process regression (GPR) techniques have emerged as the method of choice for steering many classes of experiments. We have recently demonstrated the positive impact of GPR-driven decision-making algorithms on autonomously steering experiments at a synchrotron beamline. However, due to the complexity of the experiments, GPR often cannot be used in its most basic form, but rather has to be tuned to account for the special requirements of the experiments. Two requirements seem to be of particular importance, namely inhomogeneous measurement noise (input dependent or non-i.i.d.) and anisotropic kernel functions, which are the two concepts that we tackle in this paper. Our synthetic and experimental tests demonstrate the importance of both concepts for experiments in materials science and the benefits that result from including them in the autonomous decision-making process.

physics.comp-ph↗

arXiv2019

Parallel Approximate Ideal Restriction Multigrid for Solving the S$_N$ Transport Equations

The computational kernel in solving the $S_N$ transport equations is the parallel sweep, which corresponds to directly inverting a block lower triangular linear system that arises in discretizations of the linear transport equation. Existing parallel sweep algorithms are fairly efficient on structured grids, but still have polynomial scaling, $P^{1/d}$ for $d$ dimensions and $P$ processors. Moreover, an efficient scalable parallel sweep algorithm for use on general unstructured meshes remains elusive. Recently, a classical algebraic multigrid (AMG) method based on approximate ideal restriction (AIR) was developed for nonsymmetric matrices and shown to be an effective solver for linear transport. Motivated by the superior scalability of AMG methods (logarithmic in $P$) as well as the simplicity with which AMG methods can be used in most situations, including on arbitrary unstructured meshes, this paper investigates the use of parallel AIR (pAIR) for solving the $S_N$ transport equations with source iteration in place of parallel sweeps. Results presented in this paper show that pAIR is a robust and scalable solver. Although sweeps are still shown to be much faster than pAIR on a structured mesh of a unit cube, pAIR is shown to perform similarly on both a structured and unstructured mesh, and offers a new, simple, black box alternative to parallel transport sweeps.

physics.comp-ph↗

arXiv2018

The Eigenvalues Slicing Library (EVSL): Algorithms, Implementation, and Software

This paper describes a software package called EVSL (for EigenValues Slicing Library) for solving large sparse real symmetric standard and generalized eigenvalue problems. As its name indicates, the package exploits spectrum slicing, a strategy that consists of dividing the spectrum into a number of subintervals and extracting eigenpairs from each subinterval independently. In order to enable such a strategy, the methods implemented in EVSL rely on a quick calculation of the spectral density of a given matrix, or a matrix pair. What distinguishes EVSL from other currently available packages is that EVSL relies entirely on filtering techniques. Polynomial and rational filtering are both implemented and are coupled with Krylov subspace methods and the subspace iteration algorithm. On the implementation side, the package offers interfaces for various scenarios including matrix-free modes, whereby the user can supply his/her own functions to perform matrix-vector operations or to solve sparse linear systems. The paper describes the algorithms in EVSL, provides details on their implementations, and discusses performance issues for the various methods.

math.NA↗

arXiv2017

On Parallel Solution of Sparse Triangular Linear Systems in CUDA

The acceleration of sparse matrix computations on modern many-core processors, such as the graphics processing units (GPUs), has been recognized and studied over a decade. Significant performance enhancements have been achieved for many sparse matrix computational kernels such as sparse matrix-vector products and sparse matrix-matrix products. Solving linear systems with sparse triangular structured matrices is another important sparse kernel as demanded by a variety of scientific and engineering applications such as sparse linear solvers. However, the development of efficient parallel algorithms in CUDA for solving sparse triangular linear systems remains a challenging task due to the inherently sequential nature of the computation. In this paper, we will revisit this problem by reviewing the existing level-scheduling methods and proposing algorithms with self-scheduling techniques. Numerical results have indicated that the CUDA implementations of the proposed algorithms can outperform the state-of-the-art solvers in cuSPARSE by a factor of up to $2.6$ for structured model problems and general sparse matrices.

cs.MS↗

arXiv2017

Confinement Effects on the Crystalline Features of Poly(9,9-dioctylfluorene)

