Iterative Algorithms for Symmetric Positive Semidefinite Solutions of the Lyapunov Matrix Equations

It is well-known that the stability of a first-order autonomous system can be determined by testing the symmetric positive definite solutions of associated Lyapunov matrix equations. However, the research on the constrained solutions of the Lyapunov matrix equations is quite few. In this paper, we present three iterative algorithms for symmetric positive semidefinite solutions of the Lyapunov matrix equations. The first and second iterative algorithms are based on the relaxed proximal point algorithm (RPPA) and the Peaceman–Rachford splitting method (PRSM), respectively, and their global convergence can be ensured by corresponding results in the literature. The third iterative algorithm is based on the famous alternating direction method of multipliers (ADMM), and its convergence is subsequently discussed in detail. Finally, numerical simulation results illustrate the effectiveness of the proposed iterative algorithms.

Continuous Analog of Accelerated OS-EM Algorithm for Computed Tomography

The maximum-likelihood expectation-maximization (ML-EM) algorithm is used for an iterative image reconstruction (IIR) method and performs well with respect to the inverse problem as cross-entropy minimization in computed tomography. For accelerating the convergence rate of the ML-EM, the ordered-subsets expectation-maximization (OS-EM) with a power factor is effective. In this paper, we propose a continuous analog to the power-based accelerated OS-EM algorithm. The continuous-time image reconstruction (CIR) system is described by nonlinear differential equations with piecewise smooth vector fields by a cyclic switching process. A numerical discretization of the differential equation by using the geometric multiplicative first-order expansion of the nonlinear vector field leads to an exact equivalent iterative formula of the power-based OS-EM. The convergence of nonnegatively constrained solutions to a globally stable equilibrium is guaranteed by the Lyapunov theorem for consistent inverse problems. We illustrate through numerical experiments that the convergence characteristics of the continuous system have the highest quality compared with that of discretization methods. We clarify how important the discretization method approximates the solution of the CIR to design a better IIR method.

Harmonic and biharmonic biases in potential field inversion

We found that minimum [Formula: see text]-norm and smoothness-constrained continuous solutions of the linear inverse problem of potential field data are harmonic and biharmonic, respectively. In the case of a discrete distribution, the minimum [Formula: see text]-norm and smoothness-constrained solutions become biased toward being harmonic or biharmonic, respectively. As a result, the estimated discrete distribution of density or magnetization contrast tends to be smooth and to satisfy the maximum principle, which forces the solution maxima and minima to lie on any boundary of the discretized region. The above findings were illustrated with 2D numerical examples. The harmonic or biharmonic bias is brought forth when the strengths of the minimum [Formula: see text]-norm or the smoothness constraint are enhanced (relative to all other constraints) by approximating the continuous case (a large number of discretizing cells relative to the number of independent observations) and/or by using a regularizing parameter. We discovered that, by inspecting the rearranged normal equations, it is possible to qualify three different possibilities of designing nonharmonic estimators. Then we found that all three possibilities have in fact already been implemented in the literature, reinforcing, in this way, the validity of the theoretical demonstration.

Eigenvector-Free Solutions to the Matrix EquationAXBH=Ewith Two Special Constraints

The matrix equationAXBH=EwithSX=XRorPX=sXQconstraint is considered, whereS, Rare Hermitian idempotent,P, Qare Hermitian involutory, ands=±1. By the eigenvalue decompositions ofS, R, the equationAXBH=EwithSX=XRconstraint is equivalently transformed to an unconstrained problem whose coefficient matrices contain the corresponding eigenvectors, with which the constrained solutions are constructed. The involved eigenvectors are released by Moore-Penrose generalized inverses, and the eigenvector-free formulas of the general solutions are presented. By choosing suitable matricesS, R, we also present the eigenvector-free formulas of the general solutions to the matrix equationAXBH=EwithPX=sXQconstraint.

Constrained Solutions of a System of Matrix Equations

We derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equationsAX=BandXC=D, respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skew-symmetric orthogonal solutions are respectively given. As an auxiliary, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable.