on the stability of extensions of tangent sheaves on kahler einstein fano calabi yau pairs let s be a smooth projective variety and delta a simple normal crossing mathbb q divisor with coefficients in 0 1 for any ample mathbb q line bundle l over s we denote by mathscr e l the extension sheaf of the orbifold tangent sheaf t s log delta by the structure sheaf mathcal o s with the extension class c 1 l we prove the following two results these results generalize tian s result where k s is ample and delta emptyset we give two applications of these results the first is to study a question by borbon spotti about the relationship between local euler numbers and normalized volumes of log canonical surface singularities we prove that the two invariants differ only by a factor 4 when the log canonical pair is an orbifold cone over a marked riemann surface in particular we complete the computation of langer s local euler numbers for any line arrangements in mathbb c 2 the second application is to derive miyaoka yau type inequalities on k semistable log smooth fano pairs and calabi yau pairs which generalize some chern number inequalities proved by song wang

global holomorphic 2 forms and pluricanonical systems on threefolds the change of zero locus of a global holomorphic 2 form on a threefold under birational transformations is investigated it is proved that existence of 2 forms with certain conditions on their zero loci on a threefold of nonnegative kodaira dimension limits types of terminal singularities appearing on its minimal models as a result of the restriction on the types of terminal singularities and reid s riemann roch formula a universal bound n is found such that the linear system nk defines a birational map from a threefold of general type admitting those 2 forms where k is the canonical bundle of the threefold

closed embeddings of c in c 2 part i we consider closed curves c c in the affine plane s c2 that admit a good or very good asymptote by this we mean a curve l c in s that in suitable coordinates for s is linear and tangent to c at infinity and meets c once or not at all at finite distance we classify these curves up to automorphism of s relying on the theory of open algebraic surfaces we first determine the possibilities for the singularities of c at infinity and then proceed to give explicit equations

first order system least squares for the helmholtz equation this paper develops a multilevel least squares approach for the numerical solution of the complex scalar exterior helmholtz equation this second order equation is first recast into an equivalent first order system by introducing several field variables a combination of scaled l2 and h 1 norms is then applied to the residual of this system to create a least squares functional it is shown that in an appropriate hilbert space the homogeneous part of this functional is equivalent to a squared graph norm that is a product norm on the space of individual variables this equivalence to a norm that decouples the variables means that standard finite element discretization techniques and standard multigrid solvers can be applied to obtain optimal performance however this equivalence is not uniform in the wavenumber k which can signal degrading performance of the numerical solution process as k increases to counter this difficulty we obtain a result that characterizes the error components causing performance degradation we do this by defining a finite dimensional subspace of these components on whose orthogonal complement k uniform equivalence is proved for this functional and an analogous functional that is based only on l2 norms this subspace equivalence motivates a nonstandard multigrid method that attempts to achieve optimal convergence uniformly in k we report on numerical experiments that empirically confirm k uniform optimal performance of this multigrid solver we also report on tests of the error in our discretization that seem to confirm optimal accuracy that is free of the so called pollution effect

a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

surfaces of general type with p_g q 1 k 2 8 and bicanonical map of degree 2 we classify the minimal algebraic surfaces of general type with pg q 1 k 2 8 and bicanonical map of degree 2 it will turn out that they are isogenous to a product of curves i e if s is such a surface then there exist two smooth curves c f and a finite group g acting freely on c x f such that s c x f g we describe the c f and g that occur in particular the curve c is a hyperelliptic bielliptic curve of genus 3 and the bicanonical map of s is composed with the involution induced on s by r x id c x f c x f where is the hyperelliptic involution of c in this way we obtain three families of surfaces with pg q 1 k 2 8 which yield the first known examples of surfaces with these invariants we compute their dimension and we show that they are three generically smooth irreducible components of the moduli space m of surfaces with p g q 1 k 2 8 moreover we give an alternative description of these surfaces as double covers of the plane recovering a construction proposed by du val

an optimal 9 point finite difference frequency space 2 d scalar wave extrapolator in this study a new finite difference technique is designed to reduce the number of grid points needed in frequency space domain modeling the new algorithm uses optimal nine point operators for the approximation of the laplacian and the mass acceleration terms the coefficients can be found by using the steepest descent method so that the best normalized phase curves can be obtained this method reduces the number of grid points per wavelength to 4 or less with consequent reductions of computer memory and cpu time that are factors of tens less than those involved in the conventional second order approximation formula when a band type solver is used on a scalar machine

