transformation coefficients of permutation groups the eigenfunction method is used to calculate the transformation coefficients m 1 2 m1 m2 from the yamanouchi basis of the permutation group sf1 f2 to the sf1 f2 sf1 sf2 irreducible basis a program in fortran is written and tables of the transformation coefficients for the permutation group sf up to f 6 are given possible applications of the transformation coefficients are sketched

on the linear equation method for the subduction problem in symmetric groups we focus on the transformation matrices between the standard young yamanouchi basis of an irreducible representation for the symmetric group sn and the split basis adapted to the direct product subgroups we introduce the concept of subduction graph and show that it conveniently describes the combinatorial structure of the equation system arisen from the linear equation method thus we can outline an improved algorithm to solve the subduction problem in symmetric groups by a graph searching process we conclude by observing that the general matrix form for multiplicity separations resulting from orthonormalization can be expressed in terms of sylvester matrices relative to a suitable inner product in the multiplicity space

numerical analysis of the leray reduced order model abstract standard roms generally yield spurious numerical oscillations in the simulation of convection dominated flows regularized roms use explicit rom spatial filtering to decrease these spurious numerical oscillations the leray rom is a recently introduced regularized rom that utilizes explicit rom spatial filtering of the convective term in the navier stokes equations this paper presents the numerical analysis of the finite element discretization of the leray rom error estimates for the rom differential filter which is the explicit rom spatial filter used in the leray rom are proved these rom filtering error estimates are then used to prove error estimates for the leray rom finally both the rom filtering error estimates and the leray rom error estimates are numerically investigated in the simulation of the two dimensional navier stokes equations with an analytic solution

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

split standard transformation coefficients the block selective conjecture the split basis for the symmetric group sn is adapted to sn and to the product subgroup sn a sa we consider matrices transforming between split bases and the standard young yamanouchi basis of the symmetric group sn we present a conjecture for directly relating those transformation matrices to a representation matrix in the young yamanouchi basis of a cycle permutation this block selective conjecture agrees with previously obtained solutions for the transformation matrices up to and including s6 in particular the conjecture provides a correct solution for the early occurrences of multiplicities in the product sn a sa to motivate the main result of this paper we first present transformations between permuted symmetric groups

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 reduced order extrapolated crank nicolson finite spectral element method based on pod for the 2d non stationary boussinesq equations abstract in this paper we mainly utilize proper orthogonal decomposition pod to reduce the order for the coefficient vector of the classical crank nicolson finite spectral element ccnfse method of the two dimensional 2d non stationary boussinesq equations about vorticity stream functions so that the reduced order method maintains all the advantages of the ccnfse method toward this end we first establish a ccnfse format with the second order time accuracy for the two dimensional 2d non stationary boussinesq equations about vorticity stream functions and analyze the existence stability and convergence of the ccnfse solutions and then by pod we establish a reduced order extrapolated crank nicolson finite spectral element roecnfse method and analyze the existence stability and convergence of the roecnfse solutions as well as offer the flowchart for seeking roecnfse solutions finally we use two sets of numerical experiments to validate that the numerical computational consequences are accorded with the theoretical ones such that the effectiveness and feasibility of the roecnfse method are further verified both theory and method in this paper is new and completely different from the existing reduced order methods

fast low rank modifications of the thin singular value decomposition this paper develops an identity for additive modivations of a singular value decomposition svd to reflect updates downdates shifts and edits of the data matrix this sets the stage for fast and ememory efficient sequential algorithms for tracking singular values and subspaces in conjunction with a fast solution for the pseudo inverse of a submatrix of an orthogonal matrix we develop a scheme for computing a thin svd of streaming data in a single pass with linear time complexity a rank r think svd of a p x q matrix can be computed in o pqr time for r less than or equal sqroot min p q linear algebra and its applications

split bases and multiplicity separations in symmetric group transformation coefficients we consider matrices transforming between the standard young yamanouchi basis of the symmetric group and bases adapted to the product subgroups the split basis we derive closed formulae for transformation coefficients for b 3 which includes the first cases when a choice of multiplicity separation is required we discuss considerations which can be applied to obtain a simple form for the multiplicity separation we show that the combinatorial and algebraic structure of the littlewood richardson rule also known as the biedenharn louck pattern calculus does not assist with finding a simple multiplicity separation

mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary navier stokes equations in this paper proper orthogonal decomposition pod is used for model reduction of mixed finite element mfe for the nonstationary navier stokes equations and error estimates between a reference solution and the pod solution of reduced mfe formulation are derived the basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual mfe method for the nonstationary navier stokes equations or from physics system trajectories by drawing samples of experiments and interpolation or data assimilation and then the basis functions of the usual mfe method are substituted with the pod basis functions reconstructed by the elements of the ensemble to derive the pod reduced mfe formulation for the nonstationary navier stokes equations it is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between pod solution of reduced mfe formulation and the reference solution is consistent with theoretical results moreover it is also shown that this result validates the feasibility and efficiency of the pod method

approximate partitioned method of snapshots for pod the proper orthogonal decomposition pod method has been widely used to construct efficient numerical surrogate models for computationally intensive applications in control and optimization an inherent challenge with this method is that pod basis generation can be computationally expensive due to the huge size of the input snapshot data obtained from typical high fidelity large scale dynamic system simulations however if the process can be distributed into much smaller tasks over multiple processors in parallel computational time can be drastically reduced in this paper we put forth a novel partitioned method for generating the pod basis from snapshot data this method preserves the distributed nature of the data and takes advantage of parallelism for computation additionally it greatly reduces subtask communication volume two numerical examples are presented that demonstrate the effectiveness of the new method

pod deim reduced order strategies for efficient four dimensional variational data assimilation this work studies reduced order modeling rom approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models it is shown that a key requirement for a successful reduced order solution is that reduced order karush kuhn tucker conditions accurately represent their full order counterparts in particular accurate reduced order approximations are needed for the forward and adjoint dynamical models as well as for the reduced gradient new strategies to construct reduced order based are developed for proper orthogonal decomposition pod rom data assimilation using both galerkin and petrov galerkin projections for the first time pod tensorial pod and discrete empirical interpolation method deim are employed to develop reduced data assimilation systems for a geophysical flow model namely the two dimensional shallow water equations numerical experiments confirm the theoretical framework for galerkin projection in the case of petrov galerkin projection stabilization strategies must be considered for the reduced order models the new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time

hdg pod reduced order model of the heat equation abstract we propose a new hybridizable discontinuous galerkin hdg model order reduction technique based on proper orthogonal decomposition pod we consider the heat equation as a test problem and prove error bounds that converge to zero as the number of pod modes increases we present 2d and 3d numerical results to illustrate the convergence analysis

combinatorics of transformations from standard to non standard bases in brauer algebras transformation coefficients between standard bases for irreducible representations of the brauer centralizer algebra and split bases adapted to the subalgebra f1 f2 f are considered after providing the suitable combinatorial background based on the definition of the i coupling relation on nodes of the subduction grid we introduce a generalized version of the subduction graph which extends the one given in chilla 2006 j phys a math gen 39 7657 for symmetric groups thus we can describe the structure of the subduction system arising from the linear method and give an outline of the form of the solution space an ordering relation on the grid is also given and then as in the case of symmetric groups the choices of the phases and of the free factors governing the multiplicity separations are discussed

a reduced subduction graph and higher multiplicity in sn transformation coefficients transformation coefficients between standard bases for irreducible representations of the symmetric group sn and split bases adapted to the subgroup n1 n2 n are considered we first provide a selection rule and an identity rule for the subduction coefficients which allow us to decrease the number of unknowns and equations arising from the linear method by pan and chen then using the reduced subduction graph approach we may look at higher multiplicity instances as a significant example an orthonormalized solution for the first multiplicity three case which occurs in the decomposition of the irreducible representation 4 3 2 1 of s10 into 3 2 1 3 1 of s6 s4 is presented and discussed

symmetrization procedure for structure fluid cavity system involving pseudostatic correction this paper deals with a pressure displacement formulation based on the finite element method of an elastoacoustic coupling problem since directly solving the resulting system iscpu intensive for large models the solution is usually based on only a few uncoupled structural modes in vacuum and rigid cavity modes however this classical pressuredisplacement formulation leads to nonsymmetric eigenvalue problems furthermore if this method is accurate for weak coupling models significant errors occur when the truncated modes are coupled to the ones used in the analysis the present paper suggests a method combining the pseudostatic correction of the truncated modes in modal analysis with a symmetrization technique of the eigenvalue problems it allows the calculation of the real coupled modes and frequencies as well as the computation of an accurate elastoacoustic response even for the strong coupling of a structure with a heavy fluid liquid it is less cpu expensive than the classical method because it leads to a diagonalized system

a reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for burgers equation in this article a reduced optimizing finite difference scheme fds based on singular value decomposition svd and proper orthogonal decomposition pod for burgers equation is presented also the error estimates between the usual finite difference solution and the pod solution of reduced optimizing fds are analyzed it is shown by considering the results obtained for numerical simulations of cavity flows that the error between the pod solution of reduced optimizing fds and the solution of the usual fds is consistent with theoretical results moreover it is also shown that the reduced optimizing fds is feasible and efficient

a ray based ipdg method for high frequency time domain acoustic wave propagation in inhomogeneous media abstract the numerical approximation of high frequency wave propagation in inhomogeneous media is a challenging problem in particular computing high frequency solutions by direct simulations requires several points per wavelength for stability and usually requires many points per wavelength for a satisfactory accuracy in this paper we propose a new method for the acoustic wave equation in inhomogeneous media in the time domain to achieve superior accuracy and stability without using a large number of unknowns the method is based on a discontinuous galerkin discretization together with carefully chosen basis functions to obtain the basis functions we use the idea from geometrical optics and construct the basis functions by using the leading order term in the asymptotic expansion also we use a wavefront tracking method and a dimension reduction procedure to obtain dominant rays in each cell we show numerically that the accuracy of the numerical solutions computed by our method is significantly higher than that computed by the ipdg method using polynomials moreover the relative errors of our method grow only moderately as the frequency increases

a note on the transformation coefficients between the standard and non standard representations in a physically significant case closed formulae for the transformation coefficients between the standard and non standard representations of the symmetric group sn are obtained and the results are explicitly exhibited

a reduced order extrapolation technique for solution coefficient vectors in the mixed finite element method for the 2d nonlinear rosenau equation abstract this study focuses mainly on a reduced order method for the coefficient vectors in the crank nicolson mixed finite element cnmfe solutions for the two dimensional nonlinear rosenau equation we first provide a cnmfe method for the rosenau equation and give relative theoretical results as well as rewrite the cnmfe method in matrix form we then produce two sets of proper orthogonal decomposition pod bases by means of the first few coefficient vectors of the cnmfe solutions and set up a reduced order extrapolating cnmfe roecnmfe model with the pod bases as well as discuss the existence and stability as well as errors of the roecnmfe solutions by means of the matrix tool resulting in the theoretical analysis becoming very succinct lastly we present some numerical experiments to validate the correctness of the theoretical results finding that the superiority of the roecnmfe method is further verified

transient sound fields from impacted plates comparison between numerical simulations and experiments a comparison between measured and simulated transient acoustic fields generated by a thin rectangular impacted steel plate is reported the plate vibration is modelled in the time domain using a finite difference scheme of second order in time and fourth order in space 2 4 scheme and the radiated acoustic pressure is obtained by solving numerically the rayleigh integral equation by using a simple boundary element method trapezoidal rule the simulated results are evaluated in the form of the spatial pressure distribution at times short after the impact excitation the measurements are performed using an optical method pulsed two reference beam holographic interferometry results in the form of 2 d projections of the acoustic pressure field are obtained and compared with the simulations a high degree of similarity is found between the measured and simulated sound fields

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

reduced order model feedback control design numerical implementation in a thin shell model reduced order models employing the lagrange and popular proper orthogonal decomposition pod reduced basis methods in numerical approximation and feedback control of systems are presented and numerically tested the system under consideration is a thin cylindrical shell with surface mounted piezoceramic actuators donnell mushtari equations modified to include kelvin voigt damping are used to model the system dynamics basis functions constructed from fourier polynomials tensored with cubic splines are employed in the galerkin expansion of the full order model reduced basis elements are then formed from full order approximations of the exogenously excited shell taken at different time instances numerical examples illustrating the features of the reduced basis methods are presented as a first step toward investigating the behavior of the methods when implemented in physical systems the use of reduced order model feedback control gains in the full order model is considered and numerical examples are presented

