tilting and refined donaldson thomas invariants abstract we study tilting for the heart a of the canonical t structure of the finite dimensional derived category of the ginzburg algebra for a quiver with potential q w we give conditions on that the stable objects for a central charge on a define a sequence of simple tilts from a to a 1 on that conditions the refined donaldson thomas invariant associated to q w is independent of the chosen central charge

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

adjoint based a posteriori analysis of multiscale mortar discretizations with multinumerics in this paper we derive a posteriori error estimates for linear functionals of the solution to an elliptic problem discretized using a multiscale nonoverlapping domain decomposition method the error estimates are based on the solution of an appropriately defined adjoint problem we present a general framework that allows us to consider both primal and mixed formulations of the forward and adjoint problems within each subdomain the primal subdomains are discretized using either an interior penalty discontinuous galerkin method or a continuous galerkin method with weakly imposed dirichlet conditions the mixed subdomains are discretized using raviart thomas mixed finite elements the a posteriori error estimate also accounts for the errors due to adjoint inconsistent subdomain discretizations the coupling between the subdomain discretizations is achieved via a mortar space we show that the numerical discretization error can be broken down into subdomain and mortar components which may be used to drive

multigrid on the interface for mortar mixed finite element methods for elliptic problems we consider mixed finite element approximations of second order elliptic equations on domains that can be described as a union of subdomains or blocks we assume that the subdomain grids are locally defined and need not match across the block boundaries specially chosen mortar finite element spaces are introduced on the interfaces for approximating the scalar variable pressure the mortars also serve as lagrange multipliers for imposing flux matching conditions the method is implemented by reducing the algebraic system to a positive definite interface problem in the mortar spaces this problem is then solved using a multigrid on the interface with conjugate gradient smoothing the algorithm is very efficient in a distributed parallel computing environment as only subdomain solves are required on each conjugate gradient iteration the standard variational assumptions for the multigrid are not satisfied since the interface bilinear forms vary from level to level we present theoretical results for the convergence of the v cycle and the w cycle computational results in two and three dimensions are given to illustrate and confirm the theory

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

implementation of a mortar mixed finite element method using a multiscale flux basis this paper provides a new implementation of a multiscale mortar mixed finite element method for second order elliptic problems the algorithm uses non overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem which is then solved using an iterative method the original implementation by arbogast pencheva wheeler and yotov multiscale model simul 2007 required solving one local dirichlet problem on each subdomain per interface iteration we alter this implementation by forming a multiscale flux basis this basis consists of mortar functions representing the individual flux responses for each mortar degree of freedom on each subdomain independently the computation of these basis functions requires solving a fixed number of dirichlet subdomain problems taking linear combinations of the multiscale flux basis functions replaces the need to solve any dirichlet subdomain problems during the interface iteration this new implementation yields the same solution as the original implementation and is computationally more efficient in cases where the number of interface iterations is greater than the number of mortar degrees of freedom per subdomain the gain in computational efficiency increases with the number of subdomains

robust a posteriori error control and adaptivity for multiscale multinumerics and mortar coupling we consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains termed multinumerics coupled using the mortar technique the grids need not match along the interfaces we are also interested in the multiscale setting where the subdomains are partitioned by a mesh of size h whereas the interfaces are partitioned by a mesh of much coarser size h and where lower order polynomials are used in the subdomains and higher order polynomials are used on the mortar interface mesh we derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm our estimates are also locally efficient and one of them is robust with respect to the ratio h h under an assumption of sufficient regularity of the weak solution the present approach allows bounding separately and comparing mutually the subdomain and interface errors a subdomain interface adaptive refinement s

domain decomposition methods for linear and semilinear elliptic stochastic partial differential equations in this paper we study several overlapping domain decomposition methods for the numerical solutions of some linear and semilinear elliptic stochastic partial differential equations discretized by the finite element methods in particular we show that the algorithms converge and the convergence rates are independent of the finite element mesh parameter as well as the number of subdomains used in the domain decomposition

