Block Diagonalization Method Research Articles

Scattering matrix evaluation using spatial symmetry in electromagnetic modelling

SUMMARY In this paper we discuss the possibility of a drastic computational reduction in forming the scattering matrix for electromagnetic modelling of 3-D conductivity structure embedded in a stratified and vertically anisotropic earth using integral equations. This reduction is facilitated by using the lateral homogeneity of the space and the symmetry property of the Green's functions to reduce the redundancy of calculating the scattering matrix by identifying classes of cell pairs which give either identical entries in the scattering matrix or entries that differ only in the sign. It is required that a conductivity structure be discretized into equal-size or equal-size- . based ce11s. By the latter we mean that the structure is first divided into equal-sized basic cells, and some odd numbers of the basic cells may form secondary, bigger cells where the scattering currents and other field quantities may be assumed to be constant, in order to allow the symmetry reduction while keeping the dimension of the linear system as low as possible. This method of reduction is valid for arbitrary conductivity structure. The factor of reduction depends mostly on the number of cells in the lateral direction and can be up to several hundred. the Green's tensors which are merely scalar and cannot be Electromagnetic modelling of 3-D conductivity structures vectorized. Thus, in the integral equation method the using integral equations has been widely accepted in formation of the scattering matrix can dominate the geophysics since its appearance in the early 1970s computation time. (Hohmann 1971, 1975; Weidelt 1975). Yet the quadratically With the introduction of the method of system iteration growing requirements on computer storage and on (Xiong 1992a), matrix factorization time and storage computation time in forming the scattering matrix, and the requirements were markedly reduced. In the method of cubic growth of matrix factorization time with the number of system iteration a conductivity structure is divided into discretization cells have remained the major obstacles for its many substructures. The direct matrix inversion is applied to application for about two decades. each substructure only, whereas the mutual interactions The computation of the scattering matrix elements usually among the substructures are treated as being due to consumes about 80 per cent of the total CPU time for equivalent sources. This method not only solves the problem models of moderate size. The cubically growing time of storage requirements, but also greatly reduces the consumption of the matrix factorization is dominant only for computation time needed for the solution of the matrix models with a fairly large number of cells, say over 500 cells. equation. Furthermore, the method of system iteration also With the optimization efforts of Xiong (1992a) where he allows vector and/or parallel processing. Hence, the approached the integration of the secondary parts of the scattering matrix formation time dominates the computation Green's tensors in the vertical direction analytically in the for large structures, too. Hankel transform domain and thus increased the accuracy We can use geometric model symmetries, if present, to and efficiency of the numerical integrations, the computation mitigate these computational problems. Among others, time required by matrix formation was greatly reduced. Tripp & Hohmann (1984) (see also Tripp 1990) put forward Even so, forming the scattering matrix still takes about 50 a block diagonalization method which reduces the scattering per cent of the total CPU time for structures with moderate matrix to four block diagonal matrices and reduced the discretization. Also, the computation of the scattering computation time for matrix formation by a factor of 4. The 459

Reduction, Stability, Instability and Bifurcation in Rotationally Symmetric Hamiltonian Systems

We give an explicit description of the reduced energy momentum method for determining nonlinear stability ors steadny rotating equilibrium states of Hamiltonian systems having finitely many degrees of freedom. The block-diagonalization method of Marsden et al (1989) is reviewed and illustrated by application to the heavy ton and two discrete mass-spring systems having four and six degrees of freedom respectively. We are particularly interested in cases where the stability test is inconclusive and one must examine the linearized equations of motion. Making use of the sparse structure of suitably transformed matrices, we give explicit formulae from which coefficients in the characteristic polynomial of the related eigenproblem. can be found. This permits us to give rather complete analyses of specific examples, which we also illustrate by numerical simulation. Our most interesting finding is that in both examples for which the amended potential is indefinite of index two, but the linearized analysis indicat...