calculating symmetric group split standard transformation coefficients using the block selective method a proof the split basis for the symmetric group sn is adapted to sn and to the product subgroup sn a sa the entries in the matrix that transforms between the split basis representation and the standard young yamanouchi yy basis the split standard transformation coefficients are closely related to a particle coefficients of fractional parentage recently the block selective conjecture was proposed for calculating the coefficients the conjecture implies that the transformation matrix is directly related to the representation matrix in the yy basis of a cycle permutation we present a proof of this conjecture

low rank incremental methods for computing dominant singular subspaces computing the singular values and vectors of a matrix is a crucial kernel in numerous scientific and industrial applications as such numerous methods have been proposed to handle this problem in a computationally efficient way this paper considers a family of methods for incrementally computing the dominant svd of a large matrix a specifically we describe a unification of a number of previously independent methods for approximating the dominant svd after a single pass through a we connect the behavior of these methods to that of a class of optimization based iterative eigensolvers on ata an iterative procedure is proposed which allows the computation of an accurate dominant svd using multiple passes through a we present an analysis of the convergence of this iteration and provide empirical demonstration of the proposed method on both synthetic and benchmark data

galerkin proper orthogonal decomposition methods for parabolic problems in this work error estimates for galerkin proper orthogonal decomposition pod methods for linear and certain non linear parabolic systems are proved the resulting error bounds depend on the number of pod basis functions and on the time discretization numerical examples are included

coupling subduction and induction transformations for group representations the basis of a representation space of any group may be chosen according to its decomposition into representations of the group and a subgroup chain coupling recoupling subduction and resubduction factors are each shown to arise as a result of schur s lemmas and a particular group subgroup structure when induction and mackey s subgroup theorem are considered new transformation factors occur those of induction and reinduction key properties of these factors are given together with their relationship with double coset matrix elements

an ensemble proper orthogonal decomposition method for the nonstationary navier stokes equations the definition of partial differential equation models usually involves a set of parameters whose values may vary over a wide range the solution of even a single set of parameter values may be quite expensive in many cases e g optimization control uncertainty quantification and other settings solutions are needed for many sets of parameter values we consider the case of the time dependent navier stokes equations for which a recently developed ensemble based method allows for the efficient determination of the multiple solutions corresponding to many parameter sets the method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step to significantly further reduce the costs of determining multiple solutions of the navier stokes equations we incorporate a proper orthogonal decomposition pod reduced order model into the ensemble based method the stability and convergence results for the ensemble based method are extended to the ensemble pod approach

centroidal voronoi tessellation based reduced order modeling of complex systems noindent a reduced order modeling methodology based on centroidal voronoi tessellations cvts is introduced cvts are special voronoi tessellations for which the generators of the voronoi diagram are also the centers of mass means of the corresponding voronoi cells for discrete data sets cvts are closely related to the h means and k means clustering techniques a discussion of reduced order modeling for complex systems such as fluid flows is given to provide a context for the application of reduced order bases then detailed descriptions of cvt based reduced order bases and how they can be constructed from snapshot sets and how they can be applied to the low cost simulation of complex systems are given subsequently some concrete incompressible flow examples are used to illustrate the construction and use of cvt based reduced order bases the cvt based reduced order modeling methodology is shown to be effective for these examples

a reduced order extrapolation algorithm based on sfve method and pod technique for non stationary stokes equations in this paper a reduced order extrapolation algorithm roea based on proper orthogonal decomposition pod technique and classical stabilized finite volume element sfve method for non stationary stokes equations is established the error estimates between the roea solutions and the classical sfve solutions and the implementation for solving roea are provided some numerical examples are used to verify that the results of numerical computation are consistent with theoretical conclusions moreover it is shown that roea based on sfve formulation and pod method is feasible and efficient for solving non stationary stokes equations

