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In this article, we proposed an enhancement to the convergence rate of Landweber's method by incorporating the concept of momentum acceleration. Landweber's method is commonly used to solve least squares problems of the form min x ‖ A x − b ‖. Our approach is based on Landweber's method, which is acknowledged as a particular case of the methodologies outlined in Ding and Chen (2006). Through optimizing the momentum parameter, we were able to demonstrate the superior performance of the momentum-accelerated Landweber method. Specifically, we established that when A is a nonsquare m × n matrix with Rank (A) = n and σ min (A) ≠ σ max (A) , the momentum-accelerated Landweber method with the optimal parameter consistently outperforms the standard Landweber method. Our numerical experiments have confirmed the theoretical findings, demonstrating a notable improvement in the convergence rate of the Landweber method. [ABSTRACT FROM AUTHOR]

In this work, a new algorithm for solving symmetric indefinite systems of linear equations is presented. It factorizes the matrix into the form L D L t using Jacobi rotations in order to increase the pivot´s absolute value. Furthermore, Rook´s pivoting strategy is also adapted and implemented. In determinate compatible systems, the computational cost of the algorithm was similar to the cost of the Bunch-Kaufman method, but the error was approximately 50 % smaller for intermediate and large matrices, regardless of the condition number of the coefficient matrix. Furthermore, unlike Bunch-Kaufman, the new algorithm calculates with little additional cost the fundamental basis of the null space, and obtains the minimal least squares and minimum norm solutions. In minimal least squares with minimum norm problems, the new algorithm was compared with the LAPACK Complete Orthogonal Decomposition algorithm, among others. The obtained error with both algorithms was similar but the computational cost was at least 20 % smaller with the new algorithm, even though the Complete Orthogonal Decomposition is implemented in a blocked form. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) such as those that describe linear elastic fracture mechanics admit analytic solutions with low regularity at crack tips. Existing numerical methods for such PDEs suffer from the constraint of this low regularity and fail to deliver an optimal high-order rate of convergence. We approach this problem by (i) choosing an artificial interface to enclose the center of low regularity and (ii) representing the solution in the interior of artificial interface as unknown linear combination of known low-regularity modes of the solution. This results in an interface formulation of the original PDE, and the linear combination is represented in the interface conditions. By enforcing the smooth component of the numerical solution in the interior domain to be approximately zero, a least squares problem is obtained for the unknown coefficients. The solution of this least squares problem will provide the approximate interface conditions for the numerical solution of the PDE in the exterior domain. The potential of our interface formulation is favorably demonstrated by numerical experiments on one- and two-dimensional Poisson equations with low-regularity solutions. High-order numerical solutions of unknown coefficients and PDEs are obtained. This proves the potential of the proposed interface formulation as the theoretical basis for solving linear elastic fracture mechanics problems. We address the relations between our interface formulation and domain decomposition methods, as well as a regularization strategy for the Poisson–Boltzmann equation with singular charge density. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of the large scale singular Sylvester equation of the form A X − X D ⊤ = E (or A X B ⊤ − C X D ⊤ = E), where the spectra Λ (A) and Λ (D) (or, Λ (A − λ C) and Λ (D − λ B)) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided. [ABSTRACT FROM AUTHOR]

The iterative algorithm of a class of generalized coupled Sylvester-transpose matrix equations is presented. We prove that if the system is consistent, a solution can be obtained within finite iterative steps in the absence of round-off errors for any initial matrices; if the system is inconsistent, the least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm least squares solution of the problem. Finally, numerical examples are presented to demonstrate that the algorithm is efficient. [ABSTRACT FROM AUTHOR]

Let R ∈ C m × m and S ∈ C n × n be nontrivial k -involutions if their minimal polynomials are both x k − 1 for some k ≥ 2 , i.e., R k − 1 = R − 1 ≠ ± I and S k − 1 = S − 1 ≠ ± I . We say that A ∈ C m × n is ( R , S , μ ) -symmetric if R A S − 1 = ζ μ A , and A is ( R , S , α , μ ) -symmetric if R A S − α = ζ μ A with α , μ ∈ { 0 , 1 , … , k − 1 } and α ≠ 0 . Let S be one of the subsets of all ( R , S , μ ) -symmetric and ( R , S , α , μ ) -symmetric matrices. Given X ∈ C n × r , Y ∈ C s × m , B ∈ C m × r and D ∈ C s × n , we characterize the matrices A in S that minimize ‖ A X − B ‖ 2 + ‖ Y A − D ‖ 2 (Frobenius norm) under the assumption that R and S are unitary. Moreover, among the set S ( X , Y , B , D ) ⊂ S of the minimizers of ‖ A X − B ‖ 2 + ‖ Y A − D ‖ 2 = min , we find the optimal approximate matrix A ∈ S ( X , Y , B , D ) that minimizes ‖ A − G ‖ to a given unstructural matrix G ∈ C m × n . We also present the necessary and sufficient conditions such that A X = B , Y A = D is consistent in S . If the conditions are satisfied, we characterize the consistent solution set of all such A . Finally, a numerical algorithm and some numerical examples are given to illustrate the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, by using the real representations of quaternion matrices, the particular structure of the real representations of quaternion matrices, the Kronecker product of matrices and the Moore–Penrose generalized inverse, we obtain the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation A X B + C X D = E , respectively. Our resulting formulas only involve real matrices, and therefore are simpler than those reported in Yuan (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of quaternion matrices, and therefore are very efficient and portable. Numerical examples are provided to illustrate the efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the necessary and sufficient conditions for the existence and the explicit expressions of the Hermitian { P , k + 1 } -(anti-)reflexive solutions of the matrix equation A X = B , and meanwhile the best approximation solution is considered. Then, if the solvability conditions are not satisfied, the least squares Hermitian { P , k + 1 } -(anti-)reflexive solutions and the least squares Hermitian { P , k + 1 } -(anti-)reflexive solutions with the minimum norm of the above matrix equation are respectively derived. In addition, two algorithms are shown to compute the least squares Hermitian { P , k + 1 } -(anti-)reflexive solutions, and the corresponding numerical examples are also given to illustrate the feasibility of the algorithms. [ABSTRACT FROM AUTHOR]

In this paper, by applying particular structure of the real representations of quaternion matrices and the Moore–Penrose generalized inverse, we derive the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation A X = B . The resulting formulas only involve real matrices, which are simpler than those reported in (Yuan et al., 2013). The corresponding algorithms only perform real arithmetic which also consider particular structure of the real representations of quaternion matrices, therefore are very efficient and easily understood. Numerical examples are provided to illustrate the efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper, a modified conjugate gradient iterative method for solving Sylvester equation is presented. By using this iterative method, the least squares anti-bisymmetric solution and the optimal approximation solution can be obtained. Here we present the derivation and theoretical analysis of our iterative method. Numerical results illustrate the feasibility and effectiveness of the proposed iterative method. [ABSTRACT FROM AUTHOR]

Let S and S ̂ be two sets of solutions to matrix least squares problem (LSP) A X B = Cand the perturbed matrix LSP A ̂ X ̂ B ̂ = C ̂, respectively, where A ̂ = A + Δ A, B ̂ = B + Δ B, C ̂ = C + Δ C, and Δ A, Δ B, Δ C are all small perturbation matrices. For any given X ∈ S, we deduce general formulas of the least squares solutions X ̂ ∈ S ̂ that are closest to X under appropriated norms, meanwhile, we present the corresponding distances between them. With the obtained results, we derive perturbation bounds for the nearest least squares solutions. At last, a numerical example is provided to verify our analysis. [ABSTRACT FROM AUTHOR]

This paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space, when they are pressed against each other. The graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant Poisson’s ratio. A segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction. The entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction. It is assumed that the contact region is under sliding contact conditions with the Coulomb’s law used to relate the tangential traction to the normal component. Employing Fourier integral transforms and applying the necessary boundary conditions, the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths. Ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem. The Gauss–Chebyshev quadrature-collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations. This system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously. The main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region. [ABSTRACT FROM AUTHOR]

Abstract: A better method than the least squares solution is proposed in this paper to solve an -dimensional ill-posed linear equations system in an -dimensional column subspace , which is selected in such a way that each column in is in a closer proximity to . We maximize the orthogonal projection of onto to find an approximate solution , where is a nonzero free vector. Then, we can prove that the maximal projection solution (MP) is better than the least squares solution (LS) with . Numerical examples of inverse problems under a large noise maybe up to are discussed which confirm the efficiency of presently developed MP algorithms: MPA and MPA(m). [Copyright &y& Elsevier]

Abstract: In this paper, the least squares solution of the matrix equation for symmetric arrowhead matrices with the least norm is discussed. By using Moore–Penrose inverse and the Kronecker product, the general expression of the solution to this problem is derived. A corresponding numerical algorithm and an example are also given. [Copyright &y& Elsevier]