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

a generalized block fsai preconditioner for nonsymmetric linear systems the efficient solution to nonsymmetric linear systems is still an open issue especially on parallel computers in this paper we generalize to the unsymmetric case the block factorized sparse approximate inverse block fsai preconditioner which has already proved very effective on symmetric positive definite spd problems block fsai is a hybrid approach combining an inner preconditioner with the aim of transforming the system matrix structure to block diagonal with an outer one a block diagonal incomplete or exact factorization intended to improve the conditioning of each block the proposed algorithm is experimented with in a number of large size matrices showing both a good robustness and scalability

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

mixed finite element methods on nonmatching multiblock grids we consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids a mortar finite element space is introduced on the nonmatching interfaces we approximate in this mortar space the trace of the solution and we impose weakly a continuity of flux condition a standard mixed finite element method is used within the blocks optimal order convergence is shown for both the solution and its flux moreover at certain discrete points superconvergence is obtained for the solution and also for the flux in special cases computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory

balancing domain decomposition for mixed finite elements the rate of convergence of the balancing domain decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed the balancing domain decomposition method introduced by mandel in 24 is a substructuring method that involves at each iteration the solution of a local problem with dirichlet data a local problem with neumann data and a coarse grid problem to propagate information globally and to insure the consistency of the neumann problems it is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size in both two and three dimensions and for elements of arbitrary order the bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains the key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the h 1 2 norm of an interplant of the boundary data computational results from a message passing parallel implementation on an intel delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed up for up to 64 processors

gradient bounds for wachspress coordinates on polytopes we derive upper and lower bounds on the gradients of wachspress coordinates defined over any simple convex d dimensional polytope p the bounds are in terms of a single geometric quantity h ast which denotes the minimum distance between a vertex of p and any hyperplane containing a nonincident face we prove that the upper bound is sharp for d 2 and analyze the bounds in the special cases of hypercubes and simplices additionally we provide an implementation of the wachspress coordinates on convex polyhedra using matlab and employ them in a three dimensional finite element solution of the poisson equation on a nontrivial polyhedral mesh as expected from the upper bound derivation the h 1 norm of the error in the method converges at a linear rate with respect to the size of the mesh elements

variational multiscale stabilized fem formulations for transport equations stochastic advection diffusion and incompressible stochastic navier stokes equations an extension of the deterministic variational multiscale vms approach with algebraic subgrid scale sgs modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic navier stokes equations the stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method ssfem generalized polynomial chaos and karhunen loeve expansion techniques are used for representation of uncertain quantities the proposed stabilized method is then applied to various standard advection diffusion and fluid flow examples with uncertainty in essential boundary conditions comparisons are drawn between the numerical solutions and monte carlo analytical solutions wherever possible

multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification in this paper we study multiscale finite element methods for stochastic porous media flow equations as well as applications to uncertainty quantification we assume that the permeability field the diffusion coefficient is stochastic and can be described in a finite dimensional stochastic space this is common in applications where the coefficients are expanded using chaos approximations the proposed multiscale method constructs multiscale basis functions corresponding to sparse realizations and these basis functions are used to approximate the solution on the coarse grid for any realization furthermore we apply our coarse scale model to uncertainty quantification problem where the goal is to sample the porous media properties given an integrated response such as production data our algorithm employs pre computed posterior response surface obtained via the proposed coarse scale model using fast analytical computations of the gradients of this posterior we propose approximate langevin samples these samples are further screened through the coarse scale simulation and finally used as a proposal in metropolis hasting markov chain monte carlo method numerical results are presented which demonstrate the efficiency of the proposed approach

a neumann neumann domain decomposition algorithm for solving plate and shell problems we present a new neumann neumann type preconditioner of large scale linear systems arising from plate and shell problems the advantage of the new method is a smaller coarse space than those of earlier methods of the authors this improves parallel scalability a new abstract framework for neumann neumann preconditioners is used to prove almost optimal convergence properties of the method the convergence estimates are independent of the number of subdomains coefficient jumps between subdomains and depend only polylogarithmically on the number of elements per subdomain we formulate and prove an approximate parametric variational principle for reissner mindlin elements as the plate thickness approaches zero which makes the results applicable to a large class of nonlocking elements in everyday engineering use the theoretical results are confirmed by computational experiments on model problems as well as examples from real world engineering practice