a parameterized background data weak approach to variational data assimilation formulation analysis and application to acoustics summary we present a parameterized background data weak pbdw formulation of the variational data assimilation state estimation problem for systems modeled by partial differential equations the main contributions are a constrained optimization weak framework informed by the notion of experimentally observable spaces a priori and a posteriori error estimates for the field and associated linear functional outputs weak greedy construction of prior background spaces associated with an underlying potentially high dimensional parametric manifold stability informed choice of observation functionals and related sensor locations and finally output prediction from the optimality saddle in o m3 operations where m is the number of experimental observations we present results for a synthetic helmholtz acoustics model problem to illustrate the elements of the methodology and confirm the numerical properties suggested by the theory to conclude we consider a physical raised box acoustic resonator chamber we integrate the pbdw methodology and a robotic observation platform to achieve real time in situ state estimation of the time harmonic pressure field we demonstrate the considerable improvement in prediction provided by the integration of a best knowledge model and experimental observations we extract even from these results with real data the numerical trends indicated by the theoretical convergence and stability analyses copyright 2014 john wiley sons ltd

balanced pod for model reduction of linear pde systems convergence theory we consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems the algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations the algorithm is related to the proper orthogonal decomposition and it was first proposed for systems of ordinary differential equations by rowley int j bifurc chaos appl sci eng 15 3 997 1013 2005 for the convergence analysis we consider the algorithm in terms of the hankel operator of the system rather than the product of the system gramians as originally proposed by rowley for exponentially stable systems with bounded finite rank input and output operators we prove that the balanced realization can be expressed in terms of balancing modes which are related to the hankel operator the balancing modes are required to be smooth and this can cause computational difficulties for pde systems we show how this smoothness requirement can be lessened for parabolic systems and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems we prove entrywise convergence of the matrices in the approximate reduced order models in both cases and present numerical results for two example pde systems

forward projection of transient sound pressure fields radiated by impacted plates using numerical laplace transform forward propagation of the transient sound pressure radiated by an impacted plate is presented it is shown that direct and inverse time domain discrete fourier transforms involved in fourier transform based near field acoustical holography nah lead to aliasing errors in the reconstructed time signals adding trailing zeros to the initial time signals is an inefficient way to reduce time aliasing errors hence the numerical laplace transform is introduced and a fourier transform based transient nah tnah approach is formulated an error measure is introduced to compare both nah and tnah with respect to the propagation distance and the location of the observation point in the projection plane the percentage of error with tnah is reduced by more than a factor of 10 without adding trailing zeros to the initial signals simulation results are validated experimentally using a free plexiglas plate impacted at its center

on subgroup adapted bases for representations of the symmetric group the split basis of an irreducible representation of the symmetric group sn m is the basis which is adapted to direct product subgroups of the form sn sm in this paper we have calculated symmetric group subduction coefficients relating the standard young yamanouchi basis for the symmetric group to the split basis by means of a novel version of the schur weyl duality we have also directly obtained matrix representations in the split basis using these techniques the advantages of this approach are that we obtain analytical expressions for the subduction coefficients and matrix representations and directly resolve issues of multiplicity in the subduction of sn sm irreducible representations from sn m irreducible representations our method is applicable to sn m irreducible representations labelled by young diagrams with large row length differences we provide evidence for a systematic expansion in the inverse row length difference and tools to compute the leading contribution which is exact when row length differences are infinite

low rank solvers for unsteady stokes brinkman optimal control problem with random data abstract we consider the numerical simulation of an optimal control problem constrained by the unsteady stokes brinkman equation involving random data more precisely we treat the state the control the target or the desired state as well as the viscosity as analytic functions depending on uncertain parameters this allows for a simultaneous generalized polynomial chaos approximation of these random functions in the stochastic galerkin finite element method discretization of the model the discrete problem yields a prohibitively high dimensional saddle point system with kronecker product structure we develop a new alternating iterative tensor method for an efficient reduction of this system by the low rank tensor train representation besides we propose and analyze a robust schur complement based preconditioner for the solution of the saddle point system the performance of our approach is illustrated with extensive numerical experiments based on two and three dimensional examples where the full problem size exceeds one billion degrees of freedom the developed tensor train scheme reduces the solution storage by two three orders of magnitude depending on discretization parameters

incremental proper orthogonal decomposition for pde simulation data abstract we propose an incremental algorithm to compute the proper orthogonal decomposition pod of simulation data for a partial differential equation specifically we modify an incremental matrix svd algorithm of brand to accommodate data arising from galerkin type simulation methods for time dependent pdes the algorithm is applicable to data generated by many numerical methods for pdes including finite element and discontinuous galerkin methods the algorithm initializes and efficiently updates the dominant pod eigenvalues and modes during the time stepping in a pde solver without storing the simulation data we prove that the algorithm without truncation updates the pod exactly we demonstrate the effectiveness of the algorithm using finite element computations for a 1d burgers equation and a 2d navier stokes problem

