Reduced-Basis Methods for Inverse Problems in Partial Differential Equations

We present a technique for the rapid, reliable, and accurate evaluation of functional outputs of parametrized elliptic partial differential equations. The essential ingredients are (i) rapidly globally convergent reduced-basis approximations – Galerkin projection onto a space WN spanned by the solutions of the governing partial differential equations at N selected points in parameter space; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide sharp and inexpensive bounds for the error in the output of interest; and (iii) off-line/online computational procedures – methods that decouple the generation and projection stages of the approximation process. The operation count for the online stage – in which, given a new parameter, we calculate the output of interest and associated error bounds – depends only on N (typically very small) and the parametric dependencies of the problem. In this study, we first develop rigorous a posteriori error estimators for (affine in the parameter) noncoercive problems such as the Helmholtz (reduced-wave) equation. The critical ingredients are the residual, an appropriate bound conditioner, and a piecewise-constant lower bound for the inf-sup stability factor. In addition, globally nonaffine (and nonlinear) problems are also considered: in particular, through appropriate sampling and interpolation procedures, these more difficult problems can be reduced (with very high accuracy) to the more tractable affine case. Finally, we propose a real-time - procedure for inverse problems associated with parametrized partial differential equations based on our reduced-basis approximations and error bounds. In general practice, many inverse problems are formulated as an error minimization statement relating the calculated and measured outputs. This optimization procedure requires many evaluations of the output: the reduced-basis method --- with extremely low marginal cost --- is thus very efficient for this class of problems. As an illustrative example, we consider a very important application in nondestructive evaluation: crack identification (by harmonic excitation) in a laminated plate of composite material. The numerical results demonstrate the efficiency and accuracy of the method in detecting the location and length of the crack.Singapore-MIT Alliance (SMA

Supergravities with Minkowski x Sphere Vacua

Recently the authors have introduced a new gauged supergravity theory with a positive definite potential in D=6, obtained through a generalised Kaluza-Klein reduction from D=7. Of particular interest is the fact that this theory admits certain Minkowski x Sphere vacua. In this paper we extend the previous results by constructing gauged supergravities with positive definitive potentials in diverse dimensions, together with their vacuum solutions. In addition, we prove the supersymmetry of the generalised reduction ansatz. We obtain a supersymmetric solution with no form-field fluxes in the new gauged theory in D=9. This solution may be lifted to D=10, where it acquires an interpretation as a time-dependent supersymmetric cosmological solution supported purely by the dilaton. A further uplift to D=11 yields a solution describing a pp-wave.Comment: Latex, 26 pages, typos correcte

BPS Geometries and AdS Bubbles

Recently, 1/2-BPS AdS bubble solutions have been obtained by Lin, Lunin and Maldacena, which correspond to Fermi droplets in phase space in the dual CFT picture. They can be thought of as generalisations of 1/2-BPS AdS black hole solutions in five or seven dimensional gauged supergravity. In this paper, we extend these solutions by invoking additional gauge fields and scalar fields in the supergravity Lagrangians, thereby obtaining AdS bubble generalisations of the previously-known multi-charge AdS black solutions of gauged supergravity. We also obtain analogous AdS bubble solutions in four-dimensional gauged supergravity. Our solutions generically preserve supersymmetry fractions 1/4, 1/8 and 1/8 in seven, five and four dimensions respectively. They can be lifted to M-theory or type IIB string theory, using previously known formulae for the consistent Pauli sphere reductions that yield the gauged supergravities. We also find similar solutions in six-dimensional gauged supergravity, and discuss their lift to the massive type IIA theory.Comment: Latex, 11 page

Three-dimensional character of the deformation twin in magnesium

Deformation twins are three-dimensional domains, traditionally viewed as ellipsoids because of their two-dimensional lenticular sections. In this work, we performed statistical analysis of twin shapes viewing along three orthogonal directions: the ‘dark side’ (DS) view along the twin shear direction (η1), the twinning plane normal (TPN) view (k1) and the ‘bright side’ (BS) view along the direction λ(=k1 × η1). Our electron back-scatter diffraction results show that twins in the DS and BS views normally exhibit a lenticular shape, whereas they show an irregular shape in the TPN view. Moreover, the findings in the TPN view revealed that twins grow faster along λ the lateral direction than along η1 the forward propagation direction at the initial stages of twin growth. These twin sections are irregular, indicating that growth is locally controlled and the overall shape is not perfectly ellipsoidal. We explain these findings using atomistic models, and ascribe them to differences in the mobility of the edge and screw components of the twinning dislocations

