Block Inversion Research Articles

Multigrid Preconditioning for Krylov Methods for Time-Harmonic Maxwell's Equations in Three Dimensions

We consider the rapid simulation of three-dimensional electromagnetic problems in geophysical parameter regimes, where the conductivity may vary significantly and the range of frequencies is moderate. Toward developing a multigrid preconditioner, we present a Fourier analysis based on a finite-volume discretization of a vector potential formulation of time-harmonic Maxwell's equations on a staggered grid in three dimensions. We prove grid-independent bounds on the eigenvalue and singular value ranges of the system obtained using a preconditioner based on exact inversion of the dominant diagonal blocks of the non-Hermitian coefficient matrix. This result implies that a preconditioner that uses single multigrid cycles to effect inversion of the diagonal blocks also yields a preconditioned system with an $\ell_2$-condition number bounded independent of the grid size. We then present numerical examples for more realistic situations involving large variations in conductivity (i.e.,\ jump discontinuities). Block-preconditioning with one multigrid cycle using Dendy's BOXMG solver is found to yield convergence in very few iterations, apparently independent of the grid size. The experiments show that the somewhat restrictive assumptions of the Fourier analysis do not prohibit it from describing the essential local behavior of the preconditioned operator under consideration. A very efficient, practical solver is obtained.

A coarse-mesh numerical method for one-speed neutron transport eigenvalue problems in two-dimensional Cartesian geometry

We describe a hybrid spectral nodal method applied to one-speed S N eigenvalue problems in x, y-geometry for nuclear reactor global calculations. To solve the transverse-integrated S N nodal equations, we generalize the spectral diamond (SD) method, that we developed for numerically solving slab-geometry S N eigenvalue problems with no spatial truncation error. In the present generalization, we approximate the transverse leakage through the edges of each spatial node by constants, so we call our method the SD-constant nodal (SD-CN) method that we use in the fuel regions of the nuclear reactor core. In the non-multiplying regions, e.g., reflector and baffle, we use the spectral Green's function-constant nodal (SGF-CN) method; hence the hybrid characteristic of our method. In order to converge the numerical solution for each S N “fixed source” problem (inner iterations) in each outer iteration (power method), we use the one-node block inversion (NBI) scheme. We show some numerical results to a typical model problem to illustrate the method's accuracy in coarse-mesh calculations.

A new algorithm for the frequency offset in Doppler data

Orbit improvement with Doppler data is confronted with the systematic errors caused by the frequency offset of satellite beacon and station receiver. How to estimate the frequency offset of each topocentric passage is a key problem in evaluating the accuracy of orbit determination. The method of computing the frequency offset with the Guier navigation proposed in Ref.[1] has been proved in practice to be effective. However, calculations of the Guier navigation are rather complex, so it is difficult to use this algorithm in setting up the software for computing ephemerides by numerical integration. In this paper, a new algorithm based on the block inversions of the coefficient matrix of the normal equation is proposed. This algorithm merely requires the inversion of a 6 × 6 or 7 × 7 matrix, and then it is quite simple to calculate the frequency offset. As shown by simulations, the accuracy of orbit determination with the new algorithm is close to the Cramer-Rao lower bound, i.e., the theoretically optimal accuracy, and it is higher by about 15% than that of the Guier navigation algorithm.

Use of block inversion in the 2-D interpretation of apparent resistivity data and its comparison with smooth inversion

The ability of a block inversion scheme, in which polygons are employed to define layers and/or bodies of equal resistivity, in determining the geometry and true resistivity of subsurface structures has been investigated and a simple strategy for deriving the starting model is proposed. A comparison has also been made between block inversion and smooth inversion, the latter being a cell-based scheme. The study entailed the calculation (by forward modelling) of the synthetic data over 2-D geologic models and inversion of the data. The 2-D structures modelled include vertical fault, graben and horst. The Wenner array was used. The results show that the images obtained from smooth inversion are very useful in determining the geometry; however, they can only provide guides to the true resistivity because of the smearing effects. It is shown that the starting model for block inversion can be based on a plane layer earth model. In the presence of sharp, rather than gradational, resistivity discontinuities, the model from block inversion more adequately represents the true subsurface geology, in terms of both the geometry and the formation resistivity. Field examples from a crystalline basement area of Nigeria are presented to demonstrate the versatility of the two resistivity inversion schemes.

Global Images of the Upper 200 km in the Earth by Short-Period Surface Waves.