Typical device architectures in polymer-based optoelectronic devices, such as field effect transistors organic light emitting diodes and photovoltaic cells include sub-100 nm semiconducting polymer thin-film active layers, whose microstructure is likely to be subject to finite-size effects. The aim of this study was to investigate effect of the two-dimensional spatial confinement on the internal structure of the semiconducting polymer poly(9,9-dioctylfluorene) (PFO). PFO melts were confined inside the cylindrical nanopores of anodic aluminium oxide (AAO) templates and crystallized via two crystallization strategies, namely, in the presence or in the absence of a surface bulk reservoir located at the template surface. We show that highly textured semiconducting nanowires with tuneable crystal orientation can be thus produced. Moreover, our results indicate that employing the appropriate crystallization conditions extended-chain crystals can be formed in confinement. The results presented here demonstrate the simple fabrication and crystal engineering of ordered arrays of PFO nanowires; a system with potential applications in devices where anisotropic optical properties are required, such as polarized electroluminescence, waveguiding, optical switching, lasing, etc.

cond-mat.mtrl-sci↗

arXiv2017

Fast computation of spectral densities for generalized eigenvalue problems

The distribution of the eigenvalues of a Hermitian matrix (or of a Hermitian matrix pencil) reveals important features of the underlying problem, whether a Hamiltonian system in physics, or a social network in behavioral sciences. However, computing all the eigenvalues explicitly is prohibitively expensive for real-world applications. This paper presents two types of methods to efficiently estimate the spectral density of a matrix pencil $(A, B)$ when both $A$ and $B$ are Hermitian and, in addition, $B$ is positive definite. The first one is based on the Kernel Polynomial Method (KPM) and the second on Gaussian quadrature by the Lanczos procedure. By employing Chebyshev polynomial approximation techniques, we can avoid direct factorizations in both methods, making the resulting algorithms suitable for large matrices. Under some assumptions, we prove bounds that suggest that the Lanczos method converges twice as fast as the KPM method. Numerical examples further indicate that the Lanczos method can provide more accurate spectral densities when the eigenvalue distribution is highly non-uniform. As an application, we show how to use the computed spectral density to partition the spectrum into intervals that contain roughly the same number of eigenvalues. This procedure, which makes it possible to compute the spectrum by parts, is a key ingredient in the new breed of eigensolvers that exploit "spectrum slicing".

math.NA↗

arXiv2015

A Thick-Restart Lanczos algorithm with polynomial filtering for Hermitian eigenvalue problems

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a Thick-Restart version of the Lanczos algorithm with deflation (`locking') and a new type of polynomial filters obtained from a least-squares technique. The resulting algorithm can be utilized in a `spectrum-slicing' approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different sub-intervals independently from one another.

math.NA↗

arXiv2015

Low-rank correction methods for algebraic domain decomposition preconditioners

This paper presents a parallel preconditioning method for distributed sparse linear systems, based on an approximate inverse of the original matrix, that adopts a general framework of distributed sparse matrices and exploits the domain decomposition method and low-rank corrections. The domain decomposition approach decouples the matrix and once inverted, a low-rank approximation is applied by exploiting the Sherman-Morrison-Woodbury formula, which yields two variants of the preconditioning methods. The low-rank expansion is computed by the Lanczos procedure with reorthogonalizations. Numerical experiments indicate that, when combined with Krylov subspace accelerators, this preconditioner can be efficient and robust for solving symmetric sparse linear systems. Comparisons with other distributed-memory preconditioning methods are presented.

math.NA↗

arXiv2015

Schur Complement based domain decomposition preconditioners with Low-rank corrections

This paper introduces a robust preconditioner for general sparse symmetric matrices, that is based on low-rank approximations of the Schur complement in a Domain Decomposition (DD) framework. In this "Schur Low Rank" (SLR) preconditioning approach, the coefficient matrix is first decoupled by DD, and then a low-rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface points. The method avoids explicit formation of the Schur complement matrix. We show the feasibility of this strategy for a model problem, and conduct a detailed spectral analysis for the relationship between the low-rank correction and the quality of the preconditioning. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach.

math.NA↗