certified reduced basis method for electromagnetic scattering and radar cross section estimation we study nontrivial applications of the reduced basis method rbm for electromagnetic applications with an emphasis on scattering and the estimation of radar cross section rcs the method and several extensions are explained with two examples with different characteristics parameters that are allowed to vary within the model include frequency incident angle and measurement angle as well as the geometry of the scatterers with appropriate applications of the empirical interpolation method eim transformation of the domain configuration of perfectly matched layer exponential convergence of the reduced basis solution over the entire parameter domain is achieved moreover we demonstrate that this approach allows for the effective capture of the critical behavior in this case through shapes that minimize scattering this further highlights the robustness and quality of the greedy approximation and the reduced basis method approach

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

concordance homomorphisms from knot floer homology abstract we modify the construction of knot floer homology to produce a one parameter family of homologies thfk for knots in s 3 these invariants can be used to give homomorphisms from the smooth concordance group c to z giving bounds on the four ball genus and the concordance genus of knots we give some applications of these homomorphisms

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

mathematical and numerical results on the sensitivity of the pod approximation relative to the burgers equation from mathematical and numerical points of view we study the sensitivity with respect to parametric evolutions of the error obtained by approximating a given parametric partial differential equation using a proper orthogonal decomposition pod basis determined once and for all with association to a fixed parameter more precisely we will be considering the one dimensional burgers equation with a source term

cyclic covers of rings with rational singularities we examine some recent work of phillip griffith on etale covers and fibered products from the point of view of tight closure theory while it is known that cyclic covers of gorenstein rings with rational singularities are cohen macaulay we show this is not true in general in the absence of the gorenstein hypothesis specifically we show that the canonical cover of a q gorenstein ring with rational singularities need not be cohen macaulay

natural norm a posteriori error estimators for reduced basis approximations we present a technique for the rapid and reliable prediction of linear functional outputs of coercive and noncoercive linear elliptic partial differential equations with affine parameter dependence the essential components are i rapidly convergent global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n judiciously selected points in parameter space ii a posteriori error estimation relaxations of the error residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest and iii offiine online computational procedures methods which decouple the generation and projection stages of the approximation process the operation count for the online stage in which given a new parameter value we calculate the output of interest and associated error bound depends only on n typically very small and the parametric complexity of the problem in this paper we propose a new natural norm formulation for our reduced basis error estimation framework that a greatly simplifies and improves our inf sup lower bound construction offline and evaluation online a critical ingredient of our a posteriori error estimators and b much better controls significantly sharpens our output error bounds in particular through deflation for parameter values corresponding to nearly singular solution behavior we apply the method to two illustrative problems a coercive laplacian heat conduction problem which becomes singular as the heat transfer coefficient tends to zero and a non coercive helmholtz acoustics problem which becomes singular as we approach resonance in both cases we observe very economical and sharp construction of the requisite natural norm inf sup lower bound rapid convergence of the reduced basis approximation reasonable effectivities even for near singular behavior for our deflated output error estimators and significant several order of magnitude online computational savings relative to standard finite element procedures

a high performance parallel implementation of the certified reduced basis method abstract the certified reduced basis method herein rb method is a popular approach for model reduction of parametrized partial differential equations in this paper we introduce new techniques that are required to efficiently implement the offline construction stage of the rb method on high performance parallel supercomputers this enables us to generate certified rb models for large scale three dimensional problems that can be evaluated on standard workstations and other thin computing resources with speedup of many orders of magnitude compared to the corresponding full order model we use our implementation to perform detailed numerical studies for two computationally expensive model problems a natural convection fluid flow problem and a many parameter heat transfer problem in the heat transfer problem we exploit the computational efficiency of the rb method to perform a detailed study of snapshot selection in the greedy algorithm and we also examine statistics of the output sensitivity derivatives to obtain a global view of the relative importance of the parameters

connected floer homology of covering involutions using the covering involution on the double branched cover of s 3 branched along a knot and adapting ideas of hendricks manolescu and hendricks hom lidman we define new knot concordance invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots

a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient in this work a reduced basis rb approach is used to solve a large number of boundary value problems bvps parametrized by a stochastic input expressed as a karhunen loeve expansion in order to compute outputs that are smooth functionals of the random solution fields the rb method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the rb approach developed previously for deterministic systems however the stochastic framework requires the development of new a posteriori estimates for statistical outputs such as the first two moments of integrals of the random solution fields these error bounds in turn permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many query context

multi element stochastic reduced basis methods this paper presents multi element stochastic reduced basis methods me srbms for solving linear stochastic partial differential equations in me srbms the domain of definition of the random inputs is decomposed into smaller subdomains or random elements stochastic reduced basis methods srbms are employed in each random element to evaluate the response statistics these elemental statistics are assimilated to compute the overall statistics the effectiveness of the method is demonstrated by solving the stochastic steady state heat transfer equation on two geometries involving different types of boundary conditions numerical studies are conducted to investigate the h convergence rates of global and local preconditioning strategies

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

estimation of the error in the reduced basis method solution of nonlinear equations the reduced basis method is a projection technique for approximating the solution curve of a finite system of nonlinear algebraic equations by the solution curve of a related system that is typically of much lower dimension in this paper the reduced basis error is shown to be dominated by an approximation error this in turn leads to error estimates for projection onto specific subspaces for example subspaces related to taylor lagrange and discrete least squares approximation

an hp proper orthogonal decomposition moving least squares approach for molecular dynamics simulation abstract the application of model order reduction for molecular dynamic systems exhibits intrinsic complexities due to the highly nonlinear and nonlocal properties of such systems the most costly computational part of such work is the calculation of potential energy and inter atomic force fields we introduce the so called hp proper orthogonal decomposition moving least squares hp pod mls method not only to tackle this force computation but also to improve the accuracy of reduced displacement approximations in an adaptive and controllable manner the approach combines the proper orthogonal decomposition pod method with the moving least squares mls one to interpolate the inter atomic force field at an arbitrary time instance this approximated force field is then fed into the reduced molecular dynamic md governing equation to compute the reduced order displacement field in fact this pod mls approach is already able to handle efficiently the force computation in the online computational stage i e the online force computational cost is independent of the dimension of the full md space however its corresponding approximated displacement error will be very large if the considered total time interval is long the proposed hp pod mls algorithm will divide adaptively and automatically the global temporal interval into many smaller local subintervals h refinement and then build the sets of local reduced basis functions over each of these subintervals p refinement the goal is to reduce the corresponding displacement errors to a desired tolerance over these subintervals the effectiveness of the algorithm is demonstrated by performing reduced order simulations of several molecular dynamic systems

a mathematical and numerical study of the sensitivity of a reduced order model by pod rom pod for a 2d incompressible fluid flow abstract in this work we present contributions concerning a mathematical study of the sensitivity of a reduced order model rom by the proper orthogonal decomposition pod technique applied to a quasi linear parabolic equation in particular we apply our theoretical study to the navier stokes equations for a 2d incompressible fluid flow we present a numerical test of our theoretical result for an unsteady fluid flow in a channel around a circular cylinder

reduced basis anova methods for partial differential equations with high dimensional random inputs in this paper we present a reduced basis anova approach for partial deferential equations pdes with random inputs the anova method combined with stochastic collocation methods provides model reduction in high dimensional parameter space through decomposing high dimensional inputs into unions of low dimensional inputs in this work to further reduce the computational cost we investigate spatial low rank structures in the anova collocation method and develop efficient spatial model reduction techniques using hierarchically generated reduced bases we present a general mathematical framework of the methodology validate its accuracy and demonstrate its efficiency with numerical experiments

on the upsilon invariant and satellite knots we study the effect of satellite operations on the upsilon invariant of ozsvath stipsicz szabo we obtain results concerning when a knot and its satellites are independent for example we show that the set d 2 i 1 i 1 infty is a basis for an infinite rank summand of the group of smooth concordance classes of topologically slice knots for d the positive clasped untwisted whitehead double of any knot with positive tau invariant e g the right handed trefoil we also prove that the image of the mazur satellite operator on the smooth knot concordance group contains an infinite rank subgroup of topologically slice knots

the generalized empirical interpolation method stability theory on hilbert spaces with an application to the stokes equation the generalized empirical interpolation method geim is an extension first presented by maday and mula in 1 in 2013 of the classical empirical interpolation method presented in 2004 by barrault maday nguyen and patera in 2 where the evaluation at interpolating points is replaced by the more practical evaluation at interpolating continuous linear functionals on a class of banach spaces as outlined in 1 this allows to relax the continuity constraint in the target functions and expand both the application domain and the stability of the approach in this paper we present a thorough analysis of the concept of stability condition of the generalized interpolant the lebesgue constant by relating it to an inf sup problem in the case of hilbert spaces in the second part of the paper it will be explained how geim can be employed to monitor in real time physical experiments by providing an online accurate approximation of the phenomenon that is computed by combining the acquisition of a minimal number optimally placed measurements from the processes with their mathematical models parameter dependent pdes this idea is illustrated through a parameter dependent stokes problem in which it is shown that the pressure and velocity fields can e ciently be reconstructed with a relatively low dimensional interpolation space