on block triangular preconditioned iteration methods for solving the helmholtz equation abstract to further improve the efficiency of solving the helmholtz equation in heterogeneous media with large wavenumber the krylov subspace methods incorporated with a class of inexact rotated block triangular preconditioners are presented to solve a block two by two linear system derived from the discrete helmholtz equation we further develop the eigenvalue properties of the preconditioned matrices to discuss the convergence of the corresponding preconditioned iteration methods the superiority of such preconditioned iteration methods is prominent according to the numerical results when comparing with other classical iteration methods we also investigate how the wavenumber influences the performance of the corresponding methods and it is shown that the iteration number of our proposed methods linearly increase with the wavenumber roughly furthermore the computational wave fields which conform the real physical law are exhibited to show the correctness of our proposed numerical modeling algorithm

three dimensional frequency domain full waveform inversion with an iterative solver with the acquisition of wide aperture seismic data sets full waveform inversion is an attractive method for deriving velocity models three dimensional implementations require an efficient solver for the wave equation computing 3d time harmonic responses with a frequency domain solver is complicated because a large linear system with negative and positive eigenvalues must be solved time domain schemes are an alternative nevertheless existing frequency domain iterative solvers with an efficient preconditioner are a viable option when full waveform inversion is formulated in the frequency domain an iterative solver with a multigrid preconditioner is competitive because of a high order spatial discretization numerical examples illustrated the efficiency of the iterative solvers three dimensional full waveform inversion was then studied in the context of deep water ocean bottom seismometer acquisition three dimensional synthetic data inversion results showed the behavior of full waveform inversion with respect to the initial model and the minimum frequency available in the data set results on a 3d real ocean bottom seismometer data set demonstrated the relevance of full waveform inversion especially to image the shallow part of the model

accelerating the shifted laplace preconditioner for the helmholtz equation by multilevel deflation many important physical phenomena can be described by the helmholtz equation we investigate to what extent the convergence of the shifted laplacian preconditioner for the helmholtz equation can be accelerated using deflation with multigrid vectors we therefore present a unified framework for two published algorithms the first deflates the preconditioned operator and requires no further preconditioning the second deflates the original operator and combines deflation and preconditioning in a multiplicative fashion we pursue two scientific contributions first we show using a model problem analysis that both algorithms cluster the eigenvalues the new and key insight here is that the near kernel of the coarse grid operator causes a limited set of eigenvalues to shift away from the center of the cluster with a distance proportional to the wave number this effect is less pronounced in the first algorithmic variant at the expense of a higher computational cost in the second contribution we quantify for the first time the large amount of reduction in cpu time that results from the clustering of eigenvalues and the reduction in iteration count we report to this end on the findings of an implementation in petsc on two and three dimensional problems with constant and variable wave number

generalisation of the bogomolov miyaoka yau inequality to singular surfaces the paper considers pairs x b where x is a normal projective surface over c and b is a q divisor whose coefficients are 1 or 1 1 m for some natural number m a log canonical singularity on such a pair is a quotient by a finite or infinite group so if x b has log canonical singularities the orbifold euler number eorb x b can be defined the main result is a bogomolov miyaoka yau type inequality which implies that if x b has log canonical singularities and x kx b 0 then kx b 2 3eorb x b the actual inequality proved is somewhat stronger and it also implies all the previously published versions of the bogomolov miyaoka yau inequality the proof involves the log minimal model program q sheaves when kx b is nef and a study of the changes in the two sides of the inequality under a contraction the paper also contains a further generalisation where the coefficients of b can be arbitrary rational numbers in 0 1 a different condition is imposed on the singularities and kx b is required to be nef some applications of the inequalities are also given for example estimating the number of singularities or certain kinds of configurations of curves on surfaces 1991 mathematics subject classification 14j17 14j60 14c17

a new iterative solver for the time harmonic wave equation the time harmonic wave equation also known as the helmholtz equation is obtained if the constant density acoustic wave equation is transformed from the time domain to the frequency domain its discretization results in a large sparse linear system of equations in two dimensions this system can be solved efficiently by a direct method in three dimensions direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large iterative methods are an alternative these methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence the iterative solution of the time harmonic wave equation has long been a notoriously difficult problem in numerical analysis recently a new preconditioner based on a strongly damped wave equation has heralded a breakthrough the solution of the linear system associated with the preconditioner is approximated by another iterative method the multigrid method the multigrid method fails for the original wave equation but performs well on the damped version the performance of the new iterative solver is investigated on a number of 2d test problems the results suggest that the number of required iterations increases linearly with frequency even for a strongly heterogeneous model where earlier iterative schemes fail to converge complexity analysis shows that the new iterative solver is still slower than a time domain solver to generate a full time series we compare the time domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively the new iterative solver can be applied straightforwardly to 3d problems

bounding singular surfaces via chern numbers we prove the existence of a bound on the number of steps of the minimal model program for singular surfaces in terms of discrepancies and chern numbers as an application we prove that given r in mathbb r 0 and epsilon in 0 1 the class mathcal f r epsilon of 2 dimensional pairs x d of general type with epsilon klt singularities d with standard coefficients and 4c 2 x d c 1 2 x d le r forms a bounded family

three dimensional visco acoustic modeling using a renormalized integral equation iterative solver we present a frequency domain renormalized integral equation formulation for solving a three dimensional visco acoustic medium using an iterative solver upon applying this special renormalization the resulting integral equation operator can be proven to have a contraction property hence solving the linear system of equations using a krylov optimization method will result in a good convergence rate furthermore since the matrix vector multiplication can be done using a fast fourier transform fft technique its operation is of the order of o n log n where n is the size of the discretization grid this technique also allows us to use matrix free implementation hence the memory usage is about o n numerical tests show that the computational time and memory usage of this renormalized integral equation approach can be quite competitive with the frequency domain finite difference iterative solver further the numerical examples demonstrate that it is possible to solve a problem with over 100 million unknowns using an integral equation approach

the method of polarized traces for the 2d helmholtz equation we present a solver for the 2d high frequency helmholtz equation in heterogeneous acoustic media with online parallel complexity that scales optimally as o n l where n is the number of volume unknowns and l is the number of processors as long as l grows at most like a small fractional power of n the solver decomposes the domain into layers and uses transmission conditions in boundary integral form to explicitly define polarized traces i e up and down going waves sampled at interfaces local direct solvers are used in each layer to precompute traces of local green s functions in an embarrassingly parallel way the offline part and incomplete green s formulas are used to propagate interface data in a sweeping fashion as a preconditioner inside a gmres loop the online part adaptive low rank partitioning of the integral kernels is used to speed up their application to interface data the method uses second order finite differences the complexity scalings are empirical but motivated by an analysis of ranks of off diagonal blocks of oscillatory integrals they continue to hold in the context of standard geophysical community models such as bp and marmousi 2 where convergence occurs in 5 to 10 gmres iterations while the parallelism in this paper stems from decomposing the domain we do not explore the alternative of parallelizing the systems solves with distributed linear algebra routines

enriques surfaces with eight nodes a nodal enriques surface can have at most 8 nodes we give an explicit description of enriques surfaces with 8 nodes showing that they are quotients of products of elliptic curves by a group isomorphic to or to acting freely in codimension 1 we use this result to show that if s is a minimal surface of general type with pg 0 such that the image of the bicanonical map is birational to an enriques surface then k2s 3 and the bicanonical map is a morphism of degree 2

a multigrid based preconditioned krylov subspace method for the helmholtz equation with pml abstract in this paper we generalize the complex shifted laplacian preconditioner to the complex shifted laplacian pml preconditioner for the helmholtz equation with perfectly matched layer helmholtz pml equation the helmholtz pml equation is discretized by an optimal 9 point difference scheme and the preconditioned linear system is solved by the krylov subspace method especially by the biconjugate gradient stabilized method bi cgstab the spectral analysis of the linear system is given and a new matrix based interpolation operator is proposed for the multigrid method which is used to approximately invert the preconditioner the numerical experiments are presented to illustrate the efficiency of the preconditioned bi cgstab method with the multigrid based on the new interpolation operator also numerical results are given for comparing the performance of the new interpolation operator with that of classic bilinear interpolation operator and the one suggested in erlangga et al 2006 10

some remarks on the local moduli of tangent bundles over complex surfaces using the hirzebruch s riemann roch formula for endomorphism bundles over a compact complex two fold we prove that the tangent bundle of a complex surface m of general type admits a nontrivial trace free deformation unless m is holomorphically covered by the euclidean ball it follows that the tangent bundle of the mostow siu surface which is a kahler surface with a negative definite curvature tensor does have a nontrivial trace free moduli among some other results we also point out a relationship between the kuranishi obstruction and symmetric holomorphic two tensors on a complex surface

second order absorbing boundary conditions for the wave equation a solution for the corner problem the treatment of domains with corners in the problem of absorbing boundary conditions for the wave equation is very important from a practical point of view a technical difficulty appears as soon as conditions of order greater than or equal to 2 are considered a solution is proposed for the two dimensional case when second order conditions are used this solution consists of prescribing an adequate corner condition the problem thus obtained is analyzed theoretically and the condition is proved to be optimal the results obtained here are illustrated by numerical simulations some extensions to higher space dimensions and higher order conditions are proposed

