an online method for interpolating linear parametric reduced order models a two step online method is proposed for interpolating projection based linear parametric reduced order models roms in order to construct a new rom for a new set of parameter values the first step of this method transforms each precomputed rom into a consistent set of generalized coordinates the second step interpolates the associated linear operators on their appropriate matrix manifold real time performance is achieved by precomputing inner products between the reduced order bases underlying the precomputed roms the proposed method is illustrated by applications in mechanical and aeronautical engineering in particular its robustness is demonstrated by its ability to handle the case where the sampled parameter set values exhibit a mode veering phenomenon

optimization of an acoustic horn with respect to efficiency and directivity we consider the problem of designing an acoustic horn in order to efficiently transmit the incoming wave energy and favorably distribute the energy in the far field a finite element solution of the helmholtz equation in planar or cylindrical symmetry models the wave propagation the transmission efficiency is monitored by measuring the back reflections into the feeding waveguide and the far field directivity pattern is computed using an integral expression known from scattering theory the design problem is formulated as a non linear least squares problem which is solved using a gradient based algorithm where the gradients are provided by solutions of the associated adjoint equations the results demonstrate that this approach can generate horns with almost perfect transmission in a wide frequency band due to the improved transmission properties at the lower frequency region the optimization with respect to efficiency also generates improved far field directivity patterns that is patterns that vary less with frequency it is possible to obtain even more uniform directivity patterns by explicitly including directivity requirements in the optimization however those improvements in directivity seem to be associated with a substantial loss of efficiency copyright 2007 john wiley sons ltd

tightness of slip linked polymer chains we study the interplay between entropy and topological constraints for a polymer chain in which sliding rings slip links enforce pair contacts between monomers these slip links divide a closed ring polymer into a number of subloops which can exchange length among each other in the ideal chain limit we find the joint probability density function for the sizes of segments within such a slip linked polymer chain paraknot a particular segment is tight small in size or loose of the order of the overall size of the paraknot depending on both the number of slip links it incorporates and its competition with other segments when self avoiding interactions are included scaling arguments can be used to predict the statistics of segment sizes for certain paraknot configurations

fractal time approach to dispersive transport in single species reaction diffusion the effect of dispersive transport in the single species reaction diffusion models of coagulation and annihilation is considered this transport is modelled through a fractal time random walk in which the stepping times of the walker a typical particle follows a renewal process characterized by a pausing time distribution proportional to a stable law at long times with the fractal dimension of the time this leads to a sublinear mean squared displacement for the particles the decay of the concentration of particles a t is obtained for all space dimensions d and for the whole course of the reactions the obtained results are exact for short and long times with the long time asymptotics for d 1 for d 2 and for the effect of highly non homogeneous space distributions of particles is also considered it is found that a fractal segregation of dimension with in the initial distribution of particles in the space leads to for d 1 for d 2 and for and for this shows a subordination phenomenon in the combination of space and time fractal distributions

a posteriori error estimation for deim reduced nonlinear dynamical systems in this work an efficient approach for a posteriori error estimation for pod deim reduced nonlinear dynamical systems is introduced the considered nonlinear systems may also include time and parameter affine linear terms as well as parametrically dependent inputs and outputs the reduction process involves a galerkin projection of the full system and approximation of the system s nonlinearity by the deim method s chaturantabut and d c sorensen siam j sci comput 32 2010 pp 2737 2764 the proposed a posteriori error estimator can be efficiently decomposed in an offline online fashion and is obtained by a one dimensional auxiliary ode during reduced simulations key elements for efficient online computation are partial similarity transformations and matrix deim approximations of the nonlinearity jacobians the theoretical results are illustrated by application to an unsteady burgers equation and a cell apoptosis model

