global smoothing of calabi yau threefolds yoshinori namikawa i j h m steenbrink 2 i department of mathematics sophia university kioi cho tokyo 102 japan 2 mathematical institute university of nijmegen toernooiveld 6525 ed nijmegen the netherlands oblatum 10 x 1994 13 vi 1995 introduction friedman fr has studied the relationship between local and global deforma tions of a threefold z with isolated hypersurface singularities which admits small resolutions one of his main results is as follows let z be a moishezon threefold with only ordinary double points pl p assume that the canon ical line bundle

asynchronous fast adaptive composite grid methods for elliptic problems theoretical foundations accurate numerical modeling of complex physical chemical and biological systems requires numerical simulation capability over a large range of length scales with the ability to capture rapidly varying phenomena localized in space and or time adaptive mesh refinement amr is a numerical process for dynamically introducing local fine resolution on computational grids during the solution process in response to unresolved error in a computation fast adaptive composite grid fac methods are a class of algorithms that exploit the multilevel structure of amr grids to solve elliptic problems efficiently this paper develops a theoretical foundation for afacx an asynchronous fac method a new multilevel condition number estimate establishes that the convergence rate of the afacx algorithm does not degrade as the number of refinement levels in the amr hierarchy increases

error estimates for regularization methods in hilbert scales in this paper we study regularization methods to reconstruct the solution x of the linear ill posed problem ax y a x to y from noisy data y delta in y the regularization methods are of the general form x alpha delta bar x g alpha b s a a b s a y delta a bar x where b denotes an unbounded self adjoint strictly positive definite operator in the hilbert space x assuming ax sim x a and x bar x p le e for some bar x in x a ge 0 and p ge 0 x r b r 2 x is the norm in a hilbert scale x r r in rnus we derive error estimates which show that the accuracy of the above regularization methods is order optimal provided that the function g alpha lambda the regularization parameter alpha and the parameter s are chosen properly

mordell weil lattices and toric decompositions of plane curves we extend results of cogolludo agustin and libgober relating the alexander polynomial of a plane curve c with the mordell weil rank of certain isotrivial families of jacobians over mathbf p 2 of discriminant c in the second part we introduce a height pairing on the 2 3 6 quasi toric decompositions of a plane curve we use this pairing and the results in the first part of the paper to construct a pair of degree 12 curves with 30 cusps and alexander polynomial t 2 t 1 but with distinct height pairing we use the height pairing to show that these curves from a zariski pair

on the stability of the l 2 projection in h 1 ω we prove the stability in h1 of the l2 projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions we give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions in particular stability of the l2 projection in h1 holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically

algebraic gauss manin systems and brieskorn modules we study the algebraic gauss manin system and the algebraic brieskorn module associated to a polynomial mapping with isolated singularities since the algebraic gauss manin system does not contain any information on the cohomology of singular fibers we first construct a non quasi coherent sheaf which gives the cohomology of every fiber then we study the algebraic brieskorn module and show that its position in the the algebraic gauss manin system is determined by a natural map to quotients of local analytic gauss manin systems and its pole part by the vanishing cycles at infinity comparing it with the deligne extension this implies for example a formula for the determinant of periods in the two dimensional case we can describe the global structure of the algebraic gauss manin system rather explicitly

numerical approximation of the integral fractional laplacian we propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional laplacian we first derive an integral representation of the bilinear form corresponding to the variational problem the numerical approximation of the action of the corresponding stiffness matrix consists of three steps 1 apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space 2 truncate each elliptic problem to a bounded domain 3 use the finite element method for the space approximation on each truncated domain the consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm the results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain the domain truncation parameter and the quadrature spacing parameter

adaptive boundary element methods based on computational schemes for sobolev norms multilevel methods are applied to compute local and global sobolev norms arising in adaptive boundary element methods combining this approach with an approximate computation of an error function we construct efficient and reliable a posteriori error estimators for arbitrary boundary element solutions related to either a dirichlet or neumann boundary value problem

a finite element technique for solving first order pdes in lp an approximation technique for solving first order pdes in l p omega 1 le p infty is proposed the method is a generalization of the least squares method to non hilbertian settings a priori and a posteriori error estimates are proven numerical tests in l 1 omega show that this type of technique can handle discontinuities without resorting to limiting procedures

