This paper examines the construction of rth-order truncated balanced realizations via tangential interpolation at r specified interpolation points. It is shown that when the truncated Hankel singular values are negligible-meaning the discarded states are nearly uncontrollable and unobservable-balanced truncation reduces to a bi-tangential Hermite interpolation problem. Since the required interpolation points and directions are not known apriori, we propose an iterative tangential-interpolation algorithm for balanced truncation. Upon convergence, it produces a truncated balanced realization that preserves the r largest Hankel singular values. In addition, an adaptive low-rank Lyapunov solver based on tangential interpolation is presented. Numerical tests on benchmark models confirm the effectiveness of the proposed methods.
Sparse phase retrieval is the problem of recovering an s-sparse signal x0 is an element of Fd from b = |Ax0|2 + e is an element of & Ropf;m. In this paper, we propose the 812-alpha 822 minimization models. We establish the exact recovery and the stable recovery of constrained and unconstrained models under the restricted isometry property frame with high probability, respectively. We design a fast algorithm based on ADMM to solve the unconstrained model. Experimental results show that our algorithm has better performance than other algorithms.
The temporal regularity of Maxwell's equations in dispersive media, including the Cole-Cole and Havriliak-Negami models, is investigated in this paper through an exact series representation. This representation is derived by reformulating Maxwell's equations into a single integro-differential equation, which is then analyzed using the Laplace transform. For the Cole-Cole model, our analysis shows that, when the initial electric field satisfies E-0 not equivalent to 0, the electric field E and the induced polarization field P belong to C-alpha in time, while the magnetic field H belongs to C-1,C-alpha in time, where alpha is an element of (0, 1). Moreover, when E-0 equivalent to 0, the regularity improves to E, P is an element of C-1,C-alpha and H is an element of C-2,C-alpha in time. Careful numerical experiments are conducted to verify the analytical results on temporal regularity. In addition, the analysis characterizing the singular behavior near the origin is extended to the Havriliak-Negami model in the case alpha=1 and 0 < beta < 1.
