a theory of algebraic cocycles we introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an eilenberg maclane space using these cocycles we develop a cohomology theory for complex algebraic varieties the theory is bigraded functorial and admits gysin maps it carries a natural cup product and a pairing to l homology chern classes of algebraic bundles are defined in the theory there is a natural transformation to singular integral cohomology theory that preserves cup products computations in special cases are carried out on a smooth variety it is proved that there are algebraic cocycles in each algebraic rational p p cohomology class in this announcement we present the outlines of a cohomology theory for algebraic varieties based on a new concept of an algebraic cocycle details will appear in fl our cohomology is a companion to the l homology theory recently studied in l f l f1 l f2 fm this homology is a bigraded theory based directly on the structure of the space of algebraic cycles it admits a natural transformation to integral homology that generalizes the usual map taking a cycle to its homology class our new cohomology theory is similarly bigraded and based on the structure of the space of algebraic cocycles it carries a ring structure coming from the complex join an elementary construction of projective geometry and it admits a natural transformation to integral cohomology chern classes are defined in the theory and transform under to the usual ones our definition of cohomology is very far from a duality construction on l homology nonetheless there is a natural and geometrically defined kronecker pairing between our morphic cohomology and l homology the foundation stone of our theory is the notion of an effective algebraic cocycle which is of some independent interest roughly speaking such a cocycle on a variety x with values in a projective variety y is a morphism from x to the space of cycles on y when x is normal this is equivalent by graphing to a cycle on x y with equidimensional fibres over x such cocycles abound in algebraic geometry and arise naturally in many circumstances the simplest perhaps is that of a flat morphism f x y whose corresponding cocycle associates to x x the pullback cycle f x many more arise naturally from synthetic constructions in projective geometry we show that every variety is rich in cocycles indeed if 1991 mathematics subject classification primary 14f99 14c05

algebraic cycles and infinite loop spaces in this paper we use recent results about the topology of chow varieties to answer an open question in infinite loop space theory that is we construct an infinite loop space structure on a certain product of eilenberg maclane spaces so that the total chern map is an infinite loop map an analogous result for the total stiefel whitney map is also proved further results on the structure of stabilized spaces of alebraic cycles are obtained and computational consequences are also outlined

proper orthogonal decomposition for parameter estimation in oscillating biological networks proper orthogonal decomposition pod is frequently applied to estimate parameters of partial differential equations this study examines the application of the pod method in estimating the parameters of an ordinary differential equation ode model of stable oscillating biological networks the mathematical model used to simulate molecular interactions in these oscillating networks is related to the gause lotka volterra equations the findings reveal that pod generates accurate estimates of the parameters even in the presence of experimental noise furthermore extrapolating biologically measured data points to a number of oscillations improves the curve fits c 1 approximations and parameter estimations

integral deligne cohomology for real varieties we develop an integral version of deligne cohomology for smooth proper real varieties for this purpose the role played by singular cohomology in the complex case has to be replaced by the ordinary bigraded gal mathbb c mathbb r equivariant cohomology of lewis et al bull am math soc n s 4 2 208 212 1981 the equivariant counterpart of singular cohomology the theory is aimed at giving more precise information about the 2 primary components of regulators we establish basic properties and give a geometric interpretation for the groups in dimension 2 in weights 1 and 2

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

a low cost goal oriented compact proper orthogonal decomposition basis for model reduction of static systems a novel model reduction technique for static systems is presented the method is developed using a goal oriented framework and it extends the concept of snapshots for proper orthogonal decomposition pod to include sensitivity derivatives of the state with respect to system input parameters the resulting reduced order model generates accurate approximations due to its goal oriented construction and the explicit training of the model for parameter changes the model is less computationally expensive to construct than typical pod approaches since efficient multiple right hand side solvers can be used to compute the sensitivity derivatives the effectiveness of the method is demonstrated on a parameterized aerospace structure problem

galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics error estimates for galerkin proper orthogonal decomposition pod methods for nonlinear parabolic systems arising in fluid dynamics are proved for the time integration the backward euler scheme is considered the asymptotic estimates involve the singular values of the pod snapshot set and the grid structure of the time discretization as well as the snapshot locations