A global shallow Earth structure is retrieved from short-period surface wave data (40-200 s) using a data set assembled from global digital networks. Phase velocity maps for both Love and Rayleigh waves are retrieved by spherical harmonic inversion. A comparison with results by block inversion is also presented, and the similarity of the results by the two different methods supports the belief that the data are capable of resolving most details in final velocity maps. Strong effects of crustal structure are evident in the patterns of short-period Love wave phase velocity variations, although depth resolution kernels show that the complete resolution of crustal structure is not possible from this period range of data. Two models of S-wave velocity maps are presented, one with an assumed crustal structure and topography and the other without such an assumption. Major differences between the two models occur just below the moho; the one with an assumed crustal structure shows the beginning of tectosphere-like structure immediately below the moho under cratons, while the model without it shows the beginning depths of tectosphere-like structure between 50 and 80 km. It is obvious that the short-period cut-off in this study (40 s) is not sufficiently short to resolve shallow structures in detail. A few interesting tectonic features in the models include: 1) S-wave velocity under ridge axes is slow in the upper 200 km and is very slow in the upper 100-120 km, due possibly to partial melting at shallow depths; and 2) depth slices under the Himalaya-Tibet region show fast velocity anomaly extending from India to the underside of Tibet. While it is difficult to distinguish between a subducted oceanic lithosphere and a thickened continental lid, the shape of the contour lines seems to favor the interpretation that this fast velocity anomaly is the underthrusting Indian ocean lithosphere.

The physical map of the chromosome of a serogroup A strain of Neisseria meningitidis shows complex rearrangements relative to the chromosomes of the two mapped strains of the closely related species N. gonorrhoeae.

A physical map of the chromosome of N. meningitidis Z2491 (serogroup A, subgroup IV-1) has been constructed. Z2491 DNA was digested with NheI, SpeI, SgfI, PacI, BglII, or PmeI, resulting in a limited number of fragments that were resolved by contour-clamped homogeneous electric field (CHEF) electrophoresis. The estimated genome size for this strain was 2,226 kb. To construct the map, probes corresponding to single-copy genes or sequences were used on Southern blots of chromosomal DNA digested with the different mapping enzymes and subjected to CHEF electrophoresis. By determining which fragments from different digests hybridized to each specific probe, it was possible to walk back and forth between digests to form a circular macrorestriction map. The intervals between mapped restriction sites range from 10 to 143 kb in size. A total of 117 markers have been placed on the map; 75 represent identified genes, with the remaining markers defined by anonymous cloned fragments of neisserial DNA. Comparison of the arrangement of genetic loci in Z2491 with that in gonococcal strain FA1090, for which a physical map was previously constructed, revealed complex genomic rearrangements between the two strains. Although gene order is generally conserved over much of the chromosome, a region of approximately 500 kb shows translocation and/or inversion of multiple blocks of markers between the two strains. Even within the relatively conserved portions of the maps, several genetic markers are in different positions in Z2491 and FA1090.

The Geoseismic Character of the Rift and Fault-Subsident Continental Basins of Eastern Mongolia

The geoseismic interpretation of seismic lines from the East Govi, Tamtsag, and Choibalsan basins of eastern Mongolia illustrates basin development from typical rift geometry through fault subsidence, to a phase of wrench structuration and fault-block inversion. These eastern Mongolian basins are located in an accretion zone to the south of the Siberian craton. They are close to a number of similar Chinese intraplate basins. In particular, the geology of the Songliao basin is well documented and the Mongolian basins appear to have a similar geological history. The good-quality seismic response allows identification of the following features characteristic of different phases of the evolution of the basins: (1) rift fill sequences with typical chaotic to divergent seismic configuration on the edges of V-shaped grabens, (2) the oblique and sigmoidal internal seismic configuration of a prograding delta system, (3) parallel seismic reflectors in the center of the subsiding basins (By analogy with the adjacent Chinese basins these sediments are thought to consist of low-energy lacustrine sediments. In the Songliao basin these deposits are important source beds and it is likely that they represent the source of oils known in the East Govi basin), (4) deep fossil river erosion and subsequent infill, (5)more » wrench-type fault patterns, associated intrusions, and inversion of fault blocks, and (6) limited Neogene subsidence as a result of uplift due to interplate collision.« less

On the stability of P-wave tomography at Loma Prieta: A comparison of parameterizations, linear and nonlinear inversions

We investigate the stability of tomographic analysis by comparing the results of two different methods of parameterizing the three-dimensional P -wave velocity variations in the vicinity of the 1989 Loma Prieta earthquake. A block inversion is implemented using 55 by 45 by 10 blocks of 1 km width and varying thickness to a depth of 25 km below the surface. Linear and nonlinear analysis are presented. The nonlinear analysis is achieved by iterating over three-dimensional raytracing and earthquake relocation relative to current three-dimensional models until solutions show only small improvements. A second parameterization is achieved by using cubic B-spline functions to span the space of the model, which is rotated by 46.5°. Nonlinear results are presented with several different starting models illustrating the robustness of the technique to the initial conditions. All the nonlinear results produced essentially the same final model, which was structurally the same as the model obtained by linear analysis using a reasonable starting model.

