Abstract: Abstract In this paper we introduce a new family of interpolatory subdivision schemes with the capability of reproducing trigonometric and hyperbolic functions, as well as polynomials up to second degree. It is well known that linear, non-stationary, subdivision schemes do achieve this goal, but their application requires the practical determination of the parameters defining the level-dependent rules, by preprocessing the available data. Since different conic sections require different refinement rules to guarantee exact reproduction, it is not possible to reproduce a shape composed, piecewisely, by several trigonometric functions. On the other hand, our construction is based on the design of a family of stationary nonlinear rules. We show that exact reproduction of different conic shapes may be achieved using the same nonlinear scheme, without any previous preprocessing of the data. Convergence, stability, approximation, and shape preservation properties of the new schemes are analyzed. In addition, the conditions to obtain \(\mathcal {C}^{1}\) limit functions are also studied, which are related with the monotonicity of the data. Some numerical experiments are also carried out to check the theoretical results, and a preferred nonlinear scheme in the family is identified. PubDate: 2019-12-03

Abstract: Abstract We address numerical challenges in solving hyperbolic free boundary problems described by spherically symmetric conservation laws that arise in the modeling of tumor growth due to immune cell infiltrations. In this work, we normalize the radial coordinate to transform the free boundary problem to a fixed boundary one, and utilize finite volume methods to discretize the resulting equations. We show that the conventional finite volume methods fail to preserve constant solutions and the incompressibility condition, and they typically lead to inaccurate, if not wrong, solutions even for very simple tests. These issues are addressed in a new finite volume framework with segregated flux computations that satisfy sufficient conditions for ensuring the so-called totality conservation law and the geometric conservation law. We focus on describing the new framework by enhancing the conventional first-order upwind scheme, and the methodology extends to second-order ones. The numerical performance is assessed by various benchmark tests to show that the enhanced method is able to preserve the incompressibility constraint and produce much more accurate solutions than the conventional one. PubDate: 2019-12-03

Abstract: Abstract We propose a probabilistic way for reducing the cost of classical projection-based model order reduction methods for parameter-dependent linear equations. A reduced order model is here approximated from its random sketch, which is a set of low-dimensional random projections of the reduced approximation space and the spaces of associated residuals. This approach exploits the fact that the residuals associated with approximations in low-dimensional spaces are also contained in low-dimensional spaces. We provide conditions on the dimension of the random sketch for the resulting reduced order model to be quasi-optimal with high probability. Our approach can be used for reducing both complexity and memory requirements. The provided algorithms are well suited for any modern computational environment. Major operations, except solving linear systems of equations, are embarrassingly parallel. Our version of proper orthogonal decomposition can be computed on multiple workstations with a communication cost independent of the dimension of the full order model. The reduced order model can even be constructed in a so-called streaming environment, i.e., under extreme memory constraints. In addition, we provide an efficient way for estimating the error of the reduced order model, which is not only more efficient than the classical approach but is also less sensitive to round-off errors. Finally, the methodology is validated on benchmark problems. PubDate: 2019-12-03

Abstract: Abstract In this paper, a new mixed finite element scheme in space and a linearized backward Euler scheme in time are presented and investigated for the nonlinear Schrödinger equations. By introducing a suitable time-discrete system, both the errors in L2- and H1-norms for the original variable and L2-norm for the flux variable are derived without any time-step restriction, while previous works always required certain conditions between time step and space size. Finally, some numerical results are provided to verify the theoretical analysis. PubDate: 2019-12-03

Abstract: Abstract The shallow water equations (SWE) are a widely used model for the propagation of surface waves on the oceans. We consider the problem of optimally determining the initial conditions for the one-dimensional SWE in an unbounded domain from a small set of observations of the sea surface height. In the linear case, we prove a theorem that gives sufficient conditions for convergence to the true initial conditions. At least two observation points must be used and at least one pair of observation points must be spaced more closely than half the effective minimum wavelength of the energy spectrum of the initial conditions. This result also applies to the linear wave equation. Our analysis is confirmed by numerical experiments for both the linear and nonlinear SWE data assimilation problems. These results show that convergence rates improve with increasing numbers of observation points and that at least three observation points are required for practically useful results. Better results are obtained for the nonlinear equations provided more than two observation points are used. This paper is a first step towards understanding the conditions for observability of the SWE for small numbers of observation points in more physically realistic settings. PubDate: 2019-12-03