spectral stochastic homogenization of divergence type pdes this paper presents a formulation and numerical analysis of the homogenization of stochastic pdes the framework of homogenization is adopted to describe an effective medium that is equivalent in some sense to a heterogeneous medium of interest the parameters of the resulting homogeneous medium are described as stochastic processes characterized by their polynomial chaos decomposition the formulation yields a chaos decomposition for the predicted behavior of the homogeneous medium that captures in addition to the effect of heterogeneity the effect of variability once this description has been computed various statistics of the solution can be efficiently evaluated

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

parameter multi domain hp empirical interpolation summary in this paper we introduce two parameter multi domain hp techniques for the empirical interpolation method eim in both approaches we construct a partition of the original parameter domain into parameter subdomains h refinement we apply the standard eim independently within each subdomain to yield local in parameter approximation spaces p refinement further for a particularly simple case we introduce a priori convergence theory for the partition procedure we show through two numerical examples that our approaches provide significant reduction in the eim approximation space dimension and thus significantly reduce the computational cost associated with eim approximations copyright 2012 john wiley sons ltd

holomorphic disks and three manifold invariants properties and applications in 27 we introduced floer homology theories hf y s hf y s hf y t hf y s and hf red y s associated to closed oriented three manifolds y equipped with a spiny structures s spin c y in the present paper we give calculations and study the properties of these invariants the calculations suggest a conjectured relationship with seiberg witten theory the properties include a relationship between the euler characteristics of hf and turaev s torsion a relationship with the minimal genus problem thurston norm and surgery exact sequences we also include some applications of these techniques to three manifold topology

an hp certified reduced basis method for parametrized elliptic partial differential equations we present a new hp parameter multidomain certified reduced basis rb method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations we propose and provide theoretical justification for a new procedure for adaptive partition h refinement of the parameter domain into smaller parameter subdomains we pursue a hierarchical splitting of the parameter sub domains based on proximity to judiciously chosen parameter anchor points within each subdomain subsequently we construct individual standard rb approximation spaces p refinement over each subdomain greedy parameter sampling procedures and a posteriori error estimation play important roles in both the h type and p type stages of the new algorithm we present illustrative numerical results for a convection diffusion problem the new hp approach is considerably faster respectively more costly than the standard p type reduced basis method in the online respectively offline stage

tour of bordered floer theory heegaard floer theory is a kind of topological quantum field theory tqft assigning graded groups to closed connected oriented 3 manifolds and group homomorphisms to smooth oriented four dimensional cobordisms bordered heegaard floer homology is an extension of heegaard floer homology to 3 manifolds with boundary with extended tqft type gluing properties in this survey we explain the formal structure and construction of bordered floer homology and sketch how it can be used to compute some aspects of heegaard floer theory

interpolation method for adapting reduced order models and application to aeroelasticity reduced order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real time analysis although most reduced order models can operate in near real time their construction can be computationally expensive as it requires accumulating a large number of system responses to input excitations furthermore reduced order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation together these two issues underline the need for a fast and robust method for adapting precomputed reduced order models to new sets of physical or modeling parameters to this effect this paper presents an interpolation method based on the grassmann manifold and its tangent space at a point that is applicable to structural aerodynamic aeroelastic and many other reduced order models based on projection schemes this method is illustrated here with the adaptation of computational fluid dynamics based aeroelastic reduced order models of complete fighter configurations to new values of the freestream mach number good correlations with results obtained from direct reduced order model reconstruction high fidelity nonlinear and linear simulations are reported thereby highlighting the potential of the proposed reduced order model adaptation method for near real time aeroelastic predictions using precomputed reduced order model databases

stochastic reduced basis methods stochastic reduced basis methods for solving large scale linear random algebraic systems of equations such as those obtained by discretizing linear stochastic partial differential equations in space time and the random dimension are introduced the fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients or random functions we present a theoretical justification for employing basis vectors spanning the preconditioned stochastic krylov subspace to approximate the response process subsequently variants of the bubnov galerkin scheme are employed to compute the undetermined coefficients which allow explicit expressions for the response quantities to be derived we also examine some theoretical properties of the projection scheme and procedures for computing the response statistics numerical studies are presented for static and dynamic analysis of stochastic structural systems we demonstrate that significant improvements over the neumann expansion scheme as well as other relevant techniques in the literature can be achieved

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

a dimensional reduction approach based on the application of reduced basis methods in the framework of hierarchical model reduction in this article we introduce a new dimensional reduction approach which is based on the application of reduced basis rb techniques in the hierarchical model reduction hmr frame work considering problems that exhibit a dominant spatial direction the idea of hmr is to perform a galerkin projection onto a reduced space which combines the full solution space in the dominant direction with a reduction space in the transverse direction the latter is spanned by modal or thonormal basis functions while so far the basis functions in the hmr approach have been chosen a priori s perotto a ern and a veneziani multiscale model simul 8 2010 pp 1102 1127 for instance as legendre or trigonometric polynomials in this work a highly nonlinear approximation is employed for the construction of the reduction space to this end we first derive a lower dimen sional parametrized problem in the transverse direction from the full problem where the parameters reflect the influence from the unknown solution in the dominant direction exploiting the good ap proximation properties of rb methods we then construct a reduction space by applying a proper orthogonal decomposition to a set of snapshots of the parametrized partial differential equation for an efficient construction of the snapshot set we apply adaptive refinement in parameter space based on an a posteriori error estimate that is also derived in this article we introduce our method for general elliptic problems such as advection diffusion equations in two space dimensions numerical experiments demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full dimensional problem and the computational efficiency of our new adaptive approach

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

nonlinear model reduction via a locally weighted pod method summary in this article we propose a new approach for model reduction of parameterized partial differential equations pdes by a locally weighted proper orthogonal decomposition lwpod method the presented approach is particularly suited for large scale nonlinear systems characterized by parameter variations instead of using a global basis to construct a global reduced model lwpod approximates the original system by multiple local reduced bases each local reduced basis is generated by the singular value decomposition of a weighted snapshot matrix compared with global model reduction methods such as the classical proper orthogonal decomposition lwpod can yield more accurate solutions with a fixed subspace dimension as another contribution we combine lwpod with the chord iteration to solve elliptic pdes in a computationally efficient fashion the potential of the method for achieving large speedups while maintaining good accuracy is demonstrated for both elliptic and parabolic pdes in a few numerical examples copyright 2016 john wiley sons ltd

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

element boundary terms in reduced order models for flow problems domain decomposition and adaptive coarse mesh hyper reduction abstract in this paper we present a finite element based reduced order model and in particular we consider two aspects related to the introduction of inter element boundary terms in the formulation the first is a domain decomposition strategy in which the transmission conditions involve boundary terms to account for non matching meshes and discontinuous physical properties the second is a coarse mesh hyper reduction for which we propose an adaptive refinement driven by an a posteriori error estimator that contains element boundary terms as the finite element full order model the reduced order model is based on the variational multi scale framework with sub grid scales defined not only in the element interiors but also on the inter element boundaries we present some examples of application using the incompressible navier stokes equations and the boussinesq approximation

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

a certified reduced basis method for the fokker planck equation of dilute polymeric fluids fene dumbbells in extensional flow in this paper we present a reduced basis method for the parametrized fokker planck equation associated with evolution of finitely extensible nonlinear elastic fene dumbbells in a newtonian solvent for a prescribed extensional macroscale flow we apply a proper orthogonal decomposition pod greedy sampling procedure for the stable identification of optimal reduced basis spaces and we develop a rigorous finite time a posteriori bound for the error in the reduced basis prediction of the two outputs of interest the optical anisotropy and the first normal stress difference we present numerical results for stress conformation hysteresis as a function of weissenberg number and final time that demonstrate the rapid convergence of the reduced basis approximation and the effectiveness of the a posteriori error bounds

holomorphic disks and knot invariants abstract we define a floer homology invariant for knots in an oriented three manifold closely related to the heegaard floer homologies for three manifolds defined in an earlier paper we set up basic properties of these invariants including an euler characteristic calculation and a description of the behavior under connected sums then we establish a relationship with hf for surgeries along the knot applications include calculation of hf of three manifolds obtained by surgeries on some special knots in s 3 and also calculation of hf for certain simple three manifolds which fiber over the circle

