59,391 research outputs found
Quantum-inspired sublinear classical algorithms for solving low-rank linear systems
We present classical sublinear-time algorithms for solving low-rank linear
systems of equations. Our algorithms are inspired by the HHL quantum algorithm
for solving linear systems and the recent breakthrough by Tang of dequantizing
the quantum algorithm for recommendation systems. Let be a rank- matrix, and be a vector. We
present two algorithms: a "sampling" algorithm that provides a sample from
and a "query" algorithm that outputs an estimate of an entry of
, where denotes the Moore-Penrose pseudo-inverse. Both of our
algorithms have query and time complexity , where is the condition number
of and is the precision parameter. Note that the algorithms we
consider are sublinear time, so they cannot write and read the whole matrix or
vectors. In this paper, we assume that and come with well-known
low-overhead data structures such that entries of and can be sampled
according to some natural probability distributions. Alternatively, when is
positive semidefinite, our algorithms can be adapted so that the sampling
assumption on is not required
Resolvent sampling based Rayleigh-Ritz method for large-scale nonlinear eigenvalue problems
A new algorithm, denoted by RSRR, is presented for solving large-scale
nonlinear eigenvalue problems (NEPs) with a focus on improving the robustness
and reliability of the solution, which is a challenging task in computational
science and engineering. The proposed algorithm utilizes the Rayleigh-Ritz
procedure to compute all eigenvalues and the corresponding eigenvectors lying
within a given contour in the complex plane. The main novelties are the
following. First and foremost, the approximate eigenspace is constructed by
using the values of the resolvent at a series of sampling points on the
contour, which effectively circumvented the unreliability of previous schemes
that using high-order contour moments of the resolvent. Secondly, an improved
Sakurai-Sugiura algorithm is proposed to solve the projected NEPs with
enhancements on reliability and accuracy. The user-defined probing matrix in
the original algorithm is avoided and the number of eigenvalues is determined
automatically by provided strategies. Finally, by approximating the projected
matrices with the Chebyshev interpolation technique, RSRR is further extended
to solve NEPs in the boundary element method, which is typically difficult due
to the densely populated matrices and high computational costs. The good
performance of RSRR is demonstrated by a variety of benchmark examples and
large-scale practical applications, with the degrees of freedom ranging from
several hundred up to around one million. The algorithm is suitable for
parallelization and easy to implement in conjunction with other programs and
software.Comment: 26 pages, 14 figures, 3 tables. comments and discussion to:
[email protected]
Multistep matrix splitting iteration preconditioning for singular linear systems
Multistep matrix splitting iterations serve as preconditioning for Krylov
subspace methods for solving singular linear systems. The preconditioner is
applied to the generalized minimal residual (GMRES) method and the flexible
GMRES (FGMRES) method. We present theoretical and practical justifications for
using this approach. Numerical experiments show that the multistep generalized
shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS)
iteration preconditioning are more robust and efficient compared to standard
preconditioners for some test problems of large sparse singular linear systems.Comment: 16 page
Quantum Circuit Design Methodology for Multiple Linear Regression
Multiple linear regression assumes an imperative role in supervised machine
learning. In 2009, Harrow et al. [Phys. Rev. Lett. 103, 150502 (2009)] showed
that their HHL algorithm can be used to sample the solution of a linear system
exponentially faster than any existing classical algorithm,
with some manageable caveats. The entire field of quantum machine learning
gained considerable traction after the discovery of this celebrated algorithm.
However, effective practical applications and experimental implementations of
HHL are still sparse in the literature. Here, we demonstrate a potential
practical utility of HHL, in the context of regression analysis, using the
remarkable fact that there exists a natural reduction of any multiple linear
regression problem to an equivalent linear systems problem. We put forward a
-qubit quantum circuit design, motivated from an earlier work by Cao et al.
[Mol. Phys. 110, 1675 (2012)], to solve a -variable regression problem,
using only elementary quantum gates. We also implement the Group Leaders
Optimization Algorithm (GLOA) [Mol. Phys. 109 (5), 761 (2011)] and elaborate on
the advantages of using such stochastic algorithms in creating low-cost circuit
approximations for the Hamiltonian simulation. We believe that this application
of GLOA and similar stochastic algorithms in circuit approximation will boost
time- and cost-efficient circuit designing for various quantum machine learning
protocols. Further, we discuss our Qiskit simulation and explore certain
generalizations to the circuit design.Comment: 14 pages, 7 figure
iSIRA: Integrated Shift-Invert Residual Arnoldi Method for Graph Laplacian Matrices from Big Data
The eigenvalue problem of a graph Laplacian matrix arising from a simple,
connected and undirected graph has been given more attention due to its
extensive applications, such as spectral clustering, community detection,
complex network, image processing and so on. The associated graph Laplacian
matrix is symmetric, positive semi-definite, and is usually large and sparse.
Computing some smallest positive eigenvalues and corresponding eigenvectors is
often of interest.