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

a posteriori error estimates for the mortar mixed finite element method several a posteriori error estimators for mortar mixed finite element discretizations of elliptic equations are derived a residual based estimator provides optimal upper and lower bounds for the pressure error an efficient and reliable estimator for the velocity and mortar pressure error is also derived which is based on solving local element problems in a higher order space the interface flux jump term that appears in the estimators can be used as an indicator for driving an adaptive process for the mortar grids only

the space of stability conditions on the local projective plane we study the space of stability conditions on the total space of the canonical bundle over the projective plane we explicitly describe a chamber of geometric stability conditions and show that its translates via autoequivalences cover a whole connected component we prove that this connected component is simply connected we determine the group of autoequivalences preserving this connected component which turns out to be closely related to 1 3 finally we show that there is a submanifold isomorphic to the univer sal covering of a moduli space of elliptic curves with 1 3 level struc ture the morphism is 1 3 equivariant and is given by solutions of picard fuchs equations this result is motivated by the notion of stability and by mirror symmetry

probabilistic collocation method for flow in porous media comparisons with other stochastic methods 1 an efficient method for uncertainty analysis of flow in random porous media is explored in this study on the basis of combination of karhunen loeve expansion and probabilistic collocation method pcm the random log transformed hydraulic conductivity field is represented by the karhunen loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points the procedure is straightforward and analogous to the monte carlo method but the number of simulations required in pcm is significantly reduced steady state flows in saturated random porous media are simulated with the probabilistic collocation method and comparisons are made with other stochastic methods monte carlo method the traditional polynomial chaos expansion pce approach based on galerkin scheme and the moment equation approach based on karhunen loeve expansion klme this study reveals that pcm and klme are more efficient than the galerkin pce approach while the computational efforts are greatly reduced compared to the direct sampling monte carlo method the pcm and klme approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

stochastic collocation and mixed finite elements for flow in porous media the aim of this paper is to quantify uncertainty of flow in porous media through stochastic modeling and computation of statistical moments the governing equations are based on darcy s law with stochastic permeability starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion mixed finite element approximations are used in the spatial domain and collocation at the zeros of tensor product hermite polynomials is used in the stochastic dimensions error analysis is performed and experimentally verified with numerical simulations computational results include incompressible and slightly compressible single and two phase flow

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

stability conditions on k3 surfaces this article contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic k3 surface

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

a stochastic mortar mixed finite element method for flow in porous media with multiple rock types this paper presents an efficient multiscale stochastic framework for uncertainty quantification in modeling of flow through porous media with multiple rock types the governing equations are based on darcy s law with nonstationary stochastic permeability represented as a sum of local karhunen loeve expansions the approximation uses stochastic collocation on either a tensor product or a sparse grid coupled with a domain decomposition algorithm known as the multiscale mortar mixed finite element method the latter method requires solving a coarse scale mortar interface problem via an iterative procedure the traditional implementation requires the solution of local fine scale linear systems on each iteration we employ a recently developed modification of this method that precomputes a multiscale flux basis to avoid the need for subdomain solves on each iteration in the stochastic setting the basis is further reused over multiple realizations leading to collocation algorithms that are more efficient than the traditional implementation by orders of magnitude error analysis and numerical experiments are presented

construction of polygonal interpolants a maximum entropy approach in this paper we establish a link between maximizing information theoretic entropy and the construction of polygonal interpolants the determination of shape functions on n gons n 3 leads to a non unique under determined system of linear equations the barycentric co ordinates i which form a partition of unity are associated with discrete probability measures and the linear reproducing conditions are the counterpart of the expectations of a linear function the i are computed by maximizing the uncertainty h 1 2 n i log i subject to the above constraints the description is expository in nature and the numerical results via the maximum entropy maxent formulation are compared to those obtained from a few distinct polygonal interpolants the maximum entropy formulation leads to a feasible solution for i in any convex or non convex polygon this study is an instance of the application of the maximum entropy principle wherein least biased inference is made on the basis of incomplete information copyright 2004 john wiley sons ltd

stability conditions and quantum dilogarithm identities for dynkin quivers abstract we study the fundamental groups of the exchange graphs for the bounded derived category d q of a dynkin quiver q and the finite dimensional derived category d n q of the calabi yau n ginzburg algebra associated to q in the case of d q we prove that its space of stability conditions in the sense of bridgeland is simply connected as an application we show that the donaldson thomas type invariant associated to q can be calculated as a quantum dilogarithm function on its exchange graph in the case of d n q we show that the faithfulness of the seidel thomas braid group action which is known for q of type a or n 2 implies the simply connectedness of the space of stability conditions

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

sparse grid collocation schemes for stochastic natural convection problems in recent years there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems spectral stochastic methods and monte carlo based sampling methods are two approaches that have been used to analyze these problems as the complexity of the problem or the number of random variables involved in describing the input uncertainties increases these approaches become highly impractical from implementation and convergence points of view this is especially true in the context of realistic thermal flow problems where uncertainties in the topology of the boundary domain boundary flux conditions and heterogeneous physical properties usually require high dimensional random descriptors the sparse grid collocation method based on the smolyak algorithm offers a viable alternate method for solving high dimensional stochastic partial differential equations an extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation we show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured comparison of the present method with the generalized polynomial chaos expansion and monte carlo methods are made

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

t structures on some local calabi yau varieties abstract let z be a smooth fano variety satisfying the condition that the rank of the grothendieck group of z is one more than the dimension of z let z denote the total space of the canonical line bundle of z considered as a non compact calabi yau variety we use the theory of exceptional collections to describe t structures on the derived category of coherent sheaves on z the combinatorics of these t structures is determined by a natural action of an affine braid group closely related to the well known action of the artin braid group on the set of exceptional collections on z

mutations vs seiberg duality abstract for a quiver with potential derksen weyman and zelevinsky defined in h derksen j weyman a zelevinsky quivers with potentials and their representations i mutations arxiv 0704 0649v2 math ra a combinatorial transformation mutations mukhopadhyay and ray on the other hand tell us how to compute seiberg dual quivers for some quivers with potentials through a tilting procedure thus obtaining derived equivalent algebras in this text we compare mutations with the concept of seiberg duality given by s mukhopadhyay k ray seiberg duality as derived equivalence for some quiver gauge theories arxiv hep th 0309191v2 concluding that for a certain class of potentials the good ones mutations coincide with seiberg duality therefore giving derived equivalences

a multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties abstract we here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables it is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales an efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale involving a deterministic operator and local problems at a micro scale for which we have the possibility to use fine approximation spaces global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions convergence properties of these tensor based methods which are closely related to spectral decompositions benefit from the separation of scales different types of uncertainties are considered at the micro level they may be associated with some variability in the operator or source terms or even with some geometrical variability in the latter case specific reformulations of local problems using fictitious domain methods are introduced

an efficient high order perturbation approach for flow in random porous media via karhunen loeve and polynomial expansions in this study we attempt to obtain higher order solutions of the means and co variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of karhunen loeve decomposition polynomial expansion and perturbation methods we first decompose the log hydraulic conductivity y ln ks as an infinite series on the basis of a set of orthogonal gaussian standard random variables i the coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity we then write head as an infinite series whose terms h n represent head of nth order in terms of y the standard deviation of y and derive a set of recursive equations for h n we then decompose h n with polynomial expansions in terms of the products of n gaussian random variables the coefficients in these series are determined by substituting decompositions of y and h n into those recursive equations we solve the mean head up to fourth order in y and the head variances up to third order in y2 we conduct monte carlo mc simulation and compare mc results against approximations of different orders from the moment equation approach based on karhunen loeve decomposition klme we also explore the validity of the klme approach for different degrees of medium variability and various correlation scales it is evident that the klme approach with higher order corrections is superior to the conventional first order approximations and is computationally more efficient than the monte carlo simulation

iterative substructuring preconditioners for mortar element methods in two dimensions the mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains the resulting methods are nonconforming but still yield optimal approximations in this paper we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric second order elliptic equations in two dimensions both spectral and finite element methods for geometrically conforming as well as nonconforming domain decompositions are studied in each case we obtain a polylogarithmic bound on the condition number of the preconditioned matrix

