CORE Technology and Exact Hamiltonian Real-Space Renormalization Group Transformations

The COntractor REnormalization group (CORE) method, a new approach to solving Hamiltonian lattice systems, is presented. The method defines a systematic and nonperturbative means of implementing Kadanoff-Wilson real-space renormalization group transformations using cluster expansion and contraction techniques. We illustrate the approach and demonstrate its effectiveness using scalar field theory, the Heisenberg antiferromagnetic chain, and the anisotropic Ising chain. Future applications to the Hubbard and t-J models and lattice gauge theory are discussed.Comment: 65 pages, 9 Postscript figures, uses epsf.st

Fisher Information for Inverse Problems and Trace Class Operators

This paper provides a mathematical framework for Fisher information analysis for inverse problems based on Gaussian noise on infinite-dimensional Hilbert space. The covariance operator for the Gaussian noise is assumed to be trace class, and the Jacobian of the forward operator Hilbert-Schmidt. We show that the appropriate space for defining the Fisher information is given by the Cameron-Martin space. This is mainly because the range space of the covariance operator always is strictly smaller than the Hilbert space. For the Fisher information to be well-defined, it is furthermore required that the range space of the Jacobian is contained in the Cameron-Martin space. In order for this condition to hold and for the Fisher information to be trace class, a sufficient condition is formulated based on the singular values of the Jacobian as well as of the eigenvalues of the covariance operator, together with some regularity assumptions regarding their relative rate of convergence. An explicit example is given regarding an electromagnetic inverse source problem with "external" spherically isotropic noise, as well as "internal" additive uncorrelated noise.Comment: Submitted to Journal of Mathematical Physic

Matrix-interpolation-based parametric model order reduction for multiconductor transmission lines with delays

A novel parametric model order reduction technique based on matrix interpolation for multiconductor transmission lines (MTLs) with delays having design parameter variations is proposed in this brief. Matrix interpolation overcomes the oversize problem caused by input-output system-level interpolation-based parametric macromodels. The reduced state-space matrices are obtained using a higher-order Krylov subspace-based model order reduction technique, which is more efficient in comparison to the Gramian-based parametric modeling in which the projection matrix is computed using a Cholesky factorization. The design space is divided into cells, and then the Krylov subspaces computed for each cell are merged and then truncated using an adaptive truncation algorithm with respect to their singular values to obtain a compact common projection matrix. The resulting reduced-order state-space matrices and the delays are interpolated using positive interpolation schemes, making it computationally cheap and accurate for repeated system evaluations under different design parameter settings. The proposed technique is successfully applied to RLC (R-resistor, L-inductor, C-capacitance) and MTL circuits with delays