Linear Algebra Problems Research Articles

Isometric shape interpolation

Acquiring natural-looking in-betweens is a fundamental shape interpolation problem in computer graphics. Several previous studies showed that as-isometric-as-possible interpolation is the key to natural and intuitive results. With this presumption, this paper describes a novel method for acquiring an isometric shape interpolation for given key-frame models. The technological and theoretical contributions of our method lie in the introduction of a new coordinate system, isometry-invariant intrinsic coordinates. This simplifies the nonlinear isometric interpolation into a simple linear algebraic problem. The method is shown to yield an effective isometric interpolation of the given key-frame models and is easy to implement. Experimental results confirm that the proposed method produces minimal metric distortion and exhibits reasonable efficiency.

BOUT++: Recent and current developments

BOUT++ is a 3D nonlinear finite-difference plasma simulation code, capable of solving quite general systems of Partial Differential Equations (PDEs), but targeted particularly on studies of the edge region of tokamak plasmas. BOUT++ is publicly available, and has been adopted by a growing number of researchers worldwide. Here we present improvements which have been made to the code since its original release, both in terms of structure and its capabilities. Some recent applications of these methods are reviewed, and areas of active development are discussed. We also present algorithms and tools which have been developed to enable creation of inputs from analytic expressions and experimental data, and for processing and visualisation of output results. This includes a new toolHypnotoadfor the creation of meshes from experimental equilibria. Algorithms have been implemented in BOUT++ to solve a range of linear algebraic problems encountered in the simulation of reduced Magnetohydrodynamics (MHD) and gyro-fluid models: A preconditioning scheme is presented which enables the plasma potential to be calculated efficiently using iterative methods supplied by the PETSc library (the Portable, Extensible Toolkit for Scientific Computation) (Balay et al. 2014), without invoking the Boussinesq approximation. Scaling studies are also performed of a linear solver used as part of physics-based preconditioning to accelerate the convergence of implicit time-integration schemes.

Fast Simulation of Complicated 3-D Structures Above Lossy Magnetic Media

A fast numerical method is presented for the simulation of complicated 3-D structures, such as inductors constructed from Litz or stranded wires, above or sandwiched between the planar lossy magnetic media. Making use of its smoothness, the quasi-static multilayer Green's function is numerically computed using finite differences, and its source height dependence is computed using an optimal Toeplitz-plus-Hankel decomposition. We show that a modified precorrected fast Fourier transform method can be applied to reduce the dense linear algebra problem to near-linear time, and that frequency-dependent setups can be avoided to result in a considerable speed-up. Experimental verifications are made for a 16-strand Litz wire coil realistically modeled down to each individual strand. Results are obtained in 2-3 h, showing an excellent agreement to measurements, and can be used to study the impact of transposition patterns in Litz wire construction.

A scalable approach to solving dense linear algebra problems on hybrid CPU‐GPU systems

SummaryAiming to fully exploit the computing power of all CPUs and all graphics processing units (GPUs) on hybrid CPU‐GPU systems to solve dense linear algebra problems, we design a class of heterogeneous tile algorithms to maximize the degree of parallelism, to minimize the communication volume, and to accommodate the heterogeneity between CPUs and GPUs. The new heterogeneous tile algorithms are executed upon our decentralized dynamic scheduling runtime system, which schedules a task graph dynamically and transfers data between compute nodes automatically. The runtime system uses a new distributed task assignment protocol to solve data dependencies between tasks without any coordination between processing units. By overlapping computation and communication through dynamic scheduling, we are able to attain scalable performance for the double‐precision Cholesky factorization and QR factorization. Our approach demonstrates a performance comparable to Intel MKL on shared‐memory multicore systems and better performance than both vendor (e.g., Intel MKL) and open source libraries (e.g., StarPU) in the following three environments: heterogeneous clusters with GPUs, conventional clusters without GPUs, and shared‐memory systems with multiple GPUs. Copyright © 2014 John Wiley & Sons, Ltd.

A Local Feedback Control Bringing a Linear System to Equilibrium

We design a bounded feedback control that steers a controllable linear system to equilibrium in a finite time. The procedure amounts to solving several linear-algebraic problems, including linear matrix inequalities. We solve these problems in an efficient way. The resulting steering time and the minimum time have the same order of magnitude.

Minimum number of affine simplices of given dimension

In this paper we formulate and solve extremal problems in the Euclidean space Rd and further in hypergraphs, originating from problems in stoichiometry and elementary linear algebra. The notion of affine simplex is the bridge between the original problems and the presented extremal theorem on set systems. As a sample corollary, it follows that if no triple is collinear in a set S of n points in R3, then S contains at least n4−cn3 affine simplices for some constant c. A function related to Sperner’s Theorem and its well-known extension to reciprocal sums is also considered and its relation to Turán’s hypergraph problems is discussed.

Geometric degree of nonconservativity

This paper deals with nonconservative mechanical systems subjected to nonconservative positional forces leading to nonsymmetric tangential stiffness matrices. The geometric degree of nonconservativity of such systems is then defined as the minimal number ll of kinematic constraints necessary to convert the initial system into a conservative one. Finding this number and describing the set of corresponding kinematic constraints is reduced to a linear algebra problem. This index ll of nonconservativity is the half of the rank of the skew-symmetric part KaKa of the stiffness matrix KK that is always an even number. The set of constraints is extracted from the eigenspaces of the symmetric matrix K2aKa2. Several examples including the well-known Ziegler column illustrate the results.

Is it possible to embed a 4D, $ \mathcal{N} $ = 4 supersymmetric vector multiplet within a completely off-shell adinkra hologram?

We present evidence of the existence of a 1D, N = 16 SUSY hologram that can be used to understand representation theory aspects of a 4D, $ \mathcal{N} $ = 4 supersymmetrical vector multiplet. In this context, the long-standing “off-shell SUSY” problem for the 4D, $ \mathcal{N} $ = 4 Maxwell supermultiplet is precisely formulated as a problem in linear algebra.

Communication lower bounds and optimal algorithms for numerical linear algebra

The traditional metric for the efficiency of a numerical algorithm has been the number of arithmetic operations it performs. Technological trends have long been reducing the time to perform an arithmetic operation, so it is no longer the bottleneck in many algorithms; rather, communication, or moving data, is the bottleneck. This motivates us to seek algorithms that move as little data as possible, either between levels of a memory hierarchy or between parallel processors over a network. In this paper we summarize recent progress in three aspects of this problem. First we describe lower bounds on communication. Some of these generalize known lower bounds for dense classical (O(n3)) matrix multiplication to all direct methods of linear algebra, to sequential and parallel algorithms, and to dense and sparse matrices. We also present lower bounds for Strassen-like algorithms, and for iterative methods, in particular Krylov subspace methods applied to sparse matrices. Second, we compare these lower bounds to widely used versions of these algorithms, and note that these widely used algorithms usually communicate asymptotically more than is necessary. Third, we identify or invent new algorithms for most linear algebra problems that do attain these lower bounds, and demonstrate large speed-ups in theory and practice.

Weighted least-squares finite element methods for the linearized Navier–Stokes equations

We implemented weighted least-squares finite element methods for the linearized Navier-Stokes equations based on the velocity–pressure–stress and the velocity–vorticity–pressure formulations. The least-squares functionals involve the L2-norms of the residuals of each equation multiplied by the appropriate weighting functions. The weights included a mass conservation constant, a mesh-dependent weight, a nonlinear weighting function, and Reynolds numbers. A feature of this approach is that the linearized system creates a symmetric and positive-definite linear algebra problem at each Newton iteration. We can prove that least-squares approximations converge with the linearized version solutions of the Navier–Stokes equations at the optimal convergence rate. Model problems considered in this study were the flow past a planar channel and 4-to-1 contraction problems. We presented approximate solutions of the Navier–Stokes problems by solving a sequence of the linearized Navier–Stokes problems arising from Newton iterations, revealing the convergence rates of the proposed schemes, and investigated Reynolds number effects.

Probabilistic analysis of Wiedemann's algorithm for minimal polynomial computation

Blackbox algorithms for linear algebra problems start with projection of the sequence of powers of a matrix to a sequence of vectors (Lanczos), a sequence of scalars (Wiedemann) or a sequence of smaller matrices (block methods). Such algorithms usually depend on the minimal polynomial of the resulting sequence being that of the given matrix. Here exact formulas are given for the probability that this occurs. They are based on the generalized Jordan normal form (direct sum of companion matrices of the elementary divisors) of the matrix. Sharp bounds follow from this for matrices of unknown elementary divisors. The bounds are valid for all finite field sizes and show that a small blocking factor can give high probability of success for all cardinalities and matrix dimensions.

A control Liapunov function approach to generalized and regularized descent methods for zero finding1

This paper revisits a class of recently proposed so-called invariant manifold methods for zero finding of ill-posed problems, showing that they can be profitably viewed as homotopy methods, in which the homotopy parameter is interpreted as a learning parameter. Moreover, it is shown that the choice of this learning parameter can be made in a natural manner from a control Liapunov function approach CLF. From this viewpoint, maintaining manifold invariance is equivalent to ensuring that the CLF satisfies a certain ordinary differential equation, involving the learning parameter, that allows an estimate of rate of convergence. In order to illustrate this approach, algorithms recently proposed using the invariant manifold approach, are rederived, via CLFs, in a unified manner. Adaptive regularization parameters for solving linear algebraic ill-posed problems were also proposed. This paper also shows that the discretizations of the ODEs to solve the zero finding problem, as well as the different adaptive choices of the regularization parameter, yield iterative methods for linear systems, which are also derived using the Liapunov optimizing control LOC method.

A Parallel QZ Algorithm for Distributed Memory HPC Systems

Appearing frequently in applications, generalized eigenvalue problems represent one of the core problems in numerical linear algebra. The QZ algorithm of Moler and Stewart is the most widely used algorithm for addressing such problems. Despite its importance, little attention has been paid to the parallelization of the QZ algorithm. The purpose of this work is to fill this gap. We propose a parallelization of the QZ algorithm that incorporates all modern ingredients of dense eigensolvers, such as multishift and aggressive early deflation techniques. To deal with (possibly many) infinite eigenvalues, a new parallel deflation strategy is developed. Numerical experiments for several random and application examples demonstrate the effectiveness of our algorithm on two different distributed memory HPC systems.

Definability of linear equation systems over groups and rings

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields.

Permutation polynomials over finite fields involving [formula omitted

Let p be a prime and q=ps. For integer a≥0, let Sa=x+xq+⋯+xqa−1∈Fp[x]. We present three constructions of permutation polynomials of Fqe involving Sa which generalize several recent results. When q is even, the third construction produces a large class of Dembowski–Ostrom permutation polynomials. We also discuss an interesting linear algebraic problem arising from the third construction.

Most Tensor Problems Are NP-Hard

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or the spectral norm; and determining the rank or best rank-1 approximation of a 3-tensor. Furthermore, we show that restricting these problems to symmetric tensors does not alleviate their NP-hardness. We also explain how deciding nonnegative definiteness of a symmetric 4-tensor is NP-hard and how computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.

Fast matrix rank algorithms and applications

We consider the problem of computing the rank of an m × n matrix A over a field. We present a randomized algorithm to find a set of r = rank( A ) linearly independent columns in Õ (| A | + r ω ) field operations, where | A | denotes the number of nonzero entries in A and ω < 2.38 is the matrix multiplication exponent. Previously the best known algorithm to find a set of r linearly independent columns is by Gaussian elimination, with deterministic running time O ( mnr ω-2 ). Our algorithm is faster when r < max{ m , n }, for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank in Õ ( mn ) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in exact linear algebra, combinatorial optimization and dynamic data structure.

A literature survey of low‐rank tensor approximation techniques

AbstractDuring the last years, low‐rank tensor approximation has been established as a new tool in scientific computing to address large‐scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function‐related tensors. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Interplay between discretization and algebraic computation in adaptive numerical solutionof elliptic PDE problems

AbstractThe Adaptive Finite Element Method (AFEM) for approximating solutions of PDE boundary value and eigenvalue problems is a numerical scheme that automatically and iteratively adapts the finite element space until a sufficiently accurate approximate solution is found. The adaptation process is based on a posteriori error estimators, and at each step of this process an algebraic problem (linear or nonlinear algebraic system or eigenvalue problem) has to be solved. In practical computations the solution of the algebraic problem cannot be obtained exactly. As a consequence, the algebraic error should be incorporated in the context of the AFEM and its a posteriori error estimators. The goal of this paper is to survey some existing approaches in the AFEM context that consider the interplay between the finite element discretization and the algebraic computation. We believe that a better understanding of this interplay is of great importance for the future development in the area of numerically solving large‐scale real‐world motivated problems. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Relative Perturbation Theory for Diagonally Dominant Matrices

In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.