a best points interpolation method for efficient approximation of parametrized functions we present an interpolation method for efficient approximation of parametrized functions the method recognizes and exploits the low dimensional manifold structure of the parametrized functions to provide good approximation basic ingredients include a specific problem dependent basis set defining a low dimensional representation of the parametrized functions and a set of best interpolation points capturing the spatial parameter variation of the parametrized functions the best interpolation points are defined as solution of a least squares minimization problem which can be solved efficiently using standard optimization algorithms the approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure in addition an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost numerical results are presented to demonstrate the accuracy and efficiency of the method copyright 2007 john wiley sons ltd

evaluation of galerkin and petrov galerkin model reduction for finite element approximations of the shallow water equations abstract the shallow water equations swe are widely used to model depth averaged free surface flows they are well studied and many successful numerical methods have been posited for their solution nevertheless accurate resolution of shallow water flows can still be very computationally demanding in particular computational expense is a significant hurdle for inclusion of well resolved shallow water models in engineering applications involving risk assessment optimal design or parameter estimation the swe are then a natural candidate for model reduction in previous work by lozovskiy et al 2016 we presented an approach based on proper orthogonal decomposition pod and hyper reduction for a class of semi implicit stabilized finite element approximations of the swe here we revisit this scheme to further evaluate its consistency and performance using different snapshot collection procedures we also consider a closely related petrov galerkin technique based on least squares minimization as an alternative to standard galerkin projection we formulate both schemes in a stabilized finite element context and examine their consistency rigorously both with and without hyper reduction we then evaluate their accuracy and robustness for test problems in one and two space dimensions

the singular set of ω minima we consider minima open image in new window of convex variational integrals in the vectorial case n n 2 and we provide estimates for the hausdorff dimension of their singular sets

algebraic cycles and homotopy theory examine its homotopy groups which after a certain idealization of the space turn out to be astonishingly simple in fact for complex projective space pn the structure can be understood completely this yields new information about the topology of the classical chow varieties and establishes an explicit relationship with universal cohomology operations in the general case complete computations are difficult but we shall establish a complex suspension theorem and lay the foundations for a theory based on the homotopy groups of chow varieties to state the results we must give precise meaning to the space of subvarieties we begin with the fundamental case of subvarieties in complex projective n space pln for each pair of integers p and d with d 1 and 0 p n consider the set wp d pn of all finite formal sums

equivariant cohomology and sheaves abstract ro g graded equivariant cohomology was originally defined by using equivariant spectra in this paper we construct a sheaf version of ro g graded equivariant cohomology theory by introducing a complex of presheaves we obtain an isomorphism theorem of two different cohomology theories we further apply the results to the study of real algebraic varieties

novel modal methods for transient analysis with a reduced order model based on enhanced craig bampton formulation abstract for transient analysis of structural dynamics with reduced order model rom data recovery procedures that use modal methods such as the classical mode acceleration ma and modal displacement md methods are an important step in order to increase the convergence and accuracy of the solution in this work we propose novel ma and md methods for highly accurate transient analysis with a reduced order model based on enhanced craig bampton ecb formulation which is an extension of the classical craig bampton cb method that is a way to reduce the size of a finite element fe model the performance of the proposed data recovery approach is demonstrated with two numerical examples we also investigate the relation between the proposed and classical ma and md methods

pod deim nonlinear model order reduction of an adi implicit shallow water equations model in the present paper we consider a 2 d shallow water equations swe model on a b plane solved using an alternating direction fully implicit adi finite difference scheme on a rectangular domain the scheme was shown to be unconditionally stable for the linearized equations the discretization yields a number of nonlinear systems of algebraic equations we then use a proper orthogonal decomposition pod to reduce the dimension of the swe model due to the model nonlinearities the computational complexity of the reduced model still depends on the number of variables of the full shallow water equations model by employing the discrete empirical interpolation method deim we reduce the computational complexity of the reduced order model due to its depending on the nonlinear full dimension model and regain the full model reduction expected from the pod model to emphasize the cpu gain in performance due to use of pod deim we also propose testing an explicit euler finite difference scheme ee as an alternative to the adi implicit scheme for solving the swallow water equations model we then proceed to assess the efficiency of pod deim as a function of number of spatial discretization points time steps and pod basis functions as was expected our numerical experiments showed that the cpu time performances of pod deim schemes are proportional to the number of mesh points once the number of spatial discretization points exceeded 10000 and for 90 deim interpolation points the cpu time decreased by a factor of 10 in case of pod deim implicit swe scheme and by a factor of 15 for the pod deim explicit swe scheme in comparison with the corresponding pod swe schemes moreover our numerical tests revealed that if the number of points selected by deim algorithm reached 50 the approximation errors due to pod deim and pod reduced systems have the same orders of magnitude thus supporting the theoretical results existing in the literature