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

hyperbolicity of nodal hypersurfaces we show that a nodal hypersurface x in p3 of degree d with a sufficiently large number l of nodes l 8 3 d2 5 2 d is algebraically quasi hyperbolic i e x can only have finitely many rational and elliptic curves our results use the theory of symmetric differentials and algebraic foliations and give a very striking example of the jumping of the number of symmetric differentials in families

the bogomolov miyaoka yau inequality for logarithmic surfaces in positive characteristic we generalize bogomolov s inequality for higgs sheaves and the bogomolov miyaoka yau inequality in positive characteristic to the logarithmic case we also generalize shepherd barron s results on bogomolov s inequality on surfaces of special type from rank 2 to the higher rank case we use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo p2 these examples contradict some claims by xie

high order fem domain decomposition models for high frequency wave propagation in heterogeneous media abstract large scale scientific computing models requiring iterative algebraic solvers are needed to simulate high frequency wave propagation because large degrees of freedom are needed to avoid the helmholtz computer model pollution effects in this work we investigate the use of multiple additive schwarz type domain decomposition dd approximations to efficiently simulate two and three dimensional high frequency wave propagation with high order fem we compare our dd based results with those obtained using a standard geometric multigrid approach for up to 1 000 and 300 wavelength models in two and three dimensions respectively

canonical surfaces with big cotangent bundle surfaces of general type with positive second segre number are known to have big cotangent bundle we give a new criterion ensuring that a surface of general type with canonical singularities has a minimal resolution with big cotangent bundle this provides many examples of surfaces with negative second segre number and big cotangent bundle

collapsed 5 manifolds with pinched positive sectional curvature abstract let m be a closed 5 manifold of pinched curvature 0 sec m 1 we prove that m is homeomorphic to a spherical space form if one of the following conditions holds i the center of the fundamental group has index w a constant depending on ii 1 4 1 10 6 2 and the fundamental group is a non cyclic group of order c a constant iii the volume is less than and the fundamental group is either isomorphic to a spherical 5 space group or has an odd order and it has a center of index w a constant

multifrequency viscoacoustic modeling and inversion modeling and inversion for seismic wavefields that include the attenuation and phase dispersion effects of q can be implemented in the space frequency domain the viscoacoustic wave equation is solved by the moment method absorbing boundary conditions are implemented by reducing q and adjusting the complex velocity to reduce q dependent reflectivity in a zone around the edges of the model grid nonlinear inversion is performed using iterative linearized inversions the residual wavefield at a single frequency is back projected using an anticausal green s function along the viscoacoustic wavepath in an estimate of the model to get updated velocity and q distributions the model obtained from data at one frequency becomes input to inversion at the next higher frequency velocity and q are inverted simultaneously as they are interdependent both modeling and inversion algorithms are successfully tested with synthetic examples data at two or three frequencies are sufficient to produce reliable images from noise free synthetic data

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

a dispersion minimizing finite difference scheme and preconditioned solver for the 3d helmholtz equation in this paper a new 27 point finite difference method is presented for solving the 3d helmholtz equation with perfectly matched layer pml which is a second order scheme and pointwise consistent with the equation an error analysis is made between the numerical wavenumber and the exact wavenumber and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters a full coarsening multigrid based preconditioned bi cgstab method is developed for solving the linear system stemming from the helmholtz equation with pml by the finite difference scheme the shifted laplacian is extended to precondition the 3d helmholtz equation and a spectral analysis is given the discrete preconditioned system is solved by the bi cgstab method with a multigrid method used to invert the preconditioner approximately full coarsening multigrid is employed and a new matrix based prolongation operator is constructed accordingly numerical results are presented to demonstrate the efficiency of both the new 27 point finite difference scheme with refined parameters and the preconditioned bi cgstab method with the 3d full coarsening multigrid