Block-diagonalization analysis of symmetric plates

This paper presents a block-diagonalization method to solve stiffness equations of isotropic symmetric plates. By means of a suitable “local” coordinate transformation, chosen based on group theory, the stiffness matrix is decomposed into a block-diagonal form. The stiffness equation in the local coordinate is solved block by block, thus realizing numerical efficiency and greatly reducing the requisite amount of computer memory. The efficiency has been further upgraded with the aid of the concept of augmented orbit. This method is applied to a square isotropic plate subject to “asymmetric” loads to show its usefulness.

Gauge theory and quasidiabatic states in molecular physics

Following previous works [Moody et al., Phys. Rev. Lett. 56, 893 (1986) and Zygelman, Phys. Lett. A 125, 476 (1987)], we review in some detail the gauge theoretical form of the Born–Oppenheimer description of molecules. The derivative couplings which describe nonadiabatic effects play the role of gauge potentials. Applying the gauge theoretical formalism (well known in particle physics), we rederive a theorem concerning the nonexistence of strictly diabatic states. Further we propose the Lorentz gauge condition to define quasidiabatic states. We show that the recently proposed block diagonalization method to obtain quasidiabatic states leads automatically to the Lorentz gauge locally.

FORMULATION OF CONTROL POLICIES FOR INTEGRATED RESEARCH AND DEVELOPMENT-PRODUCTION-INVENTORY SYSTEMS

Multivariable system theory is used to synthesize continuous-time control policies for research and development-production-inventory systems. The block diagona-lization method is used to study the asymptotic structure of the transfer function matrices of such systems. It is shown that the results obtained greatly facilitate the synthesis of high-grain error-actuated control policies which ensure that the closed-loop behavior of such systems becomes increasingly “tight” and nonin-teracting as the value of the gain parameter is increased. The behavior of the controlled research and development-production-inventory system is illustrated by the presentation of the results of computer simulation studies showing typical transient and steady-state response characteristics.

Large distance propagators for hadrons on a 56 × 16 3 lattice

We have computed hadron propagators in the quenched approximation to Wilson fermions at β = 6 on 24 SU(3) background fields of size 56 × 16 3. The inversion of the fermion matrix was achieved by the approximate block diagonalization method. We find a clean exponential decay of the ϱ, N and Δ propagators for time separations 8 ≲ t ⩽ 54 at quark masses am q = 0.04 and beyond.

Block diagonalization — An efficient method to compute hadrons on large lattices

We compute hadron masses from Wilson fermions in the quenched approximation. By use of our recently proposed block diagonalization method we are able to extend lattice sizes from (previously) 10 3 × 20 to 16 3 × 28, using a CYBER 205 with small memory. First results, based on 360 quark propagators, look very encouraging for our method: very close to κ critical, we achieve good stability of the conjugate gradient algorithm, so that errors can be kept small. We thus can compute quark propagators down to a pion mass of am π = 0.2. The resulting hadron masses at β = 6 match with the values obtained previously by Lipps et al. on a 10 3 × 20 lattice. We find a( β = 6) = 0.093 fm and m proton/ m p = 1.853(88).

Molecular treatment ofHe++H collisions

Total cross sections for the reactions /sup 4/He/sup +/(1s)+H(1s)..-->../sup 4/He/sup +/(1s)+H(2p), /sup 4/He/sup +/(1s) + H(1s)..-->../sup 4/He(1s2p /sup 1,3/P)+H/sup +/ are calculated in an impact-parameter formalism, for collision energies 0.5--30 keV, using a molecular approach. The molecular energies are calculated at large internuclear distances by a configuration-interaction method, and at short distances by a block-diagonalization method. The radial couplings between the molecular wave functions are evaluated exactly, and turn out to depend strongly on the choice of origin of electronic coordinates; as a consequence, the cross sections also depend on this choice. The effect of the anomalous rotational couplings which decrease like R/sup -1/ is shown to be negligible, and a method to integrate them is proposed in the Appendix. Choosing the center of mass as origin of electronic coordinates, we obtain good agreement with experimental data, but we find no a priori reason to favor this choice. As an important conclusion, the main effect of the so-called momentum-transfer problem, which is the origin dependence of the cross sections, is due to the form of the molecular wave functions at short internuclear distances.