algebraic cycles on real varieties and z 2 equivariant homotopy theory in this paper the spaces of algebraic cycles on a real projective variety x are studied as z 2 spaces under the action of the galois group gal c r in particular the equivariant homotopy type of the group of algebraic p cycles zp pc is computed a version of lawson homology for real varieties is proposed the real lawson homology groups are computed for a class of real varieties

a state space error estimate for pod deim nonlinear model reduction this paper derives state space error bounds for the solutions of reduced systems constructed using proper orthogonal decomposition pod together with the discrete empirical interpolation method deim recently developed for nonlinear dynamical systems siam j sci comput 32 2010 pp 2737 2764 the resulting error estimates are shown to be proportional to the sums of the singular values corresponding to neglected pod basis vectors both in galerkin projection of the reduced system and in the deim approximation of the nonlinear term the analysis is particularly relevant to ode systems arising from spatial discretizations of parabolic pdes the derivation clearly identifies where the parabolicity is crucial it also explains how the deim approximation error involving the nonlinear term comes into play

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

gradient estimates below the duality exponent we show sharp local a priori estimates and regularity results for possibly degenerate non linear elliptic problems with data not lying in the natural dual space we provide a precise non linear potential theoretic analog of classical potential theory results due to adams duke math j 42 765 778 1975 and adams and lewis studia math 74 169 182 1982 concerning morrey spaces imbedding regularity properties for this we introduce a technique allowing for a non local representation of solutions via riesz potentials in turn yielding optimal local estimates simultaneously in both rearrangement and non rearrangement invariant function spaces in fact we also derive sharp estimates in lorentz spaces covering borderline cases which remained open for some while

compressive projection principal component analysis principal component analysis pca is often central to dimensionality reduction and compression in many applications yet its data dependent nature as a transform computed via expensive eigendecomposition often hinders its use in severely resource constrained settings such as satellite borne sensors a process is presented that effectively shifts the computational burden of pca from the resource constrained encoder to a presumably more capable base station decoder the proposed approach compressive projection pca cppca is driven by projections at the sensor onto lower dimensional subspaces chosen at random while the cppca decoder given only these random projections recovers not only the coefficients associated with the pca transform but also an approximation to the pca transform basis itself an analysis is presented that extends existing rayleigh ritz theory to the special case of highly eccentric distributions this analysis in turn motivates a reconstruction process at the cppca decoder that consists of a novel eigenvector reconstruction based on a convex set optimization driven by ritz vectors within the projected subspaces as such cppca constitutes a fundamental departure from traditional pca in that it permits its excellent dimensionality reduction and compression performance to be realized in an light encoder heavy decoder system architecture in experimental results cppca outperforms a multiple vector variant of compressed sensing for the reconstruction of hyperspectral data

comparative numerical analysis using reduced order modeling strategies for nonlinear large scale systems we perform a comparative analysis using three reduced order strategies missing point estimation mpe method gappy pod method and discrete empirical interpolation method deim applied to a biological model describing the spatio temporal dynamics of a predator prey community the comparative study is focused on the efficiency of the reduced order approximations and the complexity reduction of the nonlinear terms different variants are discussed related to the projection based model reduction framework combined with selective spatial sampling to efficiently perform the online computations numerical results are presented

higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth abstract the aim of this paper is to derive the self improving property of integrability for the spatial gradient of solutions to degenerate parabolic obstacle problem with irregular obstacles and p x t nonstandard growth more precisely we prove that the spatial gradient of the solution is integrable to a larger power than the natural one determined by the structural assumptions on the involved differential operator

vanishing theorems for real algebraic cycles we establish the analogue of the friedlander mazur conjecture for teh s reduced lawson homology groups of real varieties which says that the reduced lawson homology of a real quasi projective variety x vanishes in homological degrees larger than the dimension of x in all weights as an application we obtain a vanishing of homotopy groups of the mod 2 topological groups of averaged cycles and a characterization in a range of indices of dos santos real lawson homology as the homotopy groups of the topological group of averaged cycles we also establish an equivariant poincare duality between equivariant friedlander walker real morphic cohomology and dos santos real lawson homology we use this together with an equivariant extension of the mod 2 beilinson lichtenbaum conjecture to compute some real lawson homology groups in terms of bredon cohomology

global gradient estimates for the parabolic p x t laplacian equation abstract in this paper we consider the global gradient estimates for weak solutions of the parabolic p x t laplacian type equation in a reifenberg flat domain

