structure preserving reduced order modeling for gradient systems abstract minimization of energy in gradient systems leads to formation of oscillatory and turing patterns in reaction diffusion systems these patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state in this paper a reduced order model rom is developed which preserves the gradient dissipative structure the coupled system of reaction diffusion equations are discretized in space by the symmetric interior penalty discontinuous galerkin sipg method the resulting system of ordinary differential equations odes are integrated in time by the average vector field avf method which preserves the energy dissipation of the gradient systems the roms are constructed by the proper orthogonal decomposition pod with galerkin projection the nonlinear reaction terms are computed efficiently by discrete empirical interpolation method deim preservation of the discrete energy of the foms and roms with pod deim ensures the long term stability of the steady state solutions numerical simulations are performed for the gradient dissipative systems with two specific equations real ginzburg landau equation and swift hohenberg equation numerical results demonstrate that the pod deim reduced order solutions preserve well the energy dissipation over time and at the steady state

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

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

a new look at proper orthogonal decomposition we investigate some basic properties of the proper orthogonal decomposition pod method as it is applied to data compression and model reduction of finite dimensional nonlinear systems first we provide an analysis of the errors involved in solving a nonlinear ode initial value problem using a pod reduced order model then we study the effects of small perturbations in the ensemble of data from which the pod reduced order model is constructed on the reduced order model we explain why in some applications this sensitivity is a concern while in others it is not we also provide an analysis of computational complexity of solving an ode initial value problem and study the computational savings obtained by using a pod reduced order model we provide several examples to illustrate our theoretical results

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

aerodynamic data reconstruction and inverse design using proper orthogonal decomposition the application of proper orthogonal decomposition for incomplete gappy data for compressible external aerodynamic problems has been demonstrated successfully in this paper for the first time using this approach it is possible to construct entire aerodynamic flowfields from the knowledge of computed aerodynamic flow data or measured flow data specified on the aerodynamic surface thereby demonstrating a means to effectively combine experimental and computational data the sensitivity of flow reconstruction results to available measurements and to experimental error is analyzed another new extension of this approach allows one to cast the problem of inverse airfoil design as a gappy data problem the gappy methodology demonstrates a great simplification for the inverse airfoil design problem and is found to work well on a range of examples including both subsonic and transonic cases

registration of translated and rotated images using finite fourier transforms a well known method for image registration is based on a conventional correlation between phase only or whitened versions of the two images to be realigned the method covering rigid translational movements is characterized by an outstanding robustness against correlated noise and disturbances such as those encountered with nonuniform time varying illumination this correspondence discusses an extension of the method to cover both translational and rotational movements

energy balance and mass conservation in reduced order models of fluid flows abstract in this paper we investigate theoretically and computationally the conservation properties of reduced order models roms for fluid flows specifically we investigate whether the roms satisfy the same or similar energy balance and mass conservation as those satisfied by the navier stokes equations all of our theoretical findings are illustrated and tested in numerical simulations of a 2d flow past a circular cylinder at a reynolds number r e 100 first we investigate the rom energy balance we show that using the snapshot average for the centering trajectory which is a popular treatment of nonhomogeneous boundary conditions in roms yields an incorrect energy balance then we propose a new approach in which we replace the snapshot average with the stokes extension theoretically the stokes extension produces an accurate energy balance numerically the stokes extension yields more accurate results than the standard snapshot average especially for longer time intervals our second contribution centers around rom mass conservation we consider roms created using two types of finite elements the standard taylor hood th element which satisfies the mass conservation weakly and the scott vogelius sv element which satisfies the mass conservation pointwise theoretically the error estimates for the sv rom are sharper than those for the th rom numerically the sv rom yields significantly more accurate results especially for coarser meshes and longer time intervals

structure preserving model reduction of port hamiltonian systems by moment matching at infinity model reduction of port hamiltonian systems by means of the krylov methods is considered aiming at port hamiltonian structure preservation it is shown how to employ the arnoldi method for model reduction in a particular coordinate system in order to preserve not only a specific number of the markov parameters but also the port hamiltonian structure for the reduced order model furthermore it is shown how the lanczos method can be applied in a structure preserving manner to a subclass of port hamiltonian systems which is characterized by an algebraic condition in fact for the same subclass of port hamiltonian systems the arnoldi method and the lanczos method turn out to be equivalent in the sense of producing reduced order port hamiltonian models with the same transfer function

non linearly stable reduced order models for incompressible flow with energy conserving finite volume methods abstract a novel reduced order model rom formulation for incompressible flows is presented with the key property that it exhibits non linearly stability independent of the mesh of the full order model the time step the viscosity and the number of modes the two essential elements to non linear stability are 1 first discretise the full order model and then project the discretised equations and 2 use spatial and temporal discretisation schemes for the full order model that are globally energy conserving in the limit of vanishing viscosity for this purpose as full order model a staggered grid finite volume method in conjunction with an implicit runge kutta method is employed in addition a constrained singular value decomposition is employed which enforces global momentum conservation the resulting velocity only rom is thus globally conserving mass momentum and kinetic energy for non homogeneous boundary conditions a one time poisson equation is solved that accounts for the boundary contribution the stability of the proposed rom is demonstrated in several test cases furthermore it is shown that explicit runge kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation

a nonlinear variational problem for image matching minimizing a nonlinear functional is presented as a way of obtaining a planar mapping that matches two similar images a smoothing term is added to the nonlinear functional to penalize discontinuous and irregular solutions one option for the smoothing term is a quadratic form generated by a linear differential operator the functional is then minimized using the fourier representation of the planar mapping with this representation the quadratic form is diagonalized another option is a quadratic form generated via a basis of compactly supported wavelets in both cases a natural approximation scheme is described both quadratic forms are shown to impose the same smoothing however in terms of the finite dimensional approximations it is easier to accommodate local deformations using the wavelet basis

alignment by maximization of mutual information a new information theoretic approach is presented for finding the pose of an object in an image the technique does not require information about the surface properties of the object besides its shape and is robust with respect to variations of illumination in our derivation few assumptions are made about the nature of the imaging process as a result the algorithms are quite general and may foreseeably be used in a wide variety of imaging situations experiments are presented that demonstrate the approach registering magnetic resonance mr images aligning a complex 3d object model to real scenes including clutter and occlusion tracking a human head in a video sequence and aligning a view based 2d object model to real images the method is based on a formulation of the mutual information between the model and the image as applied here the technique is intensity based rather than feature based it works well in domains where edge or gradient magnitude based methods have difficulty yet it is more robust than traditional correlation additionally it has an efficient implementation that is based on stochastic approximation

model reduction for compressible flows using pod and galerkin projection abstract we present a framework for applying the method of proper orthogonal decomposition pod and galerkin projection to compressible fluids for incompressible flows only the kinematic variables are important and the techniques are well known in a compressible flow both the kinematic and thermodynamic variables are dynamically important and must be included in the configuration space we introduce an energy based inner product which may be used to obtain pod modes for this configuration space we then obtain an approximate version of the navier stokes equations valid for cold flows at moderate mach number and project these equations onto a pod basis the resulting equations of motion are quadratic and are much simpler than projections of the full compressible navier stokes equations

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

entropy stable reduced order modeling of nonlinear conservation laws abstract reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model we introduce projection based hyper reduced models of nonlinear conservation laws which are globally conservative and inherit a semi discrete entropy inequality independently of the choice of basis and choice of parameters

a pyramid approach to subpixel registration based on intensity we present an automatic subpixel registration algorithm that minimizes the mean square intensity difference between a reference and a test data set which can be either images two dimensional or volumes three dimensional it uses an explicit spline representation of the images in conjunction with spline processing and is based on a coarse to fine iterative strategy pyramid approach the minimization is performed according to a new variation ml of the marquardt levenberg algorithm for nonlinear least square optimization the geometric deformation model is a global three dimensional 3 d affine transformation that can be optionally restricted to rigid body motion rotation and translation combined with isometric scaling it also includes an optional adjustment of image contrast differences we obtain excellent results for the registration of intramodality positron emission tomography pet and functional magnetic resonance imaging fmri data we conclude that the multiresolution refinement strategy is more robust than a comparable single stage method being less likely to be trapped into a false local optimum in addition our improved version of the marquardt levenberg algorithm is faster

structure preserving galerkin pod reduced order modeling of hamiltonian systems abstract the proper orthogonal decomposition reduced order model pod rom has been widely used as a computationally efficient surrogate model in large scale numerical simulations of complex systems however when it is applied to a hamiltonian system a naive application of the pod method can destroy the hamiltonian structure in the reduced order model in this paper we develop a new reduced order modeling approach for hamiltonian systems which modifies the galerkin projection based pod rom so that the appropriate hamiltonian structure is preserved since the pod truncation can degrade the approximation of the hamiltonian function we propose to use a pod basis from shifted snapshots to improve the approximation to the hamiltonian function we further derive a rigorous a priori error estimate for the structure preserving rom and demonstrate its effectiveness in several numerical examples this approach can be readily extended to dissipative hamiltonian systems port hamiltonian systems etc

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

symplectic model reduction of hamiltonian systems in this paper a symplectic model reduction technique proper symplectic decomposition psd with symplectic galerkin projection is proposed to save the computational cost for the simplification of large scale hamiltonian systems while preserving the symplectic structure as an analogy to the classical proper orthogonal decomposition pod galerkin approach psd is designed to build a symplectic subspace to fit empirical data while the symplectic galerkin projection constructs a reduced hamiltonian system on the symplectic subspace for practical use we introduce three algorithms for psd which are based upon the cotangent lift complex singular value decomposition and nonlinear programming the proposed technique has been proven to preserve system energy and stability moreover psd can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear hamiltonian systems owing to these properties the proposed technique is better suited than the classical po

structure preserving model reduction for mechanical systems this paper focuses on methods of constructing of reduced order models of mechanical systems which preserve the lagrangian structure of the original system these methods may be used in combination with standard spatial decomposition methods such as the karhunen loeve expansion balancing and wavelet decompositions the model reduction procedure is implemented for three dimensional finite element models of elasticity and we show that using the standard newmark implicit integrator significant savings are obtained in the computational costs of simulation in particular simulation of the reduced model scales linearly in the number of degrees of freedom and parallelizes well

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

a linear kernel for finding square roots of almost planar graphs abstract a graph h is a square root of a graph g if g can be obtained from h by the addition of edges between any two vertices in h that are at distance 2 from each other the square root problem is that of deciding whether a given graph admits a square root we consider this problem for planar graphs in the context of the distance from triviality framework for an integer k a planar k v graph or k apex graph is a graph that can be made planar by the removal of at most k vertices we prove that a generalization of square root in which some edges are prescribed to be either in or out of any solution has a kernel of size o k for planar k v graphs when parameterized by k our result is based on a new edge reduction rule which as we shall also show has a wider applicability for the square root problem