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

preserving lagrangian structure in nonlinear model reduction with application to structural dynamics this work proposes a model reduction methodology that preserves lagrangian structure and achieves computational efficiency in the presence of high order nonlinearities and arbitrary parameter dependence as such the resulting reduced order model retains key properties such as energy conservation and symplectic time evolution maps we focus on parameterized simple mechanical systems subjected to rayleigh damping and external forces and consider an application to nonlinear structural dynamics to preserve structure the method first approximates the system s lagrangian ingredients the riemannian metric the potential energy function the dissipation function and the external force and subsequently derives reduced order equations of motion by applying the forced euler lagrange equation with these quantities from the algebraic perspective key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness matri

operator correlations and quantum regression theorem in non markovian lindblad rate equations non markovian lindblad rate equations arise from alternative microscopic interactions such as quantum systems coupled to composite reservoirs where extra degrees of freedom mediate the interaction between the system and a markovian reservoir as well as from systems coupled to complex structured reservoirs whose action can be well approximated by a direct sum of markovian sub reservoirs budini in phys rev a 74 053815 2006 the purpose of this paper is two fold first for both kinds of interactions we find general expressions for the system operator correlations written in terms of the lindblad rate propagator secondly we find the conditions under which a quantum regression hypothesis is valid we show that a non markovian quantum regression theorem can only be granted in a stationary regime being a necessary condition the fulfillment of a detailed balance condition this result is independent of the underlying microscopic interaction providing a criterion for the validity of the regression hypothesis in non markovian lindblad like master equations as an example we study the correlations of a two level system coupled to different kind of reservoirs

efficient non linear model reduction via a least squares petrov galerkin projection and compressive tensor approximations the first author acknowledges the partial support by a national science foundation graduate fellowship and the partial support by a national defense science and engineering graduate fellowship the second and third authors acknowledge the partial support by the motor sports division of the toyota motor corporation under agreement number 48737 and the partial support by a research grant from the academic excellence alliance program between king abdullah university of science and technology kaust and stanford university all authors also acknowledge the constructive comments received during the review process

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

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

a matrix deim technique for model reduction of nonlinear parametrized problems in cardiac mechanics abstract when relying on newton iterations to solve nonlinear problems in the context of reduced basis rb methods the assembling of the rb arrays during the online stage depends on the dimension of the underlying high fidelity approximation this is more of an issue when dealing with fully nonlinear problems for which the global jacobian matrix has to be entirely reassembled at each newton step in this paper the discrete empirical interpolation method deim and its matrix version mdeim are combined to evaluate both the residual vector and the jacobian matrix very efficiently in the case of complex parametrized nonlinear mechanical problems we compare this strategy with the classical deim approach and we derive a posteriori error estimates on the solution accounting for the contribution of deim mdeim errors the effectiveness of the proposed framework is assessed on quasi static nonlinear problems in particular we consider a nonlinear elasticity problem defined on a cube and a mechanical model describing heart contraction for an idealized left ventricle geometry the latter is a coupled problem in which the activation of the heart contraction is given by the solution of an electrophysiology model our numerical results show that mdeim is preferable to the classical deim both in terms of efficiency and accuracy

a linearly scaling grid method for generalized langevin and other integrodifferential equations a grid based method for numerical solution of systems of integrodifferential dynamical equations is introduced that has computational costs which scale linearly with the grid size grid parameters are automatically generated from a user supplied error tolerance we illustrate the accuracy of the method by solving a variety of representative sets of equations the method is much faster than comparable quadratically scaling algorithms

positivity preserving non markovian master equations a general class of integrodifferential non markovian master equations is developed which is representative of the dynamics of small subsystems interacting with open reservoirs with memory conditions which guarantee positivity of the subsystem reduced density are established

average size of random polygons with fixed knot topology we have evaluated by numerical simulation the average size r k of random polygons of fixed knot topology k ensuremath emptyset 3 1 3 1 ensuremath sharp 4 1 and we have confirmed the scaling law r k 2 ensuremath sim n 2 ensuremath nu k for the number n of polygonal nodes in a wide range n 100 char21 2200 the best fit gives 2 ensuremath nu k ensuremath simeq 1 11 char21 1 16 with good fitting curves in the whole range of n the estimate of 2 ensuremath nu k is consistent with the exponent of self avoiding polygons in a limited range of n n ensuremath gtrsim 600 however we have another fit with 2 ensuremath nu k ensuremath simeq 1 01 char21 1 07 which is close to the exponent of random polygons

a posteriori error estimators for linear reduced order models using krylov based integrators summary reduced order models for linear time invariant dynamical systems are considered and the error between the full order model and the reduced order model solutions is characterized based on the analytical representation of the error an a posteriori error indicator is proposed that combines a krylov based exponential integrator and an a posteriori residual based estimate numerical experiments illustrate the quality of the error estimator copyright 2014 john wiley sons ltd

shape optimization of an acoustic horn the shape of an acoustic horn is optimized with respect to its ability to provide impedance matching to the surrounding air the resulting minimization problem is solved using a bfgs quasi newton algorithm the wave propagation is modeled by a finite element approximation of the helmholtz equation and the necessary objective function gradient is calculated from the solution of the associated adjoint equation local solutions to the optimization problem corresponding to irregular horn contours are avoided by using a filtering technique that applies smoothing to the design updates and the gradient the method converges fast to a smooth solution and the resulting horn has good impedance properties throughout the entire frequency band of interest

ab initio construction of an analytically tractable kraus map for non markovian quantum dissipation starting from a prescribed hamiltonian we construct a non markovian evolution equation for a non relativistic quantum system that exchanges energy with a large reservoir in order to create sufficient mathematical freedom the density operator is replaced by a more flexible entity that depends on two times if these times are chosen equal the density operator is recovered in deriving a non markovian integral equation for our bitemporal operator it is assumed that initially system and reservoir are completely uncorrelated furthermore in employing wick s theorem for factorization of reservoir correlation functions only those wick contractions between reservoir potentials are retained that belong to a generalized nearest neighbour class the latter is established by subjecting the set of plain nearest neighbour contractions to any cyclic permutation of reservoir potentials through generalizing the notion of nearest neighbour contraction it is ensured that the trace of the density operator is conserved by construction our bitemporal evolution equation agrees with the kraus map for quantum dissipation moreover a sound markovian limit exists that reproduces the complete van hove davies theory by making use of a rotating wave approximation and laplace transformation the density operator of a damped n level atom can be computed for large times and moderate coupling to the reservoir the atom ends up near the state of thermal equilibrium at zero temperature our non markovian integral equation gives an exact solution for the atomic density operator

pod and cvt based reduced order modeling of navier stokes flows abstract a discussion of reduced order modeling for complex systems such as fluid flows is given to provide a context for the construction and application of reduced order bases reviews of the pod proper orthogonal decomposition and cvt centroidal voronoi tessellation approaches to reduced order modeling are provided including descriptions of pod and cvt reduced order bases their construction from snapshot sets and their application to the low cost simulation of the navier stokes system some concrete incompressible flow examples are used to illustrate the construction and use of pod and cvt reduced order bases and to compare and contrast the two approaches to reduced order modeling

a method for interpolating on manifolds structural dynamics reduced order models a rigorous method for interpolating a set of parameterized linear structural dynamics reduced order models roms is presented by design this method does not operate on the underlying set of parameterized full order models hence it is amenable to an online real time implementation it is based on mapping appropriately the rom data onto a tangent space to the manifold of symmetric positive definite matrices interpolating the mapped data in this space and mapping back the result to the aforementioned manifold algorithms for computing the forward and backward mappings are offered for the case where the roms are derived from a general galerkin projection method and the case where they are constructed from modal reduction the proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input in all cases good accuracy is demonstrated at real time processing speeds copyright 2009 john wiley sons ltd

anomalous finite size effects for the mean squared gyration radius of gaussian random knots anomalously strong finite size effects have been observed for the mean square radius of gyration r2k of gaussian random polygons with a fixed knot k as a function of the number n of polygonal nodes through computer simulations with n 2000 we find that the gyration radius r2k can be approximated by a power law r2k n2 keff for several knots where the effective exponents keff are larger than 0 5 and less than 0 6 furthermore a crossover occurs for the gyration radius of the trivial knot when n is roughly equal to the characteristic length nc of random knotting assuming an asymptotic fitting formula we also discuss possible asymptotic behaviours for rk2 of gaussian random polygons

reduced basis methods from low rank matrices to low rank tensors we propose a novel combination of the reduced basis method with low rank tensor techniques for the efficient solution of parameter dependent linear systems in the case of several parameters this combination called rbtensor consists of three ingredients first the underlying parameter dependent operator is approximated by an explicit affine representation in a low rank tensor format second a standard greedy strategy is used to construct a problem dependent reduced basis third the associated reduced parametric system is solved for all parameter values on a tensor grid simultaneously via a low rank approach this allows us to explicitly represent and store an approximate solution for all parameter values at a time once this approximation is available the computation of output functionals and the evaluation of statistics of the solution become a cheap online task without requiring the solution of a linear system

a reduced order representation of the poincare steklov operator an application to coupled multi physics problems this work investigates a model reduction method applied to coupled multi physics systems the case in which a system of interest interacts with an external system is considered an approximation of the poinste operator is computed by simulating in an offline phase the external problem when the inputs are the laplace beltrami eigenfunctions defined at the interface in the online phase only the reduced representation of the operator is needed to account for the influence of the external problem on the main system an online basis enrichment is proposed in order to guarantee a precise reduced order computation several test cases are proposed on different fluid structure couplings

data driven reduced order modeling for time dependent problems abstract a data driven reduced basis rb method for parametrized time dependent problems is proposed this method requires the offline preparation of a database comprising the time history of the full order solutions at parameter locations based on the full order data a reduced basis is constructed by the proper orthogonal decomposition pod and the maps between the time parameter values and the projection coefficients onto the rb are approximated as a regression model with a natural tensor grid between the time and the parameters in the database a singular value decomposition svd is used to extract the principal components in the data of projection coefficients the regression functions are represented as the linear combinations of several tensor products of two gaussian processes one of time and the other of parameters during the online stage the solutions at new time parameter locations in the domain of interest can be recovered rapidly as outputs from the regression models featuring a non intrusive nature and the complete decoupling of the offline and online stages the proposed approach provides a reliable and efficient tool for approximating parametrized time dependent problems and its effectiveness is illustrated by non trivial numerical examples

reduced basis method for parametrized elliptic optimal control problems we propose a suitable model reduction paradigm the certied reduced basis method rb for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations pdes in particular we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable firstly we recast the optimal control problem in the framework of saddle point problems in order to take advantage of the already developed rb theory for stokes type problems then the usual ingredients of the rb methodology are called into play a galerkin projection onto a low dimensional space of basis functions properly selected by an adaptive procedure an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure an efficient and rigorous a posteriori error estimate on the state control and adjoint variables as well as on the cost functional finally we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique

non intrusive reduced order modeling of nonlinear problems using neural networks abstract we develop a non intrusive reduced basis rb method for parametrized steady state partial differential equations pdes the method extracts a reduced basis from a collection of high fidelity solutions via a proper orthogonal decomposition pod and employs artificial neural networks anns particularly multi layer perceptrons mlps to accurately approximate the coefficients of the reduced model the search for the optimal number of neurons and the minimum amount of training samples to avoid overfitting is carried out in the offline phase through an automatic routine relying upon a joint use of the latin hypercube sampling lhs and the levenberg marquardt lm training algorithm this guarantees a complete offline online decoupling leading to an efficient rb method referred to as pod nn suitable also for general nonlinear problems with a non affine parametric dependence numerical studies are presented for the nonlinear poisson equation and for driven cavity viscous flows modeled through the steady incompressible navier stokes equations both physical and geometrical parametrizations are considered several results confirm the accuracy of the pod nn method and show the substantial speed up enabled at the online stage as compared to a traditional rb strategy

numerical method for integrodifferential generalized langevin and master equations we show that the integrodifferential generalized langevin and non markovian master equations can be transformed into larger sets of ordinary differential equations on the basis of this transformation we develop a numerical method for solving such integrodifferential equations physically motivated example calculations are performed to demonstrate the accuracy and convergence of the method

parametric free form shape design with pde models and reduced basis method we present a coupling of the reduced basis methods and free form deformations for shape optimization and design of systems modelled by elliptic pdes the free form deformations give a parameterization of the shape that is independent of the mesh the initial geometry and the underlying pde model the resulting parametric pdes are solved by reduced basis methods an important role in our implementation is played by the recently proposed empirical interpolation method which allows approximating the non affinely parameterized deformations with affinely parameterized ones these ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization the proposed approach is demonstrated on an airfoil inverse design problem

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

gyration radius of a circular polymer under a topological constraint with excluded volume it is nontrivial whether the average size of a ring polymer should become smaller or larger under a topological constraint making use of some knot invariants we numerically evaluate the mean square radius of gyration for ring polymers having a fixed knot type where the ring polymers are given by self avoiding polygons consisting of freely jointed hard cylinders we obtain plots of the gyration radius versus the number of polygonal nodes for the trivial trefoil and figure eight knots we discuss possible asymptotic behaviors of the gyration radius under the topological constraint in the asymptotic limit the size of a ring polymer with a given knot is larger than that of no topological constraint when the polymer is thin and the effective expansion becomes weak when the polymer is thick enough

adaptive non intrusive reduced order modeling for compressible flows abstract an adaptive non intrusive reduced basis rb method based on gaussian process regression gpr is proposed for parametrized compressible flows adaptivity is pursued in the offline stage the reduced basis by proper orthogonal decomposition pod is constructed iteratively to achieve a specified tolerance for gpr active data selection is used at each iteration with standard deviation as the error indicator to improve accuracy for shock dominated flows a properly designed simplified problem sp is considered as input of the regression models in addition to using parameters directly furthermore a surrogate error model is constructed to serve as an efficient error estimator for the gpr models several two and three dimensional cases are conducted including the inviscid nozzle flow the inviscid naca0012 airfoil flow and the inviscid m6 wing flow for all the cases the trained models are able to make efficient predictions with reasonable accuracy in the online stage the sp based approach is observed to result in biased sampling towards transonic regions the regression models are further applied in sensitivity analysis from which the solution of the two dimensional cases are shown to be significantly more sensitive to input parameters than the wing flow this is consistent to the comparison of convergence histories between the parameter based and the sp based models for cases of high sensitivity the sp based approach is superior and can help to significantly reduce the number of required snapshots to achieve a prescribed tolerance

scattering functions of knotted ring polymers we discuss the scattering function of a gaussian random polygon with n nodes under a given topological constraint through simulation we evaluate the form factor pk q of a gaussian polygon of n 200 having a fixed knot k for some different knots such as the trivial trefoil and figure eight knots here the gaussian polygons with different knots k have distinct values of the mean square radius of gyration r2 g k we obtain the kratky plots of the form factors i e the plots of qr g k 2pk q versus qr g k for the different topological constraints and discuss nontrivial large q behavior as well as small q behavior for the scattering functions we also find that the distinct values of r2 g k play an important role in the large q and small q properties of the kratky plots

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

equilibration and asymptotic stationarity for non markovian master equations we show that certain positivity preserving non markovian generalizations of the kossakowski lindblad master equation can exhibit equilibration to an asymptotic state which is stationary with respect to the shifted system hamiltonian for general system bath coupling this is in sharp contrast to results for markovian forms which require strong relations between these operators e g commutation of isolated system hamiltonian and coupling operator we also expand the list of sufficient conditions for positivity of non markovian master equations

hyper reduction of mechanical models involving internal variables we propose to improve the efficiency of the computation of the reduced state variables related to a given reduced basis this basis is supposed to be built by using the snapshot proper orthogonal decomposition pod model reduction method in the framework of non linear mechanical problems involving internal variables the local integration of the constitutive laws can dramatically limit the computational savings provided by the reduction of the order of the model this drawback is due to the fact that using a galerkin formulation the size of the reduced basis has no effect on the complexity of the constitutive equations in this paper we show how a reduced basis approximation and a petrov galerkin formulation enable to reduce the computational effort related to the internal variables the key concept is a reduced integration domain where the integration of the constitutive equations is performed the local computations being not made over the entire domain we extrapolate the computed internal variable over the full domain using pod vectors related to the internal variables this paper shows the improvement of the computational saving obtained by the hyper reduction of the elasto plastic model of a simple structure

an efficient reduced order modeling approach for non linear parametrized partial differential equations summary for general non linear parametrized partial differential equations pdes the standard galerkin projection is no longer efficient to generate reduced order models this is because the evaluation of the integrals involving the non linear terms has a high computational complexity and cannot be pre computed this situation also occurs for linear equations when the parametric dependence is nonaffine in this paper we propose an efficient approach to generate reduced order models for large scale systems derived from pdes which may involve non linear terms and nonaffine parametric dependence the main idea is to replace the non linear and nonaffine terms with a coefficient function approximation consisting of a linear combination of pre computed basis functions with parameter dependent coefficients the coefficients are determined efficiently by an inexpensive and stable interpolation at some pre computed points the efficiency and accuracy of this method are demonstrated on several test cases which show significant computational savings relative to the standard galerkin projection reduced order approach copyright q 2008 john wiley sons ltd

manifold algorithmic errors in quantum computers with static internal imperfections the inevitable existence of static internal imperfections and residual interactions in some quantum computer architectures results in internal decoherence dissipation and destructive unitary shifts of active algorithms by exact numerical simulations we determine the relative importance and origin of these errors for a josephson charge qubit quantum computer in particular we determine that the dynamics of a cnot gate interacting with its idle neighboring qubits via native residual coupling behaves much like a perturbed kicked top in the exponential decay regime where fidelity decay is only weakly dependent on perturbation strength this means that retroactive removal of gate errors whether unitary or non unitary may not be possible and that effective error correction schemes must operate concurrently with the implementation of subcomponents of the gate

non markovian master equations from entanglement with stationary unobserved degrees of freedom we deduce a class of non markovian completely positive master equations which describe a system in a composite bipartite environment consisting of a markovian reservoir and additional stationary unobserved degrees of freedom that modulate the dissipative coupling the entanglement induced memory effects can persist for arbitrary long times and affect the relaxation to equilibrium as well as induce corrections to the quantum regression theorem by considering the extra degrees of freedom as a discrete manifold of energy levels strong non exponential behaviour can arise such as for example power law and stretched exponential decays

efficient model reduction of parametrized systems by matrix discrete empirical interpolation in this work we apply a matrix version of the so called discrete empirical interpolation mdeim for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced order models roms however in many cases such an affine decomposition is not readily available and must be recovered through often intrusive procedures such as the empirical interpolation method eim and its discrete variant deim in this paper we show that mdeim represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non intrusive efficient and purely algebraic way we propose different strategies to combine mdeim with a state approximation resulting either from a reduced basis greedy approach or proper orthogonal decomposition a posteriori error estimates accounting for the mdeim error are also developed in the case of parametrized elliptic and parabolic equations finally the capability of mdeim to generate accurate and efficient roms is demonstrated on the solution of two computationally intensive classes of problems occurring in engineering contexts namely pde constrained shape optimization and parametrized coupled problems

multi space reduced basis preconditioners for parametrized stokes equations abstract we introduce a two level preconditioner for the efficient solution of large scale saddle point linear systems arising from the finite element fe discretization of parametrized stokes equations this preconditioner extends the multi space reduced basis msrb preconditioning method proposed in dal santo et al 2018 it combines an approximated block fine grid preconditioner with a reduced basis rb solver which plays the role of coarse component a sequence of rb spaces constructed either with an enriched velocity formulation or a petrov galerkin projection is built each rb coarse component is defined to perform a single iteration of the iterative method at hand the flexible gmres fgmres algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time numerical test cases for stokes flows in three dimensional parameter dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings