Finite-dimensional Linear Operator Research Articles

A fully implicit, asymptotic-preserving, semi-Lagrangian algorithm for the time dependent anisotropic heat transport equation

In this paper, we extend the operator-split asymptotic-preserving, semi-Lagrangian algorithm for time dependent anisotropic heat transport equation proposed in Chacón et al. (2014) [18] to use a fully implicit time integration with backward differentiation formulas. The proposed implicit method can deal with arbitrary heat-transport anisotropy ratios χ∥/χ⊥⋙1 (with χ∥, χ⊥ the parallel and perpendicular heat diffusivities, respectively) in complicated magnetic field topologies in an accurate and efficient manner. The implicit algorithm is second-order accurate temporally and demonstrates an accurate treatment at boundary layers (e.g., island separatrices), which was not ensured by the operator-split implementation. The condition number of the resulting algebraic system is independent of the anisotropy ratio, and is inverted with preconditioned GMRES. We propose a simple preconditioner that renders the finite-dimensional linear operator compact, resulting in mesh-independent convergence rates for topologically simple magnetic fields, and convergence rates scaling as ∼(NΔt)1/4 (with N the total mesh size and Δt the timestep) in topologically complex magnetic-field configurations. We demonstrate the accuracy and performance of the approach with test problems of varying complexity, including an analytically tractable boundary-layer problem in a straight magnetic field, and a topologically complex magnetic field featuring magnetic islands with extreme anisotropy ratios (χ∥/χ⊥=1010).

Estimation method for inverse problems with linear forward operator and its application to magnetization estimation from magnetic force microscopy images using deep learning

ABSTRACT This study considers an inverse problem, where the corresponding forward problem is given by a finite-dimensional linear operator T. The inverse problem has the following form: It is assumed that the number of patterns that the unknown quantity can take is finite. Then, even if the unknown quantity may be uniquely determined from the data. This case is the subject of this study. We propose a method for solving this inverse problem using numerical calculations. A famous inverse problem requires the estimation of the unknown magnetization distribution or magnetic charge distribution in an anisotropic permanent magnet sample from the magnetic force microscopy images. It is known that the solution of this problem is not unique in general. In this work, we consider the case where a magnetic sample comprises cubic cells, and the unknown magnetic moment is oriented either upward or downward in each cell. This discretized problem is an example of the above-mentioned inverse problem: Numerical calculations were carried out to solve this model problem employing our method and deep learning. The experimental results show that the magnetization can be estimated roughly up to a certain depth.

Beyond the spectral theorem: Spectrally decomposing arbitrary functions of nondiagonalizable operators

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, the familiar linear operator techniques that one would then hope to use often fail since the operators cannot be diagonalized. The curse of nondiagonalizability also plays an important role even in finite-dimensional linear operators, leading to analytical impediments that occur across many scientific domains. We show how to circumvent it via two tracks. First, using the well-known holomorphic functional calculus, we develop new practical results about spectral projection operators and the relationship between left and right generalized eigenvectors. Second, we generalize the holomorphic calculus to a meromorphic functional calculus that can decompose arbitrary functions of nondiagonalizable linear operators in terms of their eigenvalues and projection operators. This simultaneously simplifies and generalizes functional calculus so that it is readily applicable to analyzing complex physical systems. Together, these results extend the spectral theorem of normal operators to a much wider class, including circumstances in which poles and zeros of the function coincide with the operator spectrum. By allowing the direct manipulation of individual eigenspaces of nonnormal and nondiagonalizable operators, the new theory avoids spurious divergences. As such, it yields novel insights and closed-form expressions across several areas of physics in which nondiagonalizable dynamics arise, including memoryful stochastic processes, open nonunitary quantum systems, and far-from-equilibrium thermodynamics. The technical contributions include the first full treatment of arbitrary powers of an operator, highlighting the special role of the zero eigenvalue. Furthermore, we show that the Drazin inverse, previously only defined axiomatically, can be derived as the negative-one power of singular operators within the meromorphic functional calculus and we give a new general method to construct it. We provide new formulae for constructing spectral projection operators and delineate the relations among projection operators, eigenvectors, and left and right generalized eigenvectors. By way of illustrating its application, we explore several, rather distinct examples. First, we analyze stochastic transition operators in discrete and continuous time. Second, we show that nondiagonalizability can be a robust feature of a stochastic process, induced even by simple counting. As a result, we directly derive distributions of the time-dependent Poisson process and point out that nondiagonalizability is intrinsic to it and the broad class of hidden semi-Markov processes. Third, we show that the Drazin inverse arises naturally in stochastic thermodynamics and that applying the meromorphic functional calculus provides closed-form solutions for the dynamics of key thermodynamic observables. Finally, we draw connections to the Ruelle–Frobenius–Perron and Koopman operators for chaotic dynamical systems and propose how to extract eigenvalues from a time-series.

