Rectangular Full Packed Format for Cholesky's Algorithm: Factorization, Solution and Inversion

We describe a new data format for storing triangular, symmetric, and Hermitian matrices called RFPF (Rectangular Full Packed Format). The standard two dimensional arrays of Fortran and C (also known as full format) that are used to represent triangular and symmetric matrices waste nearly half of the storage space but provide high performance via the use of Level 3 BLAS. Standard packed format arrays fully utilize storage (array space) but provide low performance as there is no Level 3 packed BLAS. We combine the good features of packed and full storage using RFPF to obtain high performance via using Level 3 BLAS as RFPF is a standard full format representation. Also, RFPF requires exactly the same minimal storage as packed format. Each LAPACK full and/or packed triangular, symmetric, and Hermitian routine becomes a single new RFPF routine based on eight possible data layouts of RFPF. This new RFPF routine usually consists of two calls to the corresponding LAPACK full format routine and two calls to Level 3 BLAS routines. This means {\it no} new software is required. As examples, we present LAPACK routines for Cholesky factorization, Cholesky solution and Cholesky inverse computation in RFPF to illustrate this new work and to describe its performance on several commonly used computer platforms. Performance of LAPACK full routines using RFPF versus LAPACK full routines using standard format for both serial and SMP parallel processing is about the same while using half the storage. Performance gains are roughly one to a factor of 43 for serial and one to a factor of 97 for SMP parallel times faster using vendor LAPACK full routines with RFPF than with using vendor and/or reference packed routines

Green's function multiple-scattering theory with a truncated basis set: An Augmented-KKR formalism

Korringa-Kohn-Rostoker (KKR) Green's function, multiple-scattering theory is an efficient site-centered, electronic-structure technique for addressing an assembly of $N$ scatterers. Wave-functions are expanded in a spherical-wave basis on each scattering center and indexed up to a maximum orbital and azimuthal number $L_{max}=(l,m)_{max}$, while scattering matrices, which determine spectral properties, are truncated at $L_{tr}=(l,m)_{tr}$ where phase shifts $\delta_{l>l_{tr}}$ are negligible. Historically, $L_{max}$ is set equal to $L_{tr}$; however, a more proper procedure retains free-electron and single-site contributions for $L_{max}>L_{tr}$ with $\delta_{l>l_{tr}}$ set to zero [Zhang and Butler, Phys. Rev. B {\bf 46}, 7433]. We present a numerically efficient and accurate \emph{augmented}-KKR Green's function formalism that solves the KKR secular equations by matrix inversion [$\mathcal{R}^3$ process with rank $N(l_{tr}+1)^2$] and includes higher-order $L$ contributions via linear algebra [$\mathcal{R}^2$ process with rank $N(l_{max}+1)^2$]. Augmented-KKR yields properly normalized wave-functions, numerically cheaper basis-set convergence, and a total charge density and electron count that agrees with Lloyd's formula. For fcc Cu, bcc Fe and L$1_0$ CoPt, we present the formalism and numerical results for accuracy and for the convergence of the total energies, Fermi energies, and magnetic moments versus $L_{max}$ for a given $L_{tr}$.Comment: 7 pages, 5 figure

Fractal Weyl law for chaotic microcavities: Fresnel's laws imply multifractal scattering

We demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities. As an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium. We show that the conjectured fractal Weyl law for open chaotic systems [W. T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)] is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating Fresnel's laws.Comment: 8 pages, 12 figure

Generalized contour deformation method in momentum space: two-body spectral structures and scattering amplitudes

A generalized contour deformation method (GCDM) which combines complex rotation and translation in momentum space, is discussed. GCDM gives accurate results for bound, virtual (antibound), resonant and scattering states starting with a realistic nucleon-nucleon interaction. It provides a basis for full off-shell $t$-matrix calculations both for real and complex input energies. Results for both spectral structures and scattering amplitudes compare perfectly well with exact values for the separable Yamaguchi potential. Accurate calculation of virtual states in the Malfliet-Tjon and the realistic CD-Bonn nucleon-nucleon interactions are presented. GCDM is also a promising method for the computation of in-medium properties such as the resummation of particle-particle and particle-hole diagrams in infinite nuclear matter. Implications for in-medium scattering are discussed.Comment: 15 pages, revte

Tomographic inversion using ℓ1\ell_1-norm regularization of wavelet coefficients

We propose the use of $\ell_1$ regularization in a wavelet basis for the solution of linearized seismic tomography problems $Am=d$, allowing for the possibility of sharp discontinuities superimposed on a smoothly varying background. An iterative method is used to find a sparse solution $m$ that contains no more fine-scale structure than is necessary to fit the data $d$ to within its assigned errors.Comment: 19 pages, 14 figures. Submitted to GJI July 2006. This preprint does not use GJI style files (which gives wrong received/accepted dates). Corrected typ

Physical properties and small-scale structure of the Lyman-alpha forest: Inversion of the HE 1122-1628 UVES spectrum