a robust petrov galerkin discretisation of convection diffusion equations a petrov galerkin discretisation is studied of an ultra weak variational formulation of the convection diffusion equation in a mixed form to arrive at an implementable method the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space to prevent that this latter space has to be taken increasingly large for vanishing diffusion a formulation is constructed that is well posed in the limit case of a pure transport problem numerical experiments show approximations that are very close to the best approximations to the solution from the trial space uniformly in the size of the diffusion term

robust preconditioning for coupled stokes darcy problems with the darcy problem in primal form abstract the coupled darcy stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part in this work we consider preconditioners for monolithic solution algorithms of the coupled darcy stokes problem where the darcy problem is in primal form we employ the operator preconditioning framework and utilize a fractional solver at the interface between the problems to obtain order optimal schemes that are robust with respect to the material parameters i e the permeability viscosity and beavers joseph saffman condition our approach is similar to that of holter et al 2020 but since the darcy problem is in primal form expressing mass conservation at the interface involves the normal derivative which introduces some mathematical challenges these challenges will be specifically addressed in this paper in particular we will employ fractional laplacians at the interface numerical experiments illustrating the performance are provided the preconditioner is posed in non standard sobolev spaces which may be perceived as an obstacle for its use in applications however we detail the implementational aspects and show that the preconditioner is quite feasible to realize in practice

iterative schemes for nonsymmetric and indefinite elliptic boundary value problems the purpose of this paper is twofold the first is to describe some simple and robust iterative schemes for nonsymmetric and indefinite elliptic boundary value problems the schemes are based in the sobolev space h omega and require minimal hypotheses the second is to develop algorithms utilizing a coarse grid approximation this leads to iteration matrices whose eigenvalues lie in the right half of the complex plane in fact for symmetric indefinite problems the iteration is reduced to a well conditioned symmetric positive definite system which can be solved by conjugate gradient interation applications of the general theory as well as numerical examples are given 20 refs 8 tabs

discrete interpolation norms with applications we describe norm representations for interpolation spaces generated by finite dimensional subspaces of hilbert spaces these norms are products of integer and noninteger powers of the grammian matrices associated with the generating pair of spaces for the interpolation space we include a brief description of some of the algorithms which allow the efficient computation of matrix powers we consider in some detail the case of fractional sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques numerical experiments are included

schur complements on hilbert spaces and saddle point systems for any continuous bilinear form defined on a pair of hilbert spaces satisfying the compatibility ladyshenskaya babusca brezzi condition symmetric schur complement operators can be defined on each of the two hilbert spaces in this paper we find bounds for the spectrum of the schur operators only in terms of the compatibility and continuity constants in light of the new spectral results for the schur complements we review the classical babusca brezzi theory find sharp stability estimates and improve a convergence result for the inexact uzawa algorithm we prove that for any symmetric saddle point problem the inexact uzawa algorithm converges provided that the inexact process for inverting the residual at each step has the relative error smaller than 1 3 as a consequence we provide a new type of algorithm for discretizing saddle point problems which combines the inexact uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions

efficient contact solvers based on domain decomposition techniques abstract this paper deals with the construction of efficient algorithms for the solution of the finite dimensional constrained minimization problem arising from the finite element discretization of contact problems dualization techniques have been used to decrease the problem size from the large number of unknowns in the domain to the much smaller number of inequalities at the boundary the disadvantage of this direct schur complement approach is the need of the inversion of the stiffness matrix domain decomposition techniques meet very similar requirements a global boundary value problem is decoupled into local subproblems and one interface problem at the coupling boundary two complementary approaches are the dirichlet method and the neumann method the first one requires preconditioners for local dirichlet problems and for the interface problem in h 1 2 and extension operators from the boundary into the domain the second one needs preconditioners for local neumann problems and for the interface problem in h 1 2 efficient multilevel algorithms for all components are available in literature in this paper it is shown how to use exactly these components for the construction of solvers for contact problems new results for the analysis of convergence are presented at least at uniformly refined meshes we can prove optimal time complexity numerical results show high efficiency also on adaptively refined 3d meshes

approximation of the eigenvalue problem for the time harmonic maxwell system by continuous lagrange finite elements we propose and analyze an approximation technique for the maxwell eigenvalue problem using h1 conforming finite elements the key idea consists of considering a mixed method controlling the divergence of the electric field in a fractional sobolev space h with 1 2 1 the method is shown to be convergent and spectrally correct

substructuring preconditioners for the three fields domain decomposition method we study a class of preconditioners based on substructuring for the discrete steklov poincare operator arising in the three fields formulation of domain decomposition in two dimensions under extremely general assumptions on the discretization spaces involved an upper bound is provided on the condition number of the preconditioned system which is shown to grow at most as log h h 2 h and h denoting respectively the diameter and the discretization mesh size of the subdomains extensive numerical tests performed on both a plain and a stabilized version of the method confirm the optimality of such bound

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

preconditioning a class of fourth order problems by operator splitting we develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space we consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two discrete linear second order elliptic operators and exploit this property in designing the preconditioners the underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available we propose symmetric and non symmetric preconditioners along with theory and numerical experiments they both document crucial properties of the preconditioners as well as their practical performance it is important to note that we neither need h s regularity s 1 of the continuous problem nor quasi uniform grids

perturbation from an asymmetric cubic hamiltonian this paper concerns with the number of limit cycles from an asymmetric hamiltonian of degree three under cubic perturbation eleven limit cycles are found and three different distributions are given by using the methods of bifurcation theory and qualitative analysis two of which are new

a unified approach for uzawa algorithms we present a unified approach in analyzing uzawa iterative algorithms for saddle point problems we study the classical uzawa method the augmented lagrangian method and two versions of inexact uzawa algorithms the target application is the stokes system but other saddle point systems e g arising from mortar methods or lagrange multipliers methods can benefit from our study we prove convergence of uzawa algorithms and find optimal rates of convergence in an abstract setting on finite or infinite dimensional hilbert spaces the results can be used to design multilevel or adaptive algorithms for solving saddle point problems the discrete spaces do not have to satisfy the lbb stability condition

quotient spaces and critical points of invariant functions for c actions consider a linear action of the group c on x c n 1 we study the fundamental algebraic properties of the sheaves of invariant and basic differential forms for such an action and use these to define an algebraic notion of multiplicity for critical points of functions which are invariant under the c action we also prove a theorem relating the cohomology of the milnor fibre of the critical point on the quotient space with this algebraic multiplicity we also include an appendix showing how to use cech complexes to compute local cohomology

on the mixed hodge structure on the cohomology of the milnor fibre in 18 the second author introduced a mixed hodge structure on the cohomology of the milnor fibre of an isolated hypersurface singularity for the definition of a mixed hodge structure el 17 sect 3 4 the weight filtration is essentially the monodromy weight filtration 18 sect 4 and thus simplifies the jordan normal form of the unipotent part of the monodromy the significance of the hodge filtration however is not so dear and its description seems difficult to use this paper gives another description of the hodge filtration which we hope is easier to understand and apply its definition does not use resolution of singularities instead it relies on the theory of holonomic modules in one variable with regular singularities this paper arose from conversations at the 1980 arbeitstagung in bonn where varchenko s conjectured hodge filtration 22 had been discussed in brieskorn s seminar we have since learned that varchenko obtained similar results to those in this paper in the summer and autumn of 1980 23 25 in sect 1 the milnor fibration is embedded in a family of smooth projective hypersurfaces the hodge theory of a smooth projective hypersurface is explained in the fashion of brylinski 2 in sect 2 in sect 3 we recall the description of the gauss manin system from 8 our new formula for the hodge filtration is explained in sects 4 5 in sect 6 we prove that it gives the same result as 18 we prove a thom sebastiani formula for the hodge filtration in sect 7 which leads to a proof of conjecture 5 4 of 18 in sect 8 we show how to prove varchenko s result about the jordan normal forms of multiplication by f in the jacobian ring of f and the logarithm of the unipotent monodormy of f with our method in the last chapter the mixed hodge structure is calculated for two examples

almost optimal interior penalty discontinuous approximations of symmetric elliptic problems on non matching grids we consider an interior penalty discontinuous approximation for symmetric elliptic problems of second order on non matching grids in this paper the main result is an almost optimal error estimate for the interior penalty approximation of the original problem based on partitioning of the domain into a finite number of subdomains further an error analysis for the finite element approximation of the penalty formulation is given finally numerical experiments on a series of model second order problems are presented

matrix factorizations and higher residue pairings the periodic cyclic homology of any proper dg category comes equipped with a canonical pairing we show that in the case of the dg category of matrix factorizations of an isolated singularity the canonical pairing can be identified with kyoji saito s higher residue pairing on the twisted de rham cohomology of the singularity