A Mathematical Framework for Parallel Computing of Discrete-Time Discrete-Frequency Transforms in Multi-Core Processors

This paper presents a mathematical framework for a family of discrete-time discrete-frequency transforms in terms of matrix signal algebra. The matrix signal algebra is a mathematics environment composed of a signal space, a finite dimensional linear operators and special matrices where algebraic methods are used to generate these signal transforms as computational estimators. The matrix signal algebra contribute to analysis, design and implementation of parallel algorithms in multi-core proccesors. In this work, an implementation and experimental investigation of the mathematical framework are performed using MATLAB r with the Parallel Computing Toolbox TM . We found that there is advantage to use multi-core processors and a parallel computing environment to minimize the high execution time. Also, speedup and efficiency increases when the number of logical processor and length of the signal increase. Moreover, a superlinear speedup is obtained in this ex perimental investigation.

Generalized Jordan normal forms of a finite-dimensional linear operator

We define the Mal’tsev representation of a linear operator on a finite-dimensional linear space and establish the necessary and sufficient conditions for its existence. We obtain the second step generalization of theorems on generalized Jordan normal forms of the first and second kinds.

An iterative algorithm for solving a finite-dimensional linear operator equation [formula omitted] with applications

This paper proposes an iterative algorithm for solving a general finite-dimensional linear operator equation T ( x ) = f and demonstrates that it will get the exact solution within a finite number of iteration steps. This algorithm unifies all the iterative methods in Huang (2008) [3], Peng (2005) [7] and Peng and Peng (2006) [8] and provides an iterative method for solving an inverse problem related to Hermitian-generalized Hamiltonian matrices [2,12].

Negative property of shape-preserving finite-dimensional linear operators

In this paper, we estimate the order of approximation by derivatives of shape-preserving and finite-dimensional operators defined on the space of all real-valued and κ-times continuously differentiable functions of one real variable.

Asymptotic behaviour of the partition function

Given a pair of positive integers and such that , for integer the quantity , called the partition function is considered; this by definition is equal to the cardinality of the set The properties of and its asymptotic behaviour as are studied. A geometric approach to this problem is put forward. It is shown that for sufficiently large , where and are positive constants depending on and , and and are characteristics of the exponential growth of the partition function. For some pair the exponents and are calculated as the logarithms of certain algebraic numbers; for other pairs the problem is reduced to finding the joint spectral radius of a suitable collection of finite-dimensional linear operators. Estimates of the growth exponents and the constants and are obtained.

The generalized joint spectral radius. A geometric approach

The properties of the joint spectral radius with an arbitrary exponent are investigated for a set of finite-dimensional linear operators where the summation and maximum extend over all maps Using the operation of generalized addition of convex sets, we extend the Dranishnikov-Konyagin theorem on invariant convex bodies, which has hitherto been established only for the case . The paper concludes with some assertions on the properties of invariant bodies and their relationship to the spectral radius . The problem of calculating for even integers is reduced to determining the usual spectral radius for an appropriate finite-dimensional operator. For other values of , a geometric analogue of the method with a pre-assigned accuracy is constructed and its complexity is estimated.

On the variety of invariant subspaces of a finite-dimensional linear operator

If V V is a finite-dimensional vector space over R \mathbf {R} or C \mathbf {C} and A ∈ Hom ( V ) A \in {\operatorname {Hom}}(V) , the set S A ( k ) {S_A}(k) of k k -dimensional A A -invariant subspaces is a compact subvariety of the Grassmann manifold G k ( V ) {G^k}(V) , but it need not be a Schubert variety. We study the topology of S A ( k ) {S_A}(k) . We reduce to the case where A A is nilpotent. In this case we prove that S A ( k ) {S_A}(k) is connected but need not be a manifold. However, the subset of S A ( k ) {S_A}(k) consisting of those subspaces with a fixed cyclic structure is a regular submanifold of G k ( V ) {G^k}(V) .

Variation of Grassman powers and spectra

The operator norm of the derivative of the map which takes a finite-dimensional linear operator to its kth Grassman power (the kth compound) is evaluated. This leads to a bound for the distance between the Grassman powers of two operators. As an important application, a bound for the distance between the eigenvalues of two operators is obtained.