A Spectral Nodal Method for One-GroupX, Y-Geometry Discrete Ordinates Problems

This paper reports on a nodal method that is developed for the solution of one-group discrete ordinates (S{sub N}) problems with linearly anisotropic scattering in x, y-geometry. In this method, the spectral Green's function (SGF) scheme, originally developed for solving S{sub N} problems in slab geometry with no spatial truncation error, is generalized to solve the one-dimensional transverse-integrated S{sub N} nodal equations with the constant approximation for the transverse leakage terms. The resulting SGF-constant nodal (SGF-CN) method is more accurate than conventional coarse-mesh methods for deep penetration problems because it treats the scattering source terms implicitly and exactly; the only approximation involves the transverse leakage terms. In conventional S{sub N} nodal methods, the transverse leakage terms and scattering source are both approximated. The authors solve the SGF-CN equations using the one-node block inversion iterative scheme, which uses the best available estimates for the node-entering fluxes to evaluate the node-exiting fluxes in the directions that constitute the incoming fluxes for the adjacent nodes as the equations are swept across the system. Finally, the authors give numerical results that illustrate the accuracy of the SGF-CN method for coarse-mesh calculations.

Decomposition of electron density matrix and total energy of supermolecule in the method of non-orthogonal localized molecular orbitals

The decomposition of the electron density matrix and the total energy of a supermolecule in the one-configuration approximation of the non-orthogonal localized molecular orbitals (NLMO) method based on the Frobenius block inversion formulae of the NLMO overlap matrix for two and many interacting subsystems was considered in terms of reduced one-particle density matrices of the subsystems. Formulae were derived which can be used as a theoretical basis for a qualitative and quantitative analysis of the coulombic, induction and exchange components of the intermolecular and intramolecular interaction energies and for the development of models of such interactions. This approach is related to the decomposition method developed by Morokuma and Umeyama, and to the method of non-orthogonal group functions of McWeeny and Dacre.

Multigrid and upwind viscous flow solver on three-dimensional overlapped and embedded grids

A computationally efficient method is presented for solving the three-dimensional governing equations of the viscous compressible flows about complex configurations with geometrically nonsimilar components. The physical domain is decomposed into regions for which the grid generation is relatively simple and virtually without significant restrictions. The Navier-Stokes equations are solved by an implicit, upwind, finite-volume scheme. The convergence is accelerated by a multigrid algorithm despite the holes created in the computational domain by the overlapped grids, which are completely or partially embedded within one another. The block inversions and the diagonalized scalar inversions of the coefficient matrices are modified to allow the existence of such holes. The method is tested through two demonstrative csises. The solution for a supersonic flow past a blunt-nose cylinder at a high angle of attack is obtained using a C-O grid embedded in a global Cartesian grid. This solution is successfully compared with the one obtained using a single C-O grid and the experimental data. Then, supersonic interference flows past an ogive-nose cylinder in the close proximity of a flat plate are solved using overlapped and topologically nonsimilar but simple grids.

Global Love wave phase velocity variation and its significance to plate tectonics

Global Love wave phase velocity variation was constructed for periods between 80 and 200 s by using approximately 9000 paths from 971 earthquakes (with M ≥ 5.5). The data set was from GDSN and GEOSCOPE networks between 1980 and 1987. We examined both the spherical harmonic expansion method and the block parameterization method. With a simple, constant damping parameter approach, synthetic tests showed that more accurate results were obtained by the block parameterization method than by the spherical harmonic expansion method. We adopted the block inversion method with a (nearly) equal area block (5° × 5° near the equator) discretization. The general pattern of the resulting maps were consistent with previous global and local studies. With a discretization of 5° by 5°, the maps were detailed enough to test some plate tectonic models. For the Pacific, Atlantic and Indian Oceans, surface-wave velocities increased smoothly to plate ages older than 100 Ma. Simple forward modeling showed that seismic phase velocity variation with a continuous thickening of lithosphere up to about 150 Ma fits the present observation, disagreeing with the model deduced from the heat flow and ocean depth data, which change variations at about 60–80 Ma. The seismic phase velocity variations in different oceans showed systematic differences at younger ages, and convergence beyond 100 Ma. The difference at younger ages implies a failure of scaling derived from a simple thermal boundary layer model for oceanic plates. Age-seismic phase velocity relationships on each side of ridges were also examined and asymmetric velocity variations were found, which suggests differences in thermal states from one side of the ridge to the other. In order to further examine the thermal state of the lithosphere, average age-phase velocity relations were established for each ocean, and subtracted from phase velocity variation maps. The results indicated broad, low-velocity regions in the south Pacific (super-swell region), the south and west Indian Ocean, and high-velocity regions east of the East Pacific Rise and in the north to northeast Indian Ocean. The results reflect the asymmetry of phase velocity variation about ridges. There is some correlation between hot-spot locations and low-velocity anomalies, but additional, large-scale thermal anomalies exist under old oceanic plate.

The Rio Grande rift: new electromagnetic constraints on the Socorro magma body

The Socorro ‘magma body’ in central New Mexico is a prime example of an intrusive magma body associated with continental extension in one of the major active rift zones of the world: the Rio Grande rift. To study this feature further, in particular its relation to large-scale features delineated elsewhere beneath the rift, in 1988 Brown University occupied a sequence of densely spaced (5–10 km), wide-band (10–4000 s) remote referenced, three-component magnetic variation sites along an east-west line which transected the southern margin of the magma body. Our profile was deployed so as to cross precisely above the center of what has been the most seismically active portion of the structure southwest of Socorro. Complex magnetic transfer coefficients at up to 15 periods from 21 sites along our profile have been inverted to determine two-dimensional conductivity models for vertical cross-sections transverse to the strike of the rift. Two independent generalized inverse algorithms are used; an overdetermined block inversion and a mixed-determined ‘flat’ continuous resistivity inversion. Both inversions require the presence of a conducting zone beneath the rift at intermediate depths (15–20 km) in the crust. This feature has properties which are similar to those of the intracrustal conductor seen elsewhere along the rift. In addition to this conducting zone in the deeper crust, the data require a more conductive core at shallow depth (10–15 km) which appears to be associated with the base of a zone of increased microseismicity described by Sanford and his colleagues. This latter feature might be a zone of diffuse magmatic intrusions, enhanced hydrothermal circulation, or a combination of both. Finally, our two inversions suggest that the resistivity structure beneath the western end of our profile, which is physiographically adjacent to the Colorado Plateau, has properties which are similar to those reported for interior regions of the Plateau.

島弧の深部構造

Three-dimensional velocity structure beneath island arcs, especially beneath Japan arc, has been investigated in detail by many studies in the last dacade. The low-velocity zone beneath active volcanoes and the high-velocity slab have been delineated by using the block inversion technique. The location of the upper boundary of the descending slab has been estimated precisely by the use of some kinds of converted or reflected waves; the boundary has been located just above the deep seismic zone. In some regions, an oblique thin low-velocity layer has been found on the descending high-velocity slab.

A Multigrid Acceleration Method for the One-DimensionalSNEquations with Anisotropic Scattering

In this paper a multigrid acceleration scheme for the one-dimensional slab geometry S/sub n/ equations with anisotropic scattering and linear discontinuous spatial differencing is developed. The high-frequency relaxation iteration consists of three steps: a standard source iteration, independent two-cell block inversions centered about each spatial cell edge, and an averaging of the iterates from the previous two steps. Because the linear discontinuous differencing scheme is a finite element method, fine-to-coarse projection and coarse-to-fine interpolation are straightforward.

A Linear- Eddy Model of Turbulent Scalar Transport and Mixing

Abstract Transport and mixing of diffusive scalars in turbulent flows are simulated computationally based on a novel representation of the temporal evolution along a transverse line moving with the mean fluid velocity. The scalar field along this line evolves by Fickian diffusion, representing molecular processes, and by randomly occurring events called block inversions. Block inversion, representing the effect of turbulent convection, consists of the random selection of an interval (Y0 − 1/2, Y0 + 1/2) of the line, where the interval size l may he either fixed or randomly selected, and replacement of the scalar field θ(y) within that interval by θ(2y0 For fixed l, the model requires a single input parameter, the Peclet number. To demonstrate the performance of the model, this formulation is used to compute the spatial development of diffusive scalar fields downstream of several source configurations in homogeneous turbulence. Generalization to inhomogeneous turbulence is discussed, as well as a formulati...

Block-inversion-coded QAM systems

A novel block inversion code (BIC) which is both simple and effective in controlling low-frequency components of multilevel data signals is introduced, and its power spectrum is analyzed. The BIC's effect on the performance of QAM modems is studied, and an expression for the coded modem's symbol error rate is obtained. The code allows M-ary QAM system designs to incorporate unmodulated carrier-pilot insertion techniques using high-pass filters, for more robust, unambiguous carrier recovery and automatic gain control. The effect of BIC coding on the performance of a 256 QAM modem with carrier-pilot insertion using high-pass filters is investigated. It is found that, under some circumstances, BIC coding can significantly reduce the ill-effect of the high-pass filter on the modem's performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

3-D velocity structure beneath France in different frequency bands

The 3-D velocity structure beneath France is investigated by using a method based upon teleseismic waveform analysis and a block inversion technique. In this method, the P-wave travel time is computed exactly in cach block for a spherical Earth model and the P-wave arrival times are obtained by cross-correlating the waveforms observed in 26 stations of the LDG network. One image is obtained directly from the observed signals, the other two are obtained from low-pass filtered waveforms with 0.5 and 0.25 Hz cut-off frequencies. The results show that the 3-D velocity image depends weakly but significantly on the cut-off frequency of the low-pass filter. In addition to possible changes in the signal-to-noise ratio, this may indicate a dependence of the tomographic images on the seismic wavelength. The best fit to the data is obtained from the 0.5 Hz image. The main features of this image are a clear indication of a lithospheric root beneath Switzerland, a general large scale W-E variation of velocity within the lithosphere and a diffuse low-velocity area beneath the French Massif Central and the western part of the Alps in the 100–250 km layer.

Basis factorization method for multistage linear programming problems with an application to optimal energy plant operation

An algorithm based on the basis factorization method for solving multistage linear programming problems is presented. In applying linear programming for optimal dynamic-plant operation, the constraint appears in the form of a staircase structure. When pivotings occur within each stage, the inversion of the basis matrix is easily obtained using inversions of small square diagonal blocks. In general cases, pivotings across stages occur. However, this basis matrix can be easily transformed into the former square block triangular form using the correction factor. The correction factor means multiple pivotings among the bases and contains structural columns corresponding to different columns among these bases. When these structural columns are maintained in computer storage, the relative cost factor for general bases is easily obtained using a set of elementary column operations. A program code using the above idea is developed and applied to in-plant energy operating systems. A computational example shows that the computing time of the proposed algorithm is less than 10 percent of the general purpose LP code MPS-II.

Elastic wave velocity structure of the crust and upper mantle beneath the North China Basin

Travel time residuals from locally and teleseismically recorded earthquakes are used to determine the three‐dimensional elastic wave velocity structure beneath the central North China basin and surrounding mountains. We calculated P wave and S wave travel time residuals for 200 aftershocks of the 1976 Tangshan earthquake and P wave travel time residuals for 194 teleseisms recorded by the Beijing network between 1976 and 1983. We performed iterative one‐ and three‐dimensional block inversions for crust and uppermost mantle structure based solely on the aftershock travel time residuals. We also performed iterative three‐dimensional block inversions for crust and upper mantle structure based solely on the teleseismic travel time residuals and based on the combined aftershock and teleseismic travel time residuals. We determined that the average thickness of the crust beneath the North China basin and surrounding mountains is about 35 km. Beneath the onshore portion of the North China basin, however, the crust is closer to 32 or 33 km thick. The velocity structure of the upper 20 km or so of the crust is well correlated with the surface geology and parallels the short‐wavelength (a few tens of kilometers) features of the surface topography. Rocks with low seismic velocities lie in or beneath the basin; rocks with high seismic velocities lie in or beneath the mountains. The uppermost mantle, and possibly even the lower crust, appears to be a zone of transition in velocity structure between that of the upper crust and the mantle. The trends of relative highs and lows in the velocity structure of the mantle are nearly opposite that of the crust. The mantle material underlying most of the North China basin has higher than average seismic velocities, while the material beneath the western mountains has lower than average velocities. The velocity structure of the mantle parallels the long‐wavelength (tens to hundreds of kilometers) features evident in the gravity and topography of North China. The high‐velocity mantle material presently beneath the basin probably represents mantle processes not directly related to the formation of the North China basin. We infer that the transition zone between the short‐wavelength and long‐wavelength velocity features represents a zone of decoupling between the lithosphere and asthenosphere. Specifically, the lithosphere beneath the North China block is moving eastward with respect to the asthenosphere.