on a hodge theoretic property of the kunneth map in periodic cyclic homology abstract we prove that the standard kunneth map in periodic cyclic homology of differential z 2 graded algebras is compatible with a generalization of the hodge filtration and explain how this result is related to various thom sebastiani type theorems in singularity theory

cohomological donaldson thomas theory of a quiver with potential and quantum enveloping algebras this paper concerns the cohomological aspects of donaldson thomas theory for jacobi algebras and the associated cohomological hall algebra introduced by kontsevich and soibelman we prove the hodge theoretic categorification of the integrality conjecture and the wall crossing formula and furthermore realise the isomorphism in both of these theorems as poincare birkhoff witt isomorphisms for the associated cohomological hall algebra we do this by defining a perverse filtration on the cohomological hall algebra a result of the hidden properness of the semisimplification map from the moduli stack of semistable representations of the jacobi algebra to the coarse moduli space of polystable representations this enables us to construct a degeneration of the cohomological hall algebra for generic stability condition and fixed slope to a free supercommutative algebra generated by a mixed hodge structure categorifying the bps invariants as a corollary of this construction we furthermore obtain a lie algebra structure on this mixed hodge structure the lie algebra of bps invariants for which the entire cohomological hall algebra can be seen as the positive part of a yangian type quantum group

residual reduction algorithms for nonsymmetric saddle point problems in this paper we introduce and analyze uzawa algorithms for non symmetric saddle point systems convergence for the algorithms is established based on new spectral results about schur complements a new uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework numerical results supporting the efficiency of the algorithms are presented for finite element discretization of steady state navier stokes equations

notes on a saddle point reformulation of mixed variational problems abstract we summarize some general ideas regarding approximation of mixed variational problems using saddle point reformulation we consider the concepts of optimal and almost optimal or test norm and provide estimates for the continuity and stability constants a preconditioning strategy for solving the discrete mixed formulations is used in combination with the special test norms we further provide a choice for a discrete trial space that depends on the choice of a standard test space and leads to discrete stability when using the appropriate test norm examples to illustrate how the stability of the saddle point discretization can be improved using special test norms are included

optimal order multilevel preconditioners for regularized ill posed problems in this article we design and analyze multilevel preconditioners for linear systems arising from regularized inverse problems using a scale independent distance function that measures spectral equivalence of operators it is shown that these preconditioners approximate the inverse of the operator to optimal order with respect to the spatial discretization parameter h as a consequence the number of preconditioned conjugate gradient iterations needed for solving the system will decrease when increasing the number of levels with the possibility of performing only one fine level residual computation if h is small enough the results are based on the previously known two level preconditioners of rieder 1997 see also hanke and vogel 1999 and on applying newton like methods to the operator equation x 1 a 0 we require that the associated forward problem has certain smoothing properties however only natural stability and approximation properties are assumed for the discrete operators the algorithm is applied to a reverse time parabolic equation that is the problem of finding the initial value leading to a given final state we also present some results on constructing restriction operators with preassigned approximating properties that are of independent interest

cascadic multilevel algorithms for symmetric saddle point systems abstract in this paper we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems the algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for symmetric positive definite systems on each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a schur complement is implemented the level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error we prove new estimates that relate the iteration error and the residual for the constraint equation the new estimates are the key ingredients in imposing an efficient level change criterion the first iteration on each new level uses information about the best approximation of the discrete solution from the previous level the theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a non increasing number of iterations on each level even though numerical results supporting the efficiency of the algorithms are presented for the stokes system the algorithms can be applied to a large class of boundary value problems including first order systems that can be reformulated at the continuous level as symmetric saddle point problems such as the maxwell equations

algebraic particular integrals integrability and the problem of the center in this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields we first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center these curves control the changes in the systems as parameters vary the bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered namely the real four dimensional projective space next we consider the known four algebraic conditions for the center for quadratic vector fields one of them says that the system is hamiltonian a condition which has a clear geometric meaning we determine the geometric meaning of the remaining other three algebraic conditions i ii iii

v cycle convergence of some multigrid methods for ill posed problems for ill posed linear operator equations we consider some v cycle multigrid approaches that in the framework of bramble pasciak wang and xu 1991 we prove to yield level independent contraction factor estimates consequently we can incorporate these multigrid operators in a full multigrid method that together with a discrepancy principle is shown to act as an iterative regularization method for the underlying infinite dimensional ill posed problem numerical experiments illustrate the theoretical results