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

analysis of mixed finite element method mfem for solving the generalized fractional reaction diffusion equation on nonrectangular domains abstract in the current manuscript we consider a generalized fractional reaction diffusion equation the considered model is based on the time fractional derivative the developed scheme is based on two procedures at first we obtain a semi discrete scheme for the temporal direction the time fractional derivative is discretized by using a difference scheme with second order accuracy thus the final time discrete formula has second order accuracy in the temporal direction then we obtain a fully discrete scheme by applying the mixed finite element method mfem for the constructed numerical technique we prove the unconditional stability and also obtain an error bound for the full discrete scheme by using the energy method we employ some test problems to show the accuracy of the proposed technique

nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries we demonstrate the synthesis of sparse sampling and dimensionality reduction to characterize and model nonlinear dynamical systems over a range of bifurcation parameters first we construct modal libraries using the classical proper orthogonal decomposition in order to expose the dominant low rank coherent structures here libraries of the nonlinear terms are also constructed in order to take advantage of the discrete empirical interpolation method and projection that allows for the approximation of nonlinear terms from a sparse number of grid points the selected grid points are shown to be effective sensing and measurement locations for characterizing the underlying dynamics stability and bifurcations of nonlinear dynamical systems the use of empirical interpolation points and sparse representation facilitates a family of local reduced order models for each physical regime rather than a higher order global model which has the benefit of physical interpretability of energy transfer between coherent structures the method advocated also allows for orders of magnitude improvement in computational speed and memory requirements to illustrate the method the discrete interpolation points and nonlinear modal libraries are used for sparse representation in order to classify and reconstruct the dynamic bifurcation regimes in the complex ginzburg landau equation it is also shown that point measurements of the nonlinearity are more effective than linear measurements when sensor noise is present

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

the optimal hard threshold for singular values is 4 sqrt 3 we consider recovery of low rank matrices from noisy data by hard thresholding of singular values in which empirical singular values below a threshold are set to 0 we study the asymptotic mean squared error amse in a framework where the matrix size is large compared with the rank of the matrix to be recovered and the signal to noise ratio of the low rank piece stays constant the amse optimal choice of hard threshold in the case of n by n matrix in white noise of level is simply 4 3 n 2 309 n when is known or simply 2 858 y med when is unknown where y med is the median empirical singular value for nonsquare m by n matrices with m n the thresholding coefficients 4 3 and 2 858 are replaced with different provided constants that depend on m n asymptotically this thresholding rule adapts to unknown rank and unknown noise level in an optimal manner it is always better than hard thresholding at any other value and is always better than ideal truncated singular value decomposition tsvd which truncates at the true rank of the low rank matrix we are trying to recover hard thresholding at the recommended value to recover an n by n matrix of rank r guarantees an amse at most 3 nr 2 in comparison the guarantees provided by tsvd optimally tuned singular value soft thresholding and the best guarantee achievable by any shrinkage of the data singular values are 5 nr 2 6 nr 2 and 2 nr 2 respectively the recommended value for hard threshold also offers among hard thresholds the best possible amse guarantees for recovering matrices with bounded nuclear norm empirical evidence suggests that performance improvement over tsvd and other popular shrinkage rules can be substantial for different noise distributions even in relatively small n

quaternionic algebraic cycles and reality in this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real brauer severi varieties under the action of the galois group gal c r appropriate stabilizations of these spaces yield two equivariant spectra the first one classifies dupont seymour s quaternionic k theory and the other one classifies an equivariant cohomology theory 3 which is a natural recipient of characteristic classes kh x 3 x for quaternionic bundles over real spaces x

bigraded equivariant cohomology of real quadrics we give a complete description of the bigraded bredon co homology ring of smooth projective real quadrics with coefficients in the constant mackey functor z these invariants are closely related to the integral motivic cohomology ring which is not known for these va rieties some of the results and techniques introduced can be applied to other geometrically cellular real varieties

obstacle problem for nonlinear parabolic equations abstract we show the existence of a continuous solution to a nonlinear parabolic obstacle problem with a continuous time dependent obstacle the solution is constructed by an adaptation of the schwarz alternating method moreover if the obstacle is holder continuous we prove that the solution inherits the same property