a parallel multigrid based preconditioner for the 3d heterogeneous high frequency helmholtz equation we investigate the parallel performance of an iterative solver for 3d heterogeneous helmholtz problems related to applications in seismic wave propagation for large 3d problems the computation is no longer feasible on a single processor and the memory requirements increase rapidly therefore parallelization of the solver is needed we employ a complex shifted laplace preconditioner combined with the bi cgstab iterative method and use a multigrid method to approximate the inverse of the resulting preconditioning operator a 3d multigrid method with 2d semi coarsening is employed we show numerical results for large problems arising in geophysical applications

negative curves on algebraic surfaces we study curves of negative self intersection on algebraic surfaces our main result shows there exist smooth complex projective surfaces x related to hilbert modular surfaces such that x contains reduced irreducible curves c of arbitrarily negative self intersection c 2 previously the only known examples of surfaces for which c 2 was not bounded below were in positive characteristic and the general expectation was that no examples could arise over the complex numbers indeed we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field and thus our complex examples require a different approach

multigrid simulation for high frequency solutions of the helmholtz problem in heterogeneous media the helmholtz problem is hard to solve in heterogeneous media in particular when the wave number is real and large the problem is neither coercive nor hermitian symmetric this article concerns the v cycle multigrid mg method for high frequency solutions of the helmholtz problem since we need to choose at least 10 12 grid points per wavelength for stability the coarse grid problem is still large to solve the coarse grid problem efficiently a nonoverlapping domain decomposition method is adopted without introducing another coarser subspace correction various numerical experiments have shown that the convergence rate of the resulting mg method is independent on the grid size and the wave number provided that the coarse grid problem is fine enough for the solution to capture characteristics of the physical problem

flexible multiple semicoarsening for three dimensional singularly perturbed problems we present robust parallel multigrid based solvers for 3d scalar partial differential equations the robustness is obtained by combining multiple semicoarsening strategies matrix dependent transfer operators and a krylov subspace acceleration the basis for the 3d preconditioner is a 2d method with multiple semicoarsened grids based on the mg s method from c w oosterlee appl numer math 19 1995 pp 115 128 and t washio and c w oosterlee gmd arbeitspapier 949 gmd st augustin germany 1995 the 2d method is generalized to three dimensions with a line smoother in the third dimension the method based on semicoarsening has been parallelized with the grid partitioning technique j linden b steckel and k stuben parallel comput 7 1988 pp 461 475 o a mcbryan et al impact comput sci engrg 3 1991 pp 1 75 and is evaluated as a solver and as a preconditioner on a mimd machine the robustness of the 3d method is shown for finite volume and finite difference discretizations of 3d anisotropic diffusion equations and convection dominated convection diffusion problems

a dispersion minimizing scheme for the 3 d helmholtz equation based on ray theory we develop a new dispersion minimizing compact finite difference scheme for the helmholtz equation in 2 and 3 dimensions the scheme is based on a newly developed ray theory for difference equations a discrete helmholtz operator and a discrete operator to be applied to the source and the wavefields are constructed their coefficients are piecewise polynomial functions of hk chosen such that phase and amplitude errors are minimal the phase errors of the scheme are very small approximately as small as those of the 2 d quasi stabilized fem method and substantially smaller than those of alternatives in 3 d assuming the same number of gridpoints per wavelength is used in numerical experiments accurate solutions are obtained in constant and smoothly varying media using meshes with only five to six points per wavelength and wave propagation over hundreds of wavelengths when used as a coarse level discretization in a multigrid method the scheme can even be used with down to three points per wavelength tests on 3 d examples with up to 108 degrees of freedom show that with a recently developed hybrid solver the use of coarser meshes can lead to corresponding savings in computation time resulting in good simulation times compared to the literature

when is the range of a multiplier on a banach algebra closed in this paper we prove the theorem let a be a banach algebra with a bounded approximate identity bai such that every proper closed ideal of a is contained in a proper closed ideal with a bai then a multiplier t a a has a closed range iff t factors as a product of an idempotent multiplier and an invertible multiplier

a characterization of the closed unital ideals of the fourier stieltjes algebra b g of a locally compact amenable group g abstract let g be a locally compact amenable group b g its fourier stieltjes algebra and i be a closed ideal of it in this paper we prove the following result the ideal i has a unit element iff it is principal this is the noncommutative version of the glicksberg host parreau theorem the paper also contains an abstract version of this theorem

a composite preconditioner for the electromagnetic scattering from a large cavity in this paper we propose a composite preconditioner for the electromagnetic scattering from a large cavity the electromagnetic cavity problem is described by the helmholtz equation with a nonlocal boundary condition on the aperture of the cavity and dirichlet or neumann boundary conditions on the walls of the cavity the preconditioner proposed here combines the optimal sine transform based approximation with a layered medium model using fast fourier transforms the computational cost of every iteration is o n 2logn on an nxn uniform partition of the unit square numerical results for a model problem show that the new preconditioner is more efficient than those recently considered in the literature for the cavity with a small portion of non layered media we propose a sparse preconditioned conjugate orthogonal conjugate gradient solver combined with the new preconditioner numerical results for a model problem are reported to demonstrate the efficiency of the sparse solver

velocity model building by 3d frequency domain full waveform inversion of wide aperture seismic data we assessed 3d frequency domain fd acoustic fullwaveforminversionfwidataasatooltodevelophigh resolution velocity models from low frequency global offset data the inverse problem was posed as a classic leastsquares optimization problem solved with a steepest descent method inversion was applied to a few discrete frequencies allowing management of a limited subset of the 3d data volume the forward problem was solved with a finite difference frequency domain method based on a massively paralleldirectsolver allowingefficientmultiple shotsimulations the inversion code was fully parallelized for distributedmemory platforms taking advantage of a domain decomposition of the modeled wavefields performed by the direct solver after validation on simple synthetic tests fwi was applied to two targets channel and thrust system of the 3d seg eageoverthrustmodel correspondingto3ddomains of 78 752 25 km and 13 513 54 65 km respectively the maximum inverted frequencies are 15 and 7 hz for the two applications amaximum of 30 dual core biprocessor nodes with 8 gb of shared memory per node were used for the second target the main structures were imaged successfully at a resolution scale consistent with the inverted frequencies our study confirms the feasibility of 3d frequency domain fwi of global offset data on large distributed memory platforms to develop high resolution velocity models these high velocity models may provide accurate macromodelsforwave equationprestackdepthmigration

algebraic multilevel preconditioner for the helmholtz equation in heterogeneous media an algebraic multilevel ml preconditioner is presented for the helmholtz equation in heterogeneous media it is based on a multilevel incomplete ldl t factorization and preserves the inherent complex symmetry of the helmholtz equation the ml preconditioner incorporates two key components for efficiency and numerical stability symmetric maximum weight matchings and an inverse based pivoting strategy the former increases the block diagonal dominance of the system whereas the latter controls l 1 for numerical stability when applied recursively their combined effect yields an algebraic coarsening strategy similar to algebraic multigrid methods even for highly indefinite matrices the ml preconditioner is combined with a krylov subspace method and applied as a black box solver to a series of challenging two and three dimensional test problems mainly from geophysical seismic imaging the numerical results demonstrate the robustness and efficiency of the ml preconditioner even at higher frequency regimes

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

algebraic montgomery yang problem the non cyclic case montgomery yang problem predicts that every pseudofree circle action on the 5 dimensional sphere has at most 3 non free orbits using a certain one to one correspondence kollar formulated the algebraic version of the montgomery yang problem every projective surface s with quotient singularities such that the second betti number b2 s 1 has at most 3 singular points if its smooth locus s0 is simply connected we prove the conjecture under the assumption that s has at least one non cyclic singularity in the course of the proof we classify projective surfaces s with quotient singularities such that i b2 s 1 ii h 1 s 0 mathbb z 0 and iii s has 4 or more singular points not all cyclic and prove that all such surfaces have pi 1 s 0 cong mathfrak a 5 the icosahedral group

the geometry of sporadic c embeddings into c2 abstract a closed algebraic embedding of c c 1 0 into c 2 is sporadic if for every curve a c 2 isomorphic to an affine line the intersection with c is at least 2 non sporadic embeddings have been classified there are very few known sporadic embeddings we establish geometric and algebraic tools to classify them based on the analysis of the minimal log resolution x d p 2 u where u is the closure of c on p 2 we show in particular that one can choose coordinates on c 2 in which the type at infinity of the c and the self intersection of its proper transform on x are sharply limited

harmonic metrics on higgs sheaves and uniformization of varieties of general type we prove a criterion for the existence of harmonic metrics on higgs bundles that are defined on smooth loci of klt varieties as one application we resolve the quasi etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete co compact groups of automorphisms that act freely in codimension one as a further application we establish a nonabelian hodge correspondence on smooth loci of klt varieties

bounding codimension one subvarieties and a general inequality between chern numbers we extend the miyaoka yau inequality for a surface to an arbitrary nonuniruled normal complex projective variety eliminating the hypothesis that the variety must be minimal the in equality is sharp in dimension three and is also sharp among minimal varieties for nonminimal varieties in dimension four or higher an error term is picked up which can be controlled as a consequence we bound codimension one subvarieties in a variety of general type linearly in terms of their chern numbers in particular we show that there are only a finite number of smooth fano abelian and calabi yau subvarieties of codimension one in any variety of general type

a perfectly matched layer for the helmholtz equation in a semi infinite strip the perfectly matched layer pml has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency domains we develop a discretization to solve the helmholtz equation in an infinite two dimensional strip we solve the interior equation using high order finite differences schemes the combined helmholtz pml problem is then analyzed for the parameters that give the best performance we show that the use of local high order methods in the physical domain coupled with a specific second order approximation in the pml yields global high order accuracy in the physical domain we discuss the impact of the parameters on the effectiveness of the pml numerical results are presented to support the analysis

optimal fourth order staggered grid finite difference scheme for 3d frequency domain viscoelastic wave modeling we investigate an optimal fourth order staggered grid finite difference scheme for 3d frequency domain viscoelastic wave modeling an anti lumped mass strategy is incorporated to minimize the numerical dispersion the optimal finite difference coefficients and the mass weighting coefficients are obtained by minimizing the misfit between the normalized phase velocities and the unity an iterative damped least squares method the levenberg marquardt algorithm is utilized for the optimization dispersion analysis shows that the optimal fourth order scheme presents less grid dispersion and anisotropy than the conventional fourth order scheme with respect to different poisson s ratios moreover only 3 7 grid points per minimum shear wavelength are required to keep the error of the group velocities below 1 the memory cost is then greatly reduced due to a coarser sampling a parallel iterative method named carp cg is used to solve the large ill conditioned linear system for the frequency domain modeling validations are conducted with respect to both the analytic viscoacoustic and viscoelastic solutions compared with the conventional fourth order scheme the optimal scheme generates wavefields having smaller error under the same discretization setups profiles of the wavefields are presented to confirm better agreement between the optimal results and the analytic solutions

an abstract form of a theorem of helson and applications to sets of synthesis and sets of uniqueness let e be a compact perfect subset of the real line r such that the restriction of the fourier transform a a e from l1 r into c e is onto helson proved that then for m e lim y y 0 is possible only if 0 in this paper we present an abstract version of this theorem of helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness

degree of the bicanonical map of a surface of general type let 5 be a smooth complex projective surface of general type rt 5 fpn l be the map defined by nks is called pluricanonical map for n 2 due to the works by bombieri and others in the 1970s the nature of n is determined except when n 2 and for some low values of invariants of 5 see 1 for an up to date treatment and references these maps play an important role in the classification of surfaces of general type recently the remaining case of the bicanonical map 2 for low values of invariants has aroused new research interests due to the work of francia 8 and especially the remarkable method of reider 15 see also 6 14 4 they have proved the following result among other things

on rational cuspidal projective plane curves in 2002 l nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a gorenstein surface singularity with rational homology sphere link with the seiberg witten invariant or one of its candidates of the link recently the last three authors found some counterexamples using superisolated singularities the theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves in the case when the corresponding curve has only one singular point one knows no counterexample in fact in this case the above seiberg witten conjecture led us to a very interesting and deep set of compatibility properties of these curves generalising the seiberg witten invariant conjecture but sitting deeply in algebraic geometry which seems to generalise some other famous conjectures and properties as well for example the noether nagata or the log bogomolov miyaoka yau inequalities namely we provide a set of compatibility conditions which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer d if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree d the conjectured compatibility properties have a weaker version too valid for any rational cuspidal curve with more than one singular point the goal of the present article is to formulate these conjectured properties and to verify them in all the situations when the logarithmic kodaira dimension of the complement of the corresponding plane curves is strictly less than 2

the classification of double planes of general type with k2 8 and pg 0 abstract we study minimal double planes of general type with k 2 8 and p g 0 namely pairs s where s is a minimal complex algebraic surface of general type with k 2 8 and p g 0 and is an automorphism of s of order 2 such that the quotient s is a rational surface we prove that s is a free quotient f c g where c is a curve f is an hyperelliptic curve g is a finite group that acts faithfully on f and c and is induced by the automorphism id of f c being the hyperelliptic involution of f we describe all the f c and g that occur in this way we obtain 5 families of surfaces with p g 0 and k 2 8 of which we believe only one was previously known using our classification we are able to give an alternative description of these surfaces as double covers of the plane thus recovering a construction proposed by du val in addition we study the geometry of the subset of the moduli space of surfaces of general type with p g 0 and k 2 8 that admit a double plane structure

acoustic inverse scattering via helmholtz operator factorization and optimization we present a joint acoustic seismic inverse scattering and finite frequency reflection tomography program formulated as a coupled set of optimization problems in terms of inhomogeneous helmholtz equations we use a higher order finite difference scheme for these helmholtz equations to guarantee sufficient accuracy we adapt a structured approximate direct solver for the relevant systems of algebraic equations which addresses storage requirements through compression to yield a complexity for computing the gradients or images in the optimization problems that consists of two parts viz the cost for all the matrix factorizations which is roughly o rn for example o rnlogn when d 2 times the number of frequencies and the cost for all solutions by substitution which is roughly o n for example o nlog rlogn when d 2 times the number of frequencies times the number of sources events where n n d if n is the number of grid samples in any direction and r is a parameter depending on the preset accuracy and the problem at hand with this complexity the multi frequency approach to inverse scattering and finite frequency tomography becomes computationally feasible with large data sets in dimensions d 2 and 3

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

on generalized kummer surfaces and the orbifold bogomolov miyaoka yau inequality a generalized kummer surface x km t g is the resolution of a quotient of a torus t by a finite group of symplectic automorphisms g we complete the classification of generalized kummer surfaces by studying the two last groups which have not been yet studied for these surfaces we compute the associated kummer lattice k g which is the minimal primitive sub lattice containing the exceptional curves of the resolution x t g we then prove that a k3 surface is a generalized kummer surface of type km t g if and only if its neron severi group contains k g for smooth orbifold surfaces x of kodaira dimension 0 kobayashi proved the orbifold bogomolov miyaoka yau inequality c 2 1 x 3c 2 x for kodaira dimension 2 the case of equality is characterized as x being uniformized by the complex 2 ball b 2 for smooth orbifold k3 and enriques surfaces we characterize the case of equality as being uniformized by c 2

three divisible families of skew lines on a smooth projective quintic we give an example of a family of 15 skew lines on a quintic such that its class is divisible by 3 we study properties of the codes given by arrangements of disjoint lines on quintics

on the classification of irregular surfaces of general type with nonbirational bicanonical map the present paper is devoted to the classification of irregular surfaces of general type with pg 3 and nonbirational bicanonical map our main result is that if s is such a surface and if s is minimal with no pencil of curves of genus 2 then s is the symmetric product of a curve of genus 3 and therefore pg q 3 and k2 6 furthermore we obtain some results towards the classification of minimal surfaces with pg q 3 such surfaces have 6 kz 9 and we show that kz 6 if and only if s is the symmetric product of a curve of genus 3 we also classify the minimal surfaces with pg q 3 with a pencil of curves of genus 2 proving in particular that for those one has kz 8

robust and highly scalable parallel solution of the helmholtz equation with large wave numbers numerical solution of the helmholtz equation is a challenging computational task particularly when the wave number is large for two dimensional problems direct methods provide satisfactory solutions but large three dimensional problems become unmanageable in this work the block parallel carp cg algorithm parallel computing 36 2010 is applied to the helmholtz equation with large wave numbers the effectiveness of this algorithm is shown both theoretically and practically with numerical experiments on two and three dimensional domains including heterogeneous cases and a wide range of wave numbers a second order finite difference discretization scheme is used leading to a complex nonsymmetric and indefinite linear system carp cg is both robust and efficient on the tested problems on a fixed grid its scalability improves as the wave number increases additionally when the number of grid points per wavelength is fixed the number of iterations increases linearly with the wave number convergence rates for heterogeneous cases are similar to those of homogeneous cases carp cg also outperforms at all wave numbers one of the leading methods based on the shifted laplacian preconditioner with a complex shift and solved with a multigrid

on rational cuspidal curves let c subset bold p 2 be a rational curve of degree d which has only one analytic branch at each point denote by m the maximal multiplicity of singularities of c it is proven in ms that d alpha m and this estimate is sharp the main tool used here is the logarithmic version of the bogomolov miyaoka yau inequality for curves as above we give an interpretation of this inequality in terms of the number of parameters describing curves of a given degree and the number of conditions imposed by singularity types