Generation of defects and disorder from deeply quenching a liquid to form a solid

We show how deeply quenching a liquid to temperatures where it is linearly unstable and the crystal is the equilibrium phase often produces crystalline structures with defects and disorder. As the solid phase advances into the liquid phase, the modulations in the density distribution created behind the advancing solidification front do not necessarily have a wavelength that is the same as the equilibrium crystal lattice spacing. This is because in a deep enough quench the front propagation is governed by linear processes, but the crystal lattice spacing is determined by nonlinear terms. The wavelength mismatch can result in significant disorder behind the front that may or may not persist in the latter stage dynamics. We support these observations by presenting results from dynamical density functional theory calculations for simple one- and two-component two-dimensional systems of soft core particles.Comment: 25 pages, 11 figure

Randall-Sundrum Brane Tensions

We show that the singular sources in the energy-momentum tensor for the Randall-Sundrum brane world, viewed as a solution of type IIB supergravity, are composed of two elements. One of these is a D3-brane source with tension opposite in sign to the RS tension in five dimensions; the other arises from patching two regions of flat ten-dimensional spacetime. This resolves an apparent discrepancy between supersymmetry and the sign and magnitude of the RS tension.Comment: Latex, 21 pages, 2 figure

ESSVCS: an enriched secret sharing visual cryptography

Visual Cryptography (VC) is a powerful technique that combines the notions of perfect ciphers and secret sharing in cryptography with that of raster graphics. A binary image can be divided into shares that are able to be stacked together so as to approximately recover the original image. VC is a unique technique in the sense that the encrypted message can be decrypted directly by the Human Visual System (HVS). The distinguishing characteristic of VC is the ability of secret restoration without the use of computation. However because of restrictions of the HVS, pixel expansion and alignment problems, a VC scheme perhaps can only be applied to share a small size of secret image. In this paper, we present an Enriched Secret Sharing Visual Cryptography Scheme (ESSVCS) to let the VC shares carry more secrets, the technique is to use cypher output of private-key systems as the input random numbers of VC scheme, meanwhile the encryption key could be shared, the shared keys could be associated with the VC shares. After this operation, VC scheme and secret sharing scheme are merged with the private-key system. Under this design, we implement a (k; t; n)-VC scheme. Compared to those existing schemes, our scheme could greatly enhance the ability of current VC schemes and could cope with pretty rich secrets

Nonlinear Volatility of River Flux Fluctuations

We study the spectral properties of the magnitudes of river flux increments, the volatility. The volatility series exhibits (i) strong seasonal periodicity and (ii) strongly power-law correlations for time scales less than one year. We test the nonlinear properties of the river flux increment series by randomizing its Fourier phases and find that the surrogate volatility series (i) has almost no seasonal periodicity and (ii) is weakly correlated for time scales less than one year. We quantify the degree of nonlinearity by measuring (i) the amplitude of the power spectrum at the seasonal peak and (ii) the correlation power-law exponent of the volatility series.Comment: 5 revtex pages, 6 page

Characterizing the boundary lateral to the shear direction of deformation twins in magnesium

The three-dimensional nature of twins, especially the atomic structures and motion mechanisms of the boundary lateral to the shear direction of the twin, has never been characterized at the atomic level, because such boundary is, in principle, crystallographically unobservable.We thus refer to it here as the dark side of the twin. Here, using high-resolution transmission electron microscopy and atomistic simulations, we characterize the dark side of {1012} deformation twins in magnesium. It is found that the dark side is serrated and comprised of {1012} coherent twin boundaries and semi-coherent twist prismatic–prismatic {2110} boundaries that control twin growth. The conclusions of this work apply to the same twin mode in other hexagonal close-packed materials, and the conceptual ideas discussed here should hold for all twin modes in crystalline materials

Characterizing the boundary lateral to the shear direction of deformation twins in magnesium

The three-dimensional nature of twins, especially the atomic structures and motion mechanisms of the boundary lateral to the shear direction of the twin, has never been characterized at the atomic level, because such boundary is, in principle, crystallographically unobservable.We thus refer to it here as the dark side of the twin. Here, using high-resolution transmission electron microscopy and atomistic simulations, we characterize the dark side of {1012} deformation twins in magnesium. It is found that the dark side is serrated and comprised of {1012} coherent twin boundaries and semi-coherent twist prismatic–prismatic {2110} boundaries that control twin growth. The conclusions of this work apply to the same twin mode in other hexagonal close-packed materials, and the conceptual ideas discussed here should hold for all twin modes in crystalline materials