a weighted reduced basis method for elliptic partial differential equations with random input data in this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations pdes with random input data the pdes are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure a priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems

an online manifold learning approach for model reduction of dynamical systems this article discusses a newly developed online manifold learning method subspace iteration using reduced models sirm for the dimensionality reduction of dynamical systems this method may be viewed as subspace iteration combined with a model reduction procedure specifically starting with a test solution the method solves a reduced model to obtain a more precise solution and it repeats this process until sufficient accuracy is achieved the reduced model is obtained by projecting the full model onto a subspace that is spanned by the dominant modes of an extended data ensemble the extended data ensemble in this article contains not only the state vectors of some snapshots of the approximate solution from the previous iteration but also the associated tangent vectors therefore the proposed manifold learning method takes advantage of the information of the original dynamical system to reduce the dynamics moreover the learning procedure is computed in the online stage as opposed to being compute

localized discrete empirical interpolation method this paper presents a new approach to construct more efficient reduced order models for nonlinear partial differential equations with proper orthogonal decomposition and the discrete empirical interpolation method deim whereas deim projects the nonlinear term onto one global subspace our localized discrete empirical interpolation method ldeim computes several local subspaces each tailored to a particular region of characteristic system behavior then depending on the current state of the system ldeim selects an appropriate local subspace for the approximation of the nonlinear term in this way the dimensions of the local deim subspaces and thus the computational costs remain low even though the system might exhibit a wide range of behaviors as it passes through different regimes ldeim uses machine learning methods in the offline computational phase to discover these regions via clustering local deim approximations are then computed for each cluster in the online computational phase machine

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

a reduced basis method for parametrized variational inequalities reduced basis methods are an efficient tool for significantly reducing the computational complexity of solving parametrized pdes originally introduced for elliptic equations they have been generalized during the last decade to various types of elliptic parabolic and hyperbolic systems in this article we extend the reduction technique to parametrized variational inequalities first we propose a reduced basis variational inequality scheme in a saddle point form and prove existence and uniqueness of the solution we state some elementary analytical properties of the scheme such as reproduction of solutions a priori stability with respect to the data and lipschitz continuity with respect to the parameters an offline online decomposition guarantees an efficient assembling of the reduced scheme which can be solved by constrained quadratic programming second we provide rigorous a posteriori error bounds with a partial offline online decomposition the reduction scheme is applied to one dimensional o

on cobordisms between knots braid index and the upsilon invariant we use ozsvath stipsicz and szabo s upsilon invariant to provide bounds on cobordisms between knots that contain full twists in particular we recover and generalize a classical consequence of the morton franks williams inequality for knots positive braids that contain a positive full twist realize the braid index of their closure we also establish that quasi positive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure finally we provide inductive formulas for the upsilon invariant of torus knots and compare it to the levine tristram signature profile

multi symbolic rees algebras and strong f regularity let i be a divisorial ideal of a strongly f regular ring a k i watanabe raised the question whether the symbolic rees algebra mathcal r s i oplus n ge 0 i n is cohen macaulay whenever it is noetherian we develop the notion of multi symbolic rees algebras and use this to show that mathcal r s i is indeed cohen macaulay whenever a certain auxiliary ring is finitely generated over a

new quantum obstructions to sliceness it is well known that generic perturbations of the complex frobenius algebra used to define khovanov cohomology each give rise to rasmussen s concordance invariant ss this gives a concordance homomorphism to the integers and a strong lower bound on the smooth slice genus of a knot similar behavior has been observed in sl n sl n khovanov rozansky cohomology where a perturbation gives rise to the concordance homomorphisms snsn for each n 2n 2 and where we have s2 ss2 s we demonstrate that snsn for n 3n 3 does not in fact arise generically and that varying the chosen perturbation gives rise both to new concordance homomorphisms and to new sliceness obstructions that are not equivalent to concordance homomorphisms

absolutely graded floer homologies and intersection forms for four manifolds with boundary abstract in ozsvath and szabo holomorphic triangles and invariants for smooth four manifolds math sg 0110169 2001 we introduced absolute gradings on the three manifold invariants developed in ozsvath and szabo holomorphic disks and topological invariants for closed three manifolds math sg 0101206 ann of math 2001 to appear coupled with the surgery long exact sequences we obtain a number of three and four dimensional applications of this absolute grading including strengthenings of the complexity bounds derived in ozsvath and szabo holomorphic disks and three manifold invariants properties and applications math sg 0105202 ann of math 2001 to appear restrictions on knots whose surgeries give rise to lens spaces and calculations of hf for a variety of three manifolds moreover we show how the structure of hf constrains the exoticness of definite intersection forms for smooth four manifolds which bound a given three manifold in addition to these new applications the techniques also provide alternate proofs of donaldson s diagonalizability theorem and the thom conjecture for cp 2

a surgery sequence in dimension four the relations with knot concordance we present a systematic treatment of the classification problem for compact smooth 4 manifolds m it is modeled on the surgery exact sequence the central theorem in the classification of n manifolds n 5 the price for the extension to dimension 4 is a hole in m where a homotopy 1 skeleton should be there is no homotopy theoretic or surgical obstruction to complet ing m with a wedge of circles so that the completion has the topology of a compact smooth manifold this point set problem is all that stands between 4 manifolds and the tranquility that prevails in higher dimensions when m is simply connected only a point is missing from the model the applications of this are discussed in f and fq 1 the general theory has application to knot and link theory in particular knots with alexander polynomial 1 are characterized geometrically as knots admitting a certain

khovanov homology and the slice genus we use lee s work on the khovanov homology to define a knot invariant s we show that s k is a concordance invariant and that it provides a lower bound for the smooth slice genus of k as a corollary we give a purely combinatorial proof of the milnor conjecture

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

reduced basis approximation and a posteriori error estimation for a boltzmann model abstract we present a reduced basis approximation and associated rigorous a posteriori output flow rate error bound for a simple linearized bgk boltzmann model for flow in a channel we consider two parameters the knudsen number and the accommodation coefficient the reduced basis approximation is built upon and the reduced basis error is measured relative to a truth mapped weighted spectral discretization of a stabilized petrov galerkin weak formulation we exploit the affine nature of the parametric dependence to provide an offline online computational procedure that decouples the spectral and reduced basis quantities the computational complexity of the online stage in which given a new parameter value we evaluate the reduced basis flow rate prediction and associated error bound depends only on the typically very small dimension of the reduced basis space n and not on the typically very large dimension of the truth spectral approximation n we demonstrate that the reduced basis approximation converges very rapidly with n and that the error estimator effectivities remain bounded albeit somewhat large in the online stage the reduced basis approximation achieves a certifiable relative accuracy of 10 2 with respect to the truth spectral result at only 1 100th the computational effort the method is appropriate in the many query and real time e g parameter estimation contexts in which there is a premium on reduced marginal cost and a tolerance for significant offline pre processing effort

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

stochastic finite elements as a bridge between random material microstructure and global response this paper presents a finite element method aimed at the introduction of microstructural material randomness below the level of a single finite element a consideration of dependence of effective moduli on scale and on the defining boundary conditions leads to an identification of a finite element as a mesoscale window or a mesoscale finite element in a stochastic finite element method sfe an estimation of the global response can be obtained through bounds stemming from minimum potential energy and complementary energy principles which involve dirichlet and neumann boundary conditions on all the mesoscale finite elements respectively while in the classical case of a homogeneous material these two bounds converge to each other as the finite element mesh becomes sufficiently fine an optimal mesoscale with respect to the difference between both bounds may exist in the case of a heterogeneous material the proposed sfe method is illustrated with a numerical example of a sample two phase voronoi composite with some 26 000 grains where a reference solution taking into account the entire microstructure without any smearing out is shown to fall between both energy bounds a generalization to an ensemble response is straightforward

f rationality of rees algebras in this paper using the notion of the tight integral closure we will give a criterion for f rationality of rees algebras of m primary ideals in a cohen macaulay local ring as its application we prove the following results 1 in dimension two if a is f rational and i is integrally closed then the rees algebra r i is f rational on the other hand in higher dimensions we construct many examples of cohen macaulay normal rees algebras which are not f rational 2 if both a and r i are f rational then so is the extended rees algebra r i 3 if r i is f rational and a g i 1 then a is f rational on the other hand using resolution of singularities we will prove that a two dimensional rational singularity always admits f rational rees algebras in particular this theorem gives another way than that devised by watanabe 1997 j pure appl algebra122 323 328 to construct counterexamples to the boutot type theorem for f rational rings