However, the singularity of makes the classical eigensolvers inefficient
since we need to factorize for the purpose of solving large and sparse
linear systems exactly. The next difficulty is that it is usually time
consuming or even unavailable to factorize a large and sparse matrix arising
from real network problems from big data such as social media transactional
databases, and sensor systems because there is in general not only local
connections.
In this paper, we propose an eignsolver based on the inexact residual Arnoldi
method together with an implicit remedy of the singularity and an effective
deflation for convergent eigenvalues. Numerical experiments reveal that the
integrated eigensolver outperforms the classical Arnoldi/Lanczos method for
computing some smallest positive eigeninformation provided the LU factorization
is not available
A Constraint-Reduced MPC Algorithm for Convex Quadratic Programming, with a Modified Active Set Identification Scheme
A constraint-reduced Mehrotra-Predictor-Corrector algorithm for convex
quadratic programming is proposed. (At each iteration, such algorithms use only
a subset of the inequality constraints in constructing the search direction,
resulting in CPU savings.) The proposed algorithm makes use of a regularization
scheme to cater to cases where the reduced constraint matrix is rank deficient.
Global and local convergence properties are established under arbitrary
working-set selection rules subject to satisfaction of a general condition. A
modified active-set identification scheme that fulfills this condition is
introduced. Numerical tests show great promise for the proposed algorithm, in
particular for its active-set identification scheme. While the focus of the
present paper is on dense systems, application of the main ideas to large
sparse systems is briefly discussed
On GMRES for singular EP and GP systems
In this contribution, we study the numerical behavior of the Generalized
Minimal Residual (GMRES) method for solving singular linear systems. It is
known that GMRES determines a least squares solution without breakdown if the
coefficient matrix is range-symmetric (EP), or if its range and nullspace are
disjoint (GP) and the system is consistent. We show that the accuracy of GMRES
iterates may deteriorate in practice due to three distinct factors: (i) the
inconsistency of the linear system; (ii) the distance of the initial residual
to the nullspace of the coefficient matrix; (iii) the extremal principal angles
between the ranges of the coefficient matrix and its transpose. These factors
lead to poor conditioning of the extended Hessenberg matrix in the Arnoldi
decomposition and affect the accuracy of the computed least squares solution.
We also compare GMRES with the range restricted GMRES (RR-GMRES) method.
Numerical experiments show typical behaviors of GMRES for small problems with
EP and GP matrices.Comment: 16 pages, 18 figure
Solving Splitted Multi-Commodity Flow Problem by Efficient Linear Programming Algorithm
Column generation is often used to solve multi-commodity flow problems. A
program for column generation always includes a module that solves a linear
equation. In this paper, we address three major issues in solving linear
problem during column generation procedure which are (1) how to employ the
sparse property of the coefficient matrix; (2) how to reduce the size of the
coefficient matrix; and (3) how to reuse the solution to a similar equation. To
this end, we first analyze the sparse property of coefficient matrix of linear
equations and find that the matrices occurring in iteration are very sparse.
Then, we present an algorithm locSolver (for localized system solver) for
linear equations with sparse coefficient matrices and right-hand-sides. This
algorithm can reduce the number of variables. After that, we present the
algorithm incSolver (for incremental system solver) which utilizes similarity
in the iterations of the program for a linear equation system. All three
techniques can be used in column generation of multi-commodity problems.
Preliminary numerical experiments show that the incSolver is significantly
faster than the existing algorithms. For example, random test cases show that
incSolver is at least 37 times and up to 341 times faster than popular solver
LAPACK.Comment: 27 page
Quantum Regularized Least Squares Solver with Parameter Estimate
In this paper we propose a quantum algorithm to determine the Tikhonov
regularization parameter and solve the ill-conditioned linear equations, for
example, arising from the finite element discretization of linear or nonlinear
inverse problems. For regularized least squares problem with a fixed
regularization parameter, we use the HHL algorithm and work on an extended
matrix with smaller condition number. For the determination of the
regularization parameter, we combine the classical L-curve and GCV function,
and design quantum algorithms to compute the norms of regularized solution and
the corresponding residual in parallel and locate the best regularization
parameter by Grover's search. The quantum algorithm can achieve a quadratic
speedup in the number of regularization parameters and an exponential speedup
in the dimension of problem size
Broyden's method for nonlinear eigenproblems
Broyden's method is a general method commonly used for nonlinear systems of
equations, when very little information is available about the problem. We
develop an approach based on Broyden's method for nonlinear eigenvalue
problems. Our approach is designed for problems where the evaluation of a
matrix vector product is computationally expensive, essentially as expensive as
solving the corresponding linear system of equations. We show how the structure
of the Jacobian matrix can be incorporated into the algorithm to improve
convergence. The algorithm exhibits local superlinear convergence for simple
eigenvalues, and we characterize the convergence. We show how deflation can be
integrated and combined such that the method can be used to compute several
eigenvalues. A specific problem in machine tool milling, coupled with a PDE is
used to illustrate the approach. The simulations are done in the julia
programming language, and are provided as publicly available module for
reproducability
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