Abstract: Abstract Kolmogorov n-widths and Hankel singular values are two commonly used concepts in model reduction. Here, we show that for the special case of linear time-invariant (LTI) dynamical systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov n-width of an LTI system equals its (n + 1)st Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov n-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov n-width. PubDate: 2019-11-29

Abstract: Abstract In this paper, we consider a total variation–based image denoising model that is able to alleviate the well-known staircasing phenomenon possessed by the Rudin-Osher-Fatemi model (Rudin et al., Phys. D 60, 259–268, 30). To minimize this variational model, we employ augmented Lagrangian method (ALM). Convergence analysis is established for the proposed algorithm. Numerical experiments are presented to demonstrate the features of the proposed model and also show the efficiency of the proposed numerical method. PubDate: 2019-11-29

Abstract: Abstract In this paper, we propose a reduced order approach for 3D variational data assimilation governed by parameterized partial differential equations. In contrast to the classical 3D-VAR formulation that penalizes the measurement error directly, we present a modified formulation that penalizes the experimentally observable misfit in the measurement space. Furthermore, we include a model correction term that allows to obtain an improved state estimate. We begin by discussing the influence of the measurement space on the amplification of noise and prove a necessary and sufficient condition for the identification of a “good” measurement space. We then propose a certified reduced basis (RB) method for the estimation of the model correction, the state prediction, the adjoint solution, and the observable misfit with respect to the true state for real-time and many-query applications. A posteriori bounds are proposed for the error in each of these approximations. Finally, we introduce different approaches for the generation of the reduced basis spaces and the stability-based selection of measurement functionals. The 3D-VAR method and the associated certified reduced basis approximation are tested in a parameter and state estimation problem for a steady-state thermal conduction problem with unknown parameters and unknown Neumann boundary conditions. PubDate: 2019-11-29

Abstract: Abstract In this paper, using the first-order Nédélec conforming edge element space of the second type, we develop and analyze a continuous interior penalty finite element method (CIP-FEM) for the time-harmonic Maxwell equation in the three-dimensional space. Compared with the standard finite element methods, the novelty of the proposed method is that we penalize the jumps of the tangential component of its vorticity field. It is proved that if the penalty parameter is a complex number with negative imaginary part, then the CIP-FEM is well-posed without any mesh constraint. The error estimates for the CIP-FEM are derived. Numerical experiments are presented to verify our theoretical results. PubDate: 2019-11-29

Abstract: Abstract We study the information-based complexity of approximating integrals of smooth functions at absolute precision ε > 0 with confidence level 1 − δ ∈ (0, 1) using function evaluations within randomized algorithms. The probabilistic error criterion is new in the context of integrating smooth functions. In previous research, Monte Carlo integration was studied in terms of the expected error (or the root mean squared error), for which linear methods achieve optimal rates of the error e(n) in terms of the number n of function evaluations. In our context, usually methods that provide optimal confidence properties exhibit non-linear features. The optimal probabilistic error rate e(n,δ) for multivariate functions from classical isotropic Sobolev spaces \({W_{p}^{r}}(G)\) with sufficient smoothness on bounded Lipschitz domains \(G \subset {\mathbb R}^{d}\) is determined. It turns out that the integrability index p has an effect on the influence of the uncertainty δ in the complexity. In the limiting case p = 1, we see that deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the additional effort for diminishing the uncertainty. Finally, we add a discussion about this problem for function spaces with mixed smoothness. PubDate: 2019-11-29

Abstract: Abstract Smooth orthogonal projections with good localization properties were originally studied in the wavelet literature as a way to both understand and generalize the construction of smooth wavelet bases on \(L^{2}(\mathbb {R})\). Smoothness plays a critical role in the construction of wavelet bases and their generalizations as it is instrumental to achieve excellent approximation properties. In this paper, we extend the construction of smooth orthogonal projections to higher dimensions, a challenging problem in general for which relatively few results are found in the literature. Our investigation is motivated by the study of multidimensional nonseparable multiscale systems such as shearlets. Using our new class of smooth orthogonal projections, we construct new smooth Parseval frames of shearlets in \(L^{2}(\mathbb {R}^{2})\) and \(L^{2}(\mathbb {R}^{3})\). PubDate: 2019-11-29

Abstract: Parametric models in vector spaces are shown to possess an associated linear map, leading directly to reproducing kernel Hilbert spaces and affine/linear representations in terms of tensor products. From this map, analogues of correlation operators can be formed such that the associated linear map factorises the correlation. Its spectral decomposition and the associated Karhunen-Loève- or proper orthogonal decomposition in a tensor product follow directly, including an extension to continuous spectra. It is shown that all factorisations of a certain class are unitarily equivalent, as well as that every factorisation induces a different representation, and vice versa. No particular assumptions are made on the parameter set, other than that the vector space of real valued functions on this set allows an appropriate inner product on a subspace. A completely equivalent spectral and factorisation analysis can be carried out in kernel space. The relevance of these abstract constructions is shown on a number of mostly familiar examples, thus unifying many such constructions under one theoretical umbrella. From the factorisation, one obtains tensor representations, which may be cascaded, leading to tensors of higher degree. When carried over to a discretised level in the form of a model order reduction, such factorisations allow sparse low-rank approximations which lead to very efficient computations especially in high dimensions. PubDate: 2019-11-28

Abstract: Abstract In this work, we set up a new, general, and computationally efficient way to tackle parametrized fluid flows modeled through unsteady Navier-Stokes equations defined on domains with variable shape, when relying on the reduced basis method. We easily describe a domain by flexible boundary parametrizations, and generate domain (and mesh) deformations by means of a solid extension, obtained by solving a harmonic extension or a linear elasticity problem. The proposed procedure is built over a two-stage reduction: (i) first, we construct a reduced basis approximation for the mesh motion problem, irrespective of the fluid flow problem we focus on; (ii) then, we generate a reduced basis approximation of the unsteady Navier-Stokes problem, relying on finite element snapshots evaluated over a set of reduced deformed configuration, and approximating both velocity and pressure fields simultaneously. To deal with unavoidable nonaffine parametric dependencies arising in both the mesh motion and the state problem, we apply a matrix version of the discrete empirical interpolation method, allowing treating a wide range of geometrical deformations in an efficient and purely algebraic way. The same strategy is used to perform hyper-reduction of nonlinear terms. To assess the numerical performances of the proposed technique, we address the solution of parametrized fluid flows where the parameters describe both the shape of the domain and relevant physical features. Complex flow patterns such as the ones appearing in a patient-specific carotid bifurcation are accurately approximated, as well as derived quantities of potential clinical interest. PubDate: 2019-11-27

Abstract: Abstract We present a model reduction formulation for parametrized nonlinear partial differential equations (PDEs) associated with steady hyperbolic and convection-dominated conservation laws. Our formulation builds on three ingredients: a discontinuous Galerkin (DG) method which provides stability for conservation laws, reduced basis (RB) spaces which provide low-dimensional approximations of the parametric solution manifold, and the empirical quadrature procedure (EQP) which provides hyperreduction of the Galerkin-projection-based reduced model. The hyperreduced system inherits the stability of the DG discretization: (i) energy stability for linear hyperbolic systems, (ii) symmetry and non-negativity for steady linear diffusion systems, and hence (iii) energy stability for linear convection-diffusion systems. In addition, the framework provides (a) a direct quantitative control of the solution error induced by the hyperreduction, (b) efficient and simple hyperreduction posed as a ℓ1 minimization problem, and (c) systematic identification of the reduced bases and the empirical quadrature rule by a greedy algorithm. We demonstrate the formulation for parametrized aerodynamics problems governed by the compressible Euler and Navier-Stokes equations. PubDate: 2019-11-23

Abstract: Abstract The purpose of this work is to present different reduced order model strategies starting from full order simulations stabilized using a residual-based variational multiscale (VMS) approach. The focus is on flows with moderately high Reynolds numbers. The reduced order models (ROMs) presented in this manuscript are based on a POD-Galerkin approach. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case, the VMS stabilization method is used at both the full order and the reduced order levels. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark. PubDate: 2019-11-23

Abstract: Abstract A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work, we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings. PubDate: 2019-11-19

Abstract: Abstract In this contribution, we are concerned with tight a posteriori error estimation for projection-based model order reduction of \(\inf \)-\(\sup \) stable parameterized variational problems. In particular, we consider the Reduced Basis Method in a Petrov-Galerkin framework, where the reduