In this paper, the problem of full state approximation by model reduction is studied for stochastic and bilinear systems. Our proposed approach relies on identifying the dominant subspaces based on the reachability Gramian of a system. Once the desired subspace is computed, the reduced order model is then obtained by a Galerkin projection. We prove that, in the stochastic case, this approach either preserves mean square asymptotic stability or leads to reduced models whose minimal realization is mean square asymptotically stable. This stability preservation guarantees the existence of the reduced system reachability Gramian which is the basis for the full state error bounds that we derive. This error bound depends on the neglected eigenvalues of the reachability Gramian and hence shows that these values are a good indicator for the expected error in the dimension reduction procedure. Subsequently, we establish the stability preservation result and the error bound for a full state approximation to bilinear systems in a similar manner. These latter results are based on a recently proved link between stochastic and bilinear systems. We conclude the paper by numerical experiments using a benchmark problem. We compare this approach with balanced truncation and show that it performs well in reproducing the full state of the system. \end{abstract}
We propose and analyze a pressure-stabilized projection Lagrange--Galerkin scheme for the transient Oseen problem. The proposed scheme inherits the following advantages from the projection Lagrange--Galerkin scheme. The first advantage is computational efficiency. The scheme decouples the computation of each component of the velocity and pressure. The other advantage is essential unconditional stability. Here we also use the equal-order approximation for the velocity and pressure, and add a symmetric pressure stabilization term. This enriched pressure space enables us to obtain accurate soluti…
In this paper we study the conforming Galerkin approximation of the problem: find u $\in$ U such that a(u, v) = <L, v> for all v $\in$ V, where U and V are Hilbert or Banach spaces, a is a continuous bilinear or sesquilinear form and L $\in$ V' a given data. The approximate solution is sought in a finite dimensional subspace of U, and test functions are taken in a finite dimensional subspace of V. We provide a necessary and sufficient condition on the form a for convergence of the Galerkin approximation, which is also equivalent to convergence of the Galerkin approximation for the adjoint prob…
A priori analysis for a generalized local projection stabilized conforming finite element approximation of Darcy flow and Stokes problems is presented in this paper. A first-order conforming P1 finite element space is used to approximate both the velocity and the pressure. It is shown that the stabilized discrete bilinear form satisfy the inf-sup condition with respect to a generalized local projection norm. Moreover, a priori error estimates are derived for both problems. Finally, the validation of the proposed stabilization scheme is demonstrated with appropriate numerical examples.
Sets of bilinear constraints are important in various machine learning models. Mathematically, they are hyperbolas in a product space. In this paper, we give a complete formula for projections onto sets of bilinear constraints or hyperbolas in a general Hilbert space.
High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable DG methods which incorporate an "entropy projection" are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.
We propose, analyze mathematically, and study numerically a novel approach for the finite element approximation of the spectrum of second-order elliptic operators. The main idea is to reduce the stiffness of the problem by subtracting a least-squares penalty on the gradient jumps across the mesh interfaces from the standard stiffness bilinear form. This penalty bilinear form is similar to the known technique used to stabilize finite element approximations in various contexts. The penalty term is designed to dampen the high frequencies in the spectrum and so it is weighted here by a negative co…
On Hyperbolic Stochastic Galerkin Projections of Shallow Water Linearised Moment Equations
Bernoulli Institute, University of Groningen（格罗宁根大学伯努利研究所）；CogniGron (Groningen Cognitive Systems and Materials Centre), University of Groningen（格罗宁根大学认知系统材料中心（CogniGron））；Department of Mathematics, Computer Science and Statistics, Ghent University（根特大学数学、计算机科学与统计系）
Galerkin and Petrov-Galerkin projection-based reduced-order models (ROMs) of transient partial differential equations are typically obtained by performing a dimension reduction and projection process that is defined at either the spatially continuous or spatially discrete level. In both cases, it is common to add stabilization to the resulting ROM to increase the stability and accuracy of the method; the addition of stabilization is particularly common for advection-dominated systems when the ROM is under-resolved. While these two approaches can be equivalent in certain settings, differing tec…
The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall type result, optimal error estimates are derived, when piecewise polynomials of degree $r\geq 1$ are used, which improve upon earlier results of Axelsson [Numer. Math. 28 (1977), pp. 1-14] requiring for 2d $r\geq 2$ and for 3d $r\geq 3.$ Based on quasi-projection technique introduced by Douglas {\it et al.} [Math. Comp.32 (1978),pp. 345-362], superconvergence …
We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction-diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and "balanced" norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the "balanced" norm is artifically introduced to capture the boundary layer contribution. We establish a uniform converg…
In this paper, we use Fourier analysis to study the superconvergence of the semi-discrete discontinuous Galerkin method for scalar linear advection equations in one spatial dimension. The error bounds and asymptotic errors are derived for initial discretization by $L_2$ projection, Gauss-Radau projection, and other projections proposed by Cao et. al. For pedagogical purpose, the errors are computed in two different ways. In the first approach, we compute the difference between the numerical solution and a special interpolation of the exact solution, and show that it consists of an asymptotic e…
For singularly perturbed reaction-diffusion problems in 1D and 2D, we study a local discontinuous Galerkin (LDG) method on a Shishkin mesh. In these cases, the standard energy norm is too weak to capture adequately the behavior of the boundary layers that appear in the solutions. To deal with this deficiency, we introduce a balanced norm stronger than the energy norm. In order to achieve optimal convergence under the balanced norm in one-dimensional case, we design novel numerical fluxes and propose a special interpolation that consists of a Gauss-Radau projection and a local $L^2$ projection.…
We study dynamical Galerkin schemes for evolutionary partial differential equations (PDEs), where the projection operator changes over time. When selecting a subset of basis functions, the projection operator is non-differentiable in time and an integral formulation has to be used. We analyze the projected equations with respect to existence and uniqueness of the solution and prove that non-smooth projection operators introduce dissipation, a result which is crucial for adaptive discretizations of PDEs, e.g., adaptive wavelet methods. For the Burgers equation we illustrate numerically that thr…
The discontinuous Galerkin (DG) time-stepping method applied to abstract evolution equation of parabolic type is studied using a variational approach. We establish the inf-sup condition or Babuška--Brezzi condition for the DG bilinear form. Then, a nearly best approximation property and a nearly symmetric error estimate are obtained as corollaries. Moreover, the optimal order error estimates under appropriate regularity assumption on the solution are derived as direct applications of the standard interpolation error estimates. Our method of analysis is new for the DG time-stepping method; it d…
We study in this paper new developments of the Lagrange-Galerkin method for the advection equation. In the first part of the article we present a new improved error estimate of the conventional Lagrange-Galerkin method. In the second part, we introduce a new local projection stabilized Lagrange-Galerkin method, whereas in the third part we introduce and analyze a discontinuity-capturing Lagrange-Galerkin method. Also, attention has been paid to the influence of the quadrature rules on the stability and accuracy of the methods via numerical experiments.
The differential Riccati equation appears in different fields of applied mathematics like control and system theory. Recently Galerkin methods based on Krylov subspaces were developed for the autonomous differential Riccati equation. These methods overcome the prohibitively large storage requirements and computational costs of the numerical solution. In view of memory efficient approximation, we review and extend known solution formulas and identify invariant subspaces for a possibly low-dimensional solution representation. Based on these theoretical findings, we propose a Galerkin projection …
Evaluating High Order Discontinuous Galerkin Discretization of the Boltzmann Collision Integral in $O(N^2)$ Operations Using the Discrete Fourier Transform
Department of Mathematics, California State University Nortrhidge, Nortrhdige, CA 91330, USA
This work proposes a windowed least-squares (WLS) approach for model-reduction of dynamical systems. The proposed approach sequentially minimizes the time-continuous full-order-model residual within a low-dimensional space-time trial subspace over time windows. The approach comprises a generalization of existing model reduction approaches, as particular instances of the methodology recover Galerkin, least-squares Petrov-Galerkin (LSPG), and space-time LSPG projection. In addition, the approach addresses key deficiencies in existing model-reduction techniques, e.g., the dependence of LSPG and s…
Several nonlinear model reduction techniques are compared for the three cases of the non-parallel version of the Kuramoto-Sivashinsky equation, the transient regime of flow past a cylinder at $Re=100$ and fully developed flow past a cylinder at the same Reynolds number. The linear terms of the governing equations are reduced by Galerkin projection onto a POD basis of the flow state, while the reduced nonlinear convection terms are obtained either by a Galerkin projection onto the same state basis, by a Galerkin projection onto a POD basis representing the nonlinearities or by applying the Disc…
Discontinuous Galerkin methods of higher order are applied as temporal discretizations for the transient Navier--Stokes equations. The spatial discretization based on inf-sup stable pairs of finite element spaces is stabilised using a one-level local projection stabilisation method. Optimal error bounds for the velocity with constants independent of the viscosity parameter are obtained for the semi-discrete case. For the fully discrete case, error estimates for both velocity and pressure are given. Numerical results support the theoretical predictions.
A Reduced-Order Model for Nonlinear Radiative Transfer Problems Based on Moment Equations and POD-Petrov-Galerkin Projection of the Normalized Boltzmann Transport Equation
A modified primal-dual weak Galerkin (M-PDWG) finite element method is designed for the second order elliptic equation in non-divergence form. Compared with the existing PDWG methods proposed in \cite{wwnondiv}, the system of equations resulting from the M-PDWG scheme could be equivalently simplified into one equation involving only the primal variable by eliminating the dual variable (Lagrange multiplier). The resulting simplified system thus has significantly fewer degrees of freedom than the one resulting from existing PDWG scheme. In addition, the condition number of the simplified system …
Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields, and derive a-priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neurosc…
Stability conditions for the explicit integration of projection based nonlinear reduced-order and hyper reduced structural mechanics finite element models
BMW Group, Research and Innovation Centre（宝马集团研究与创新中心）；Dynamore GmbH（Dynamore有限公司）；Queen Mary, University of London（伦敦大学女王玛丽学院）
This paper addresses the design of linear and nonlinear stabilization procedures for high-order continuous Galerkin (CG) finite element discretizations of scalar conservation laws. We prove that the standard CG method is entropy conservative for the square entropy. In general, the rate of entropy production/dissipation depends on the residual of the governing equation and on the accuracy of the finite element approximation to the entropy variable. The inclusion of linear high-order stabilization generates an additional source/sink in the entropy budget equation. To balance the amount of entrop…
Superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional linear time-dependent fourth-order equations
Pod-Galerkin Reduced Order Methods for CFD Using Finite Volume Discretisation: Vortex Shedding Around a Circular Cylinder
SISSA, International School for Advanced Studies, Mathematics Area, mathLab Trieste, Italy（国际高等研究院）；Department of Energy, Politecnico di Milano, Italy（米兰理工大学能源系）
Conservative model reduction for finite-volume models
Sandia National Laboratories（桑迪亚国家实验室）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；University of Washington（华盛顿大学）；HERE Technologies（HERE科技公司）