Abstract: In this paper, we construct an iterative method to solve the general coupled matrix equations where is a reflexive matrix with a specified central principal submatrix. The algorithm produces suitable such that within finite iteration steps in the absence of roundoff errors. We show that the algorithm is stable any case. The algorithm requires little storage capacity. Given numerical examples show that the algorithm is efficient. [Copyright &y& Elsevier]

Abstract: By using the complex representation of quaternion matrices, and the Moore–Penrose generalized inverse, we derive the expressions of the least squares solution with the least norm, the least squares pure imaginary solution with the least norm, and the least squares real solution with the least norm for the quaternion matrix equation , respectively. Finally, we discuss their applications in color image restoration. [Copyright &y& Elsevier]

Abstract: In this paper, the solvability conditions and the explicit expressions of the generalized bisymmetric and bi-skew-symmetric solutions of the matrix equation are respectively established by applying two methods. Then the maximal and minimal ranks of the solutions are derived. If the solvability conditions are not satisfied, the generalized bisymmetric and bi-skew-symmetric least squares solutions of the matrix equation are considered, and the generalized bisymmetric and bi-skew-symmetric least squares solutions with the minimum norm are also obtained. In addition, two algorithms are provided to compute the generalized bi (skew-) symmetric least squares solution, and some examples are given to illustrate that the algorithms are feasible. [Copyright &y& Elsevier]

Abstract: Linear discriminant analysis (LDA) is a popular supervised dimension reduction algorithm, which projects the data into an effective low-dimensional linear subspace while the separation between the projected data from different classes is improved. While this subspace is typically determined by solving a generalized eigenvalue decomposition problem, its high computation costs prohibit the use of LDA especially when the scale and the dimensionality of the data are large. Based on the recent success of least squares LDA (LSLDA), we propose a novel rank-one update method with a simplified class indicator matrix. Using the proposed algorithm, we are able to derive the LSLDA model efficiently. Moreover, our LSLDA model can be extended to address the learning task of concept drift, in which the recently received data exhibit with gradual or abrupt changes in distribution. In other words, our LSLDA is able to observe and model the data distribution changes, while the dependency on outdated data will be suppressed. This proposed LSLDA will benefit applications of streaming data classification or mining, and it can recognize data with newly added class labels during the learning process. Experimental results on both synthetic and real datasets (with and without concept drift) confirm the effectiveness of our propose LSLDA. [Copyright &y& Elsevier]

Abstract: Within the framework of the theory of the column and row determinants, we obtain explicit representation formulas (analogs of Cramer’s rule) for the minimum norm least squares solutions of quaternion matrix equations and . [Copyright &y& Elsevier]

Abstract: Surface developability is a key feature in many industrial product designs. In this paper, based on a measure defined in an extended Gaussian image, an efficient least squares solution is proposed to achieve an optimal quasi-developable mesh patch. For a free-form mesh model assembled by a set of patches, each patch can be deformed using the least squares solution to obtain the best developability within a user-specified tolerance while the patch boundary remains continuous with neighboring patches. The proposed least squares scheme formulates the quasi-developable mesh approximation problem as a large sparse linear system with its coefficient matrix independent of the mesh verticesʼ new positions. We show that this linear system can be efficiently solved by a least squares direct matrix solver. Experimental results and applications are provided to demonstrate the controllability of shape change as well as the effectiveness of mesh developability improvement provided by the proposed solution. [Copyright &y& Elsevier]

Abstract: This paper gives some closed-form formulas for computing the maximal and minimal ranks and inertias of with respect to , where is given, and is a Hermitian least squares solution to the matrix equation . We derive, as applications, necessary and sufficient conditions for in the Löwner partial ordering. In addition, we give necessary and sufficient conditions for the existence of a Hermitian positive (negative, nonpositive, nonnegative) definite least squares solution to . [Copyright &y& Elsevier]

Abstract: A stochastic conjugate gradient method for the approximation of a function is proposed. The proposed method avoids computing and storing the covariance matrix in the normal equations for the least squares solution. In addition, the method performs the conjugate gradient steps by using an inner product that is based on stochastic sampling. Theoretical analysis shows that the method is convergent in probability. The method has applications in such fields as predistortion for the linearization of power amplifiers. [Copyright &y& Elsevier]

Abstract: The least squares solutions with the minimum norm of the matrix equations and are considered in this paper. We use the determinantal representations of the Moore–Penrose inverse obtained earlier by the author and get analogs of the Cramer rule for the minimum norm least squares solutions of these matrix equations. [Copyright &y& Elsevier]

Abstract: In this article we give some formulas for the maximal and minimal ranks of the submatrices in a least squares solution X to AXB = C. From these formulas, we derive necessary and sufficient conditions for the submatrices to be zero and other special forms, respectively. Finally, some Hermitian properties for least squares solution to matrix equation AXB = C are derived. [Copyright &y& Elsevier]

Abstract: Let R and S be m × m and n × n nontrivial real symmetric involutions. An m × n complex matrix A is termed (R, S)-conjugate if , where denotes the conjugate of A. In this paper, necessary and sufficient conditions are established for the existence of the (R, S)-conjugate solution to the system of matrix equations AX = C and XB = D. The expression is also presented for such solution to this system. In addition, the explicit expression of this solution to the corresponding optimal approximation problem is obtained. Furthermore, the least squares (R, S)-conjugate solution with least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally an algorithm and numerical examples are given. [ABSTRACT FROM AUTHOR]

Abstract: Least squares solution of linear inequalities appears in many disciplines such as linear separability problems and inconsistency correction. In this paper we consider this problem with uncertainty in its data. Then we prove that its robust counterpart is equivalent to a second order conic linear optimization problem, which can be efficiently solved using interior point methods. [Copyright &y& Elsevier]

Abstract: By using the complex representations of quaternion matrices, Moore–Penrose generalized inverse and the Kronecker product of matrices, we derive the expression of the least squares Hermitian solution of the matrix equation with the least norm over the skew field of quaternions, and provide a numerical algorithm to calculate the forementioned solution. [Copyright &y& Elsevier]

Abstract: In topology optimization, the notion of topology is introduced into the analysis by assigning density design variables to positions throughout the design domain. The standard practice is to set the lower bound values of the density design variables so that they are small but nonzero values. The intent of the nonvanishing lower bound has been to ensure that the (tangent) stiffness matrix of the finite element analysis remains nonsingular, and the lower bound values are selected so that the low density elements are structurally insignificant to the extent that the structural analysis can be performed. However, from a pragmatic point of view, we should desire that void material be represented by regions of zero density elements. Here, a simple algorithm that does not requiring remeshing and accommodates the zero density design variable values is presented, and the ramifications of the zero density lower bounds on the topology optimization are discussed. [Copyright &y& Elsevier]

This paper presents a numerical scheme to evaluate the effects of the communication cable attached to an underwater flight vehicle. Both simulation and model validation results show that the numerical scheme is effective and provides a means for developing a feed-forward controller to compensate for the cable effects when developing an autopilot for the tethered vehicle. Moreover, the numerical scheme can also be applied to predict the effects of the ROVs umbilical during its deployment. [Copyright &y& Elsevier]

In this paper we are concerned with the robust calibration of light sources in the scene from a known shape. The image of a 3-D object depends on the light source(s), its 3-D geometry, and its surface reflectance properties (Robot Vision, MIT Press, Cambridge, MA, 1986). In the last two decades in the computer vision research communities intensive researches have been conducted along the line of shape from shading (Shape from Shading, MIT Press, Cambridge, MA, 1989), where great efforts are made to recover the 3-D geometry with a priori knowledge regarding the illumination and surface reflectance properties. However, as pointed out by Sato et al. (Proceedings of CVPR’99, 1999, pp. 306), little progress has been made for the recovery of light source(s) with known shape and surface reflectance properties. In a recent paper (IEEE Trans PAMI 23 (2001) 915), Zhang and Yang achieved multiple illuminant direction recovery based on the critical points with impressive performance. In this paper we first formulate the local light source constant constraint, i.e., in a local area of smooth lightness it is likely that the corresponding 3-D world points on the object are illuminated by the same light sources. Based on this constraint we develop an algorithm to recover the multiple illuminants: first based on the Lambertian irradiance formula a linear system is formulated for a local area, the local illuminant direction is then reconstructed by a least-squares solution. To effect insensitivity to noises, the least trimmed square method is carried out. Next a dense set of candidate critical points is obtained as a result of a two-step robust processing, which is used to arrive at the directions of multiple illuminants with an adaptive Hough transform. The magnitude for each light source is then computed by solving an over-determined linear system which is formed by pooling pixels illuminated by the same combined light vector. Initial experimental results based on synthetic and real world images suggested encouraging performance. [Copyright &y& Elsevier]

Let X and Y be Hilbert spaces, and let T : X → Y be a bounded linear operator with closed range. In this paper, we present an optimal perturbation result on the least squares solutions to the operator equation Tx = y under the most general condition. [Copyright &y& Elsevier]