a reduced order dg formulation based on pod method for the time domain maxwell s equations in dispersive media abstract in this work a proper orthogonal decomposition pod method is applied to time domain maxwell s equations coupled to a drude dispersion model which are discretized in space by a discontinuous galerkin dg method an auxiliary differential equation ade method is used to represent the constitutive relation for the dispersive medium a pod dgtd formulation with lower dimension and sufficiently high accuracy is established together with the description of the pod reduced order basis its construction from a snapshot set and its application to the solution of the time domain maxwell s equations the overall goal is to reduce the computational time while maintaining an acceptable level of accuracy in order to obtain an efficient time domain solver to be used as a starting point for an optimization strategy we provide results from numerical experiments for two dimensional problems that illustrate the capabilities of the proposed pod dgtd formulation and assess its efficiency

irreducible representations of hecke algebras in the nonstandard basis and subduction coefficients irreducible representations of hecke algebras of the type an 1 hn q are discussed in the nonstandard basis some subduction coefficients i e the transformation coefficients between the standard and nonstandard base of hn q are derived by using the linear equation method

new pod error expressions error bounds and asymptotic results for reduced order models of parabolic pdes the derivations of existing error bounds for reduced order models of time varying partial differential equations pdes constructed using proper orthogonal decomposition pod have relied on bounding the error between the pod data and various pod projections of that data furthermore the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the pod data approximation error bounds we consider time varying data taking values in two different hilbert spaces h and v with v subset h and prove exact expressions for the pod data approximation errors considering four different pod projections and the two different hilbert space error norms furthermore the exact error expressions can be computed using only the pod eigenvalues and modes and we prove the errors converge to zero as the number of pod modes increases we consider the pod error estimation approaches of kunisch and volkwein siam j numer anal 40 2002 pp 492 515 and chapelle gariah an

reduced order control based on approximate inertial manifolds a reduced order method based on approximate inertial manifolds is applied to optimal control problems in infinite dimensional state spaces a detailed analysis of the method is given for the linear quadratic regulator problem the method can also be applied to higher order control systems with an appropriate decomposition of the state space in terms of slow and fast exponential decay

proper orthogonal decomposition with supg stabilized isogeometric analysis for reduced order modelling of unsteady convection dominated convection diffusion reaction problems abstract we consider reduced order modelling of unsteady convection dominated convection diffusion reaction problems with proper orthogonal decomposition pod in combination with isogeometric analysis isogeometric analysis has potential advantages in exact geometry representations efficient mesh generation different h p and k refinements and smooth bspline nurbs basis functions in order to compensate the oscillations caused by the convection dominated effect the streamline upwind petrov galerkin supg stabilization method is used both in generation of snapshots and pod galerkin method based on the recent novel and promising discretization method isogeometric analysis we propose a new fully discrete supg stabilized scheme the associated numerical error features three components due to spatial discretization by isogeometric analysis with supg stabilization time discretization with the crank nicolson scheme and modes truncation by pod we show a priori error estimates of the fully discrete scheme and give suitable stabilization parameters numerically a variety of two and three dimensional benchmark tests and numerical examples are provided to show the effectiveness accuracy and efficiency of the reduced order modelling methods by virtue of potential advantages of isogeometric analysis

a forward approach to numerical data assimilation variational data assimilation problems are concerned with computing unknown initial values for the simulation and prediction of natural phenomena most notably in weather prediction and are usually solved via an ill posed optimal control problem for the initial state at the time of the first available measurements an alternative forward approach focuses on computation of the final state after this interval which is just as suitable for prediction purposes and is well posed without additional regularization specifically it is possible to compute projections of the unknown final state on all elements of an orthonormal basis which theoretically allows for the complete reconstruction of the final state in this paper an efficient numerical method for linear evolution equations of diffusive type is presented and convergence of the numerical approximation based on a discontinuous galerkin discretization is proved the key of this method is the computation of an adaptively ordered orthonormal basis using proper orthogonal decomposition numerical examples for a scalar convection diffusion equation in two and three dimensions show the effectiveness of the method