We study the physical properties of the Lyman-alpha forest by applying the inversion method described by Pichon et al. (2001) to the high resolution and high S/N ratio spectrum of the z_em=2.40 quasar HE 1122-1628 obtained during Science Verification of UVES at the VLT. We compare the column densities obtained with the new fitting procedure with those derived using standard Voigt profile methods. The agreement is good and gives confidence in the new description of the Lyman-alpha forest as a continuous field as derived from our method. We show that the observed number density of lines with logN>13 and 14 is, respectively, 50 and 250 per unit redshift at z~2. We study the physical state of the gas, neglecting peculiar velocities, assuming a relation between the overdensity and the temperature. T=Tbar * rho^(2beta). There is an intrinsic degeneracy between the parameters beta and Tbar. We demonstrate that, at a fixed beta, the temperature at mean density, Tbar, can be uniquely extracted however. While applying the method to HE 1122-1628, we conclude that for 0.2<beta<0.3, 6000<Tbar<15000 K at z~2. We investigate the small scale structure of strong absorption lines using the information derived from the Lyman-beta, Lyman-gamma and Civ profiles. Introducing the Lyman-beta line in the fit allows us to reconstruct the density field up to rho~10 instead of 5 for the Lyman-alpha line only. There may be small velocity shifts ~10km.s^{-1} between the peaks in the Civ and Hi density profiles. Although the statistics is small, it seems that Civ/Hi and n_HI are anti-correlated. This could be a consequence of the high sensitivity of the Civ/Hi ratio to temperature. The presence of associated Ovi absorption, with similar profile, confirms that the gas is photo-ionized and at a temperature of T~10^5 K.Comment: 15 pages, 21 figures, accepted for publication in A&A. Quoted results from other papers in section 4.2 have been modifie

Phases in the gaugino sector: direct reconstruction of the basic parameters and impact on the neutralino pair production

We consider recovering analytically the (generally complex) parameters $\mu$, $M_1$ and $M_2$ of the gaugino and Higgsino Lagrangian, from appropriate physical input in the chargino and neutralino sectors. For given $\tan\beta$, we obtain very simple analytic solutions for $M_2$, $| \mu|$, $Arg[\mu]$ in the chargino sector and a twofold $| M_1 |$, $Arg[M_1]$ analytic solution in the neutralino sector, assuming two chargino, two neutralino masses, and one of the chargino mixing angles as physical input. The twofold ambiguity in the neutralino parameters reconstruction may be essentially resolved by measuring the $e^+e^- \to \chi^0_1 \chi^0_2$ production cross-section at future linear collider energies, which we study explicitly with the phase dependences. Some salient features and specific properties of this complex case gaugino "spectrum inversion" are illustrated and compared with the similar inversion in the real case. In particular, our algorithms exhibit in a direct and transparent way the non-trivial theoretical correlation among the chargino and neutralino parameters, and the resulting allowed domains when only a subset of the required physical input masses and production cross-sections is known.Comment: Latex, 28 pages, 10 figure

Infinite Products of Large Random Matrices and Matrix-valued Diffusion

We use an extension of the diagrammatic rules in random matrix theory to evaluate spectral properties of finite and infinite products of large complex matrices and large hermitian matrices. The infinite product case allows us to define a natural matrix-valued multiplicative diffusion process. In both cases of hermitian and complex matrices, we observe an emergence of "topological phase transition" in the spectrum, after some critical diffusion time $\tau_{\rm crit}$ is reached. In the case of the particular product of two hermitian ensembles, we observe also an unusual localization-delocalization phase transition in the spectrum of the considered ensemble. We verify the analytical formulae obtained in this work by numerical simulation.Comment: 39 pages, 12 figures; v2: references added; v3: version to appear in Nucl. Phys.

Characterisation of the transmissivity field of a fractured and karstic aquifer, Southern France

International audienceGeological and hydrological data collected at the Terrieu experimental site north of Montpellier, in a confined carbonate aquifer indicates that both fracture clusters and a major bedding plane form the main flow paths of this highly heterogeneous karst aquifer. However, characterising the geometry and spatial location of the main flow channels and estimating their flow properties remain difficult. These challenges can be addressed by solving an inverse problem using the available hydraulic head data recorded during a set of interference pumping tests.We first constructed a 2D equivalent porous medium model to represent the test site domain and then employed regular zoning parameterisation, on which the inverse modelling was performed. Because we aim to resolve the fine-scale characteristics of the transmissivity field, the problem undertaken is essentially a large-scale inverse model, i.e. the dimension of the unknown parameters is high. In order to deal with the high computational demands in such a large-scale inverse problem, a gradient-based, non-linear algorithm (SNOPT) was used to estimate the transmissivity field on the experimental site scale through the inversion of steady-state, hydraulic head measurements recorded at 22 boreholes during 8 sequential cross-hole pumping tests. We used the data from outcrops, borehole fracture measurements and interpretations of inter-well connectivities from interference test responses as initial models to trigger the inversion. Constraints for hydraulic conductivities, based on analytical interpretations of pumping tests, were also added to the inversion models. In addition, the efficiency of the adopted inverse algorithm enables us to increase dramatically the number of unknown parameters to investigate the influence of elementary discretisation on the reconstruction of the transmissivity fields in both synthetic and field studies.By following the above approach, transmissivity fields that produce similar hydrodynamic behaviours to the real head measurements were obtained. The inverted transmissivity fields show complex, spatial heterogeneities with highly conductive channels embedded in a low transmissivity matrix region. The spatial trend of the main flow channels is in a good agreement with that of the main fracture sets mapped on outcrops in the vicinity of the Terrieu site suggesting that the hydraulic anisotropy is consistent with the structural anisotropy. These results from the inverse modelling enable the main flow paths to be located and their hydrodynamic properties to be estimated