Developing an Improved Shift-and-Invert Arnoldi Method

An algorithm has been developed for finding a number of eigenvalues close to a given shift and in interval [ Lb,Ub ] of a large unsymmetric matrix pair. The algorithm is based on the shift-andinvert Arnoldi with a block matrix method. The block matrix method is simple and it uses for obtaining the inverse matrix. This algorithm also accelerates the shift-and-invert Arnoldi Algorithm by selecting a suitable shift. We call this algorithm Block Shift-and-Invert or BSI. Numerical examples are presented and a comparison has been shown with the results obtained by Sptarn Algorithm in Matlab. The results show that the method works well

The Mixed Type Splitting Methods for Solving Fuzzy Linear Systems

We consider a class of fuzzy linear systems (FLS) and demonstrate some of the existing methods using the embedding approach for calculating the solution. The main aim in this paper is to design a class of mixed type splitting iterative methods for solving FLS. Furthermore, convergence analysis of the method is proved. Numerical example is illustrated to show the applicability of the methods and to show the efficiency of proposed algorithm

The Mixed Type Splitting Methods for Solving Fuzzy Linear Systems

We consider a class of fuzzy linear systems (FLS) and demonstrate some of the existing methods using the embedding approach for calculating the solution. The main aim in this paper is to design a class of mixed type splitting iterative methods for solving FLS. Furthermore, convergence analysis of the method is proved. Numerical example is illustrated to show the applicability of the methods and to show the efficiency of proposed algorithm

Stability Analysis of Distributed Order Fractional Differential Equations

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure

Out-Of-Domain Unlabeled Data Improves Generalization

We propose a novel framework for incorporating unlabeled data into semi-supervised classification problems, where scenarios involving the minimization of either i) adversarially robust or ii) non-robust loss functions have been considered. Notably, we allow the unlabeled samples to deviate slightly (in total variation sense) from the in-domain distribution. The core idea behind our framework is to combine Distributionally Robust Optimization (DRO) with self-supervised training. As a result, we also leverage efficient polynomial-time algorithms for the training stage. From a theoretical standpoint, we apply our framework on the classification problem of a mixture of two Gaussians in $\mathbb{R}^d$, where in addition to the $m$ independent and labeled samples from the true distribution, a set of $n$ (usually with $n\gg m$) out of domain and unlabeled samples are given as well. Using only the labeled data, it is known that the generalization error can be bounded by $\propto\left(d/m\right)^{1/2}$. However, using our method on both isotropic and non-isotropic Gaussian mixture models, one can derive a new set of analytically explicit and non-asymptotic bounds which show substantial improvement on the generalization error compared to ERM. Our results underscore two significant insights: 1) out-of-domain samples, even when unlabeled, can be harnessed to narrow the generalization gap, provided that the true data distribution adheres to a form of the ``cluster assumption", and 2) the semi-supervised learning paradigm can be regarded as a special case of our framework when there are no distributional shifts. We validate our claims through experiments conducted on a variety of synthetic and real-world datasets.Comment: Published at ICLR 2024 (Spotlight), 29 pages, no figure

Analytic study on linear systems of distributed order fractional differential equations

In this paper we introduce the distributed order fractional  differential equations (DOFDE) with respect to the nonnegative density function. We generalize the inertia and characteristics polynomial  concepts of pair $(A,B)$ with respect to the nonnegative density function. We also give generalization of the invariant factors of a matrix and some inertia theorems for analyzing the stability of the DOFDE systems

A NEW FRACTIONAL MODEL OF SINGLE DEGREE OF FREEDOM SYSTEM, BY USING GENERALIZED DIFFERENTIAL TRANSFORM METHOD

Generalized differential transform method (GDTM) is a powerful method to solve the fractional differential equations. In this paper, a new fractional model for systems with single degree of freedom (SDOF) is presented, by using the GDTM. The advantage of this method compared with some other numerical methods has been shown. The analysis of new approximations, damping and acceleration of systems are also described. Finally, by reducing damping and analysis of the errors, in one of the fractional cases, we have shown that in addition to having a suitable solution for the displacement close to the exact one, the system enjoys acceleration once crossing the equilibrium point

GMRES implementations and residual smoothing techniques for solving ill-posed linear systems

AbstractThere are verities of useful Krylov subspace methods to solve nonsymmetric linear system of equations. GMRES is one of the best Krylov solvers with several different variants to solve large sparse linear systems. Any GMRES implementation has some advantages. As the solution of ill-posed problems are important. In this paper, some GMRES variants are discussed and applied to solve these kinds of problems. Residual smoothing techniques are efficient ways to accelerate the convergence speed of some iterative methods like CG variants. At the end of this paper, some residual smoothing techniques are applied for different GMRES methods to test the influence of these techniques on GMRES implementations