new estimates for multilevel algorithms including the v cycle the purpose of this paper is to provide new estimates for certain multilevel algorithms in particular we are concerned with the simple additive multilevel algorithm discussed recently together with j xu and the standard v cycle algorithm with one smoothing step per grid we shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity for example the theory applies to the standard multigrid vcycle on the l shaped domain or a domain with a crack and yields a uniform convergence rate we also prove uniform convergence rates for the multigrid v cycle for problems with nonuniformly refined meshes finally we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid v cycle algorithms

computational scales of sobolev norms with application to preconditioning this paper provides a framework for developing computationally efficient multilevel preconditioners and representations for sobolev norms specifically given a hilbert space v and a nested sequence of subspaces v 1 c v 2 c c v we construct operators which are spectrally equivalent to those of the form a k k q k q k 1 here k k 1 2 are positive numbers and q k is the orthogonal projector onto v k with q 0 0 we first present abstract results which show when a is spectrally equivalent to a similarly constructed operator a defined in terms of an approximation q k of q k for k 1 2 we show that these results lead to efficient preconditioners for discretizations of differential and pscudo differential operators of positive and negative order these results extend to sums of operators for example singularly perturbed problems such as i e can be preconditioned uniformly independently of the parameter e we also show how to precondition an operator which results from tikhonov regularization of a problem with noisy data finally we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition

wavelet in time multigrid in space preconditioning of parabolic evolution equations two space time variational formulations of linear parabolic evolution equations are considered one is symmetric and elliptic on the trial space while the other is not in each case a space time petrov galerkin discretization using suitable tensor product trial and test functions leads to a large linear system of equations the well posedness of this system with respect to parabolic norms induces a canonical preconditioner for the algebraic equations that arise after a choice of basis for the iterative resolution of this algebraic system with parallelization in the temporal direction we propose a sparse algebraic wavelet in time transformation on possibly nonuniform temporal meshes this transformation approximately block diagonalizes the preconditioner and the individual spatial blocks can then be inverted for instance by standard spatial multigrid methods in parallel the performance of the preconditioner is documented in a series of numerical experiments

regularity estimates for elliptic boundary value problems in besov spaces we consider the dirichlet problem for poisson s equation on a nonconvex plane polygonal domain new regularity estimates for its solution in terms of besov and sobolev norms of fractional order are proved the analysis is based on new interpolation results and multilevel representations of norms on sobolev and besov spaces the results can be extended to a large class of elliptic boundary value problems some new sharp finite element error estimates are deduced

a robust hierarchical basis preconditioner on general meshes in this paper we introduce a multi level direct sum space decomposition of general possibly locally refined linear or multi linear finite element spaces the resulting additive schwarz preconditioner is optimal for symmetric second order elliptic problems moreover it turns out to be robust with respect to coefficient jumps over edges in the coarsest mesh perturbations with positive zeroth order terms and after a further decomposition of the spaces also with respect to anisotropy along the grid lines important for an efficient implementation is that stable bases of the subspaces defining our decomposition consisting of functions having small supports can be easily constructed

uniform preconditioners for problems of positive order uniform preconditioners for operators of negative order discretized by dis continuous piecewise polynomials of any order are constructed from a boundedly invertible operator of opposite order discretized by continuous piecewise linears besides the cost of the application of the latter discretized operator the other cost of the preconditioner scales linearly with the number of mesh cells compared to earlier proposals the preconditioner has the following advantages it does not require the inverse of a non diagonal matrix it applies without any mildly grading assumption on the mesh and it does not require a barycentric refinement of the mesh underlying the trial space

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets

torelli theorems for singularities in this paper we study how const deformations of an isolated hypersurface singularity affect the mixed hodge structure mhs on the cohomology of the milnor fiber using scherk and steenbrink sd description of this mhs and the related approach of varchenko we give a necessary and sufficient condition for the infinitesimal torelli theorem to hold and identify the const stratum in the discriminant of the miniversal deformation after examining the geometric reasons causing torelli to fail for certain deformations we define a local moduli space and prove a local torelli theorem for singularities in the process we complete the work of m saito and malgrange on k saito s conjecture about primitive forms

analysis of a stabilized three fields domain decomposition method in this paper we prove that for suitable choices of the bilinear form involved in the stabilization procedure the stabilized three fields domain decomposition method proposed in 8 is stable and convergent uniformly in the number of subdomains and with respect to their sizes under quite general assumptions on the decomposition and on the discretization spaces the same is proven to hold for the resulting discrete steklov poincare operator