parallel domain decomposition methods for stochastic elliptic equations we present parallel schwarz type domain decomposition preconditioned recycling krylov subspace methods for the numerical solution of stochastic elliptic problems whose coefficients are assumed to be a random field with finite variance karhunen loeve kl expansion and double orthogonal polynomials are used to reformulate the stochastic elliptic problem into a large number of related but uncoupled deterministic equations the key to an efficient algorithm lies in recycling computed subspaces based on a careful analysis of the kl expansion we propose and test a grouping algorithm that tells us when to recycle and when to recompute some components of the expensive computation we show theoretically and experimentally that the schwarz preconditioned recycling gmres method is optimal for the entire family of linear systems a fully parallel implementation is provided and scalability results are reported in the paper

the harder narasimhan system in quantum groups and cohomology of quiver moduli methods of harder and narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations using the hall algebra approach to quantum groups an analog of the harder narasimhan recursion is constructed inside the quantized enveloping algebra of a kac moody algebra this leads to a canonical orthogonal system the hn system in this algebra using a resolution of the recursion an explicit formula for the hn system is given as an application explicit formulas for betti numbers of the cohomology of quiver moduli are derived generalizing several results on the cohomology of quotients in linear algebra type situations

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

a stochastic variational multiscale method for diffusion in heterogeneous random media a stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media the exact solution of the governing equations is split into two components a coarse scale solution that can be captured on a coarse mesh and a subgrid solution a localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse scale solution the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model the generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models finally various numerical examples are considered for evaluating the method

a stochastic multiscale framework for modeling flow through random heterogeneous porous media flow through porous media is ubiquitous occurring from large geological scales down to the microscopic scales several critical engineering phenomena like contaminant spread nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the system any realistic modeling of these transport phenomena has to resolve two key issues i the multi length scale variations in permeability that these systems exhibit and ii the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes a stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features a stochastic analogue to a mixed multiscale finite element framework is used to formulate the physical stochastic multiscale process recent developments in linear and non linear model reduction techniques are used to convert the limited information available about the permeability variation into a viable stochastic input model an adaptive sparse grid collocation strategy is used to efficiently solve the resulting stochastic partial differential equations spdes the framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given

a mixed formulation of mortar based frictionless contact abstract a class of mortar based frictionless contact formulations is derived based on a classical three field mixed variational framework within a penalty regularization complemented by uzawa augmentations discrete mortar constraints are naturally induced by the variational setting major aspects of earlier mortar approaches are obtained through constrained lumped or unconstrained recovery procedures for the mixed kinematic and kinetic mortar quantities from their projected counterparts two and three dimensional examples at the infinitesimal and finite deformation regimes highlight the local and global quality of the contact interactions

a stochastic analysis of steady state two phase flow in heterogeneous media 1 we present a novel approach to modeling stochastic multiphase flow problems for example nonaqueous phase liquid flow in a heterogeneous subsurface medium with random soil properties in particular with randomly heterogeneous intrinsic permeability and soil pore size distribution a stochastic numerical model for steady state water oil flow in a random soil property field is developed using the karhunen loeve moment equation klme approach and is numerically implemented an exponential model is adopted to define the constitutive relationship between phase relative permeability and capillary pressure the log transformed intrinsic permeability y x and soil pore size distribution b x are assumed to be gaussian random functions with a separable exponential covariance function the perturbation part of these two log transformed soil properties is then decomposed into an infinite series based on a set of orthogonal normal random variables xn the phase pressure capillary pressure and phase mobility are decomposed by polynomial expansions and the perturbation method combining these expansions of y x b x and dependent pressures the steady state water oil flow equations and corresponding boundary conditions are reformulated as a series of differential equations up to second order these differential equations are solved numerically and the solutions are directly used to construct moments of phase pressure and capillary pressure we demonstrate the validity of the proposed klme model by favorably comparing firstand second order approximations to monte carlo simulations the significant computational efficiency of the klme approach over monte carlo simulation is also illustrated

a multiscale stochastic finite element method on elliptic problems involving uncertainties in this study a multiscale stochastic finite element method mssfem is developed to resolve scale coupling stochastic elliptic problems based on formulations of a stochastic variational approach and scale bridging multiscale shape functions by employing polynomial chaos expansions and lagrange polynomials stochastic galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods i e variational multiscale method and multiscale finite element method the resulting spectral and pseudo spectral stochastic finite element methods are incorporated into the fine and coarse mesh computation of the mssfem a benchmark multiscale model and the numerical experiment based on the formulated mssfem are provided for illustration it is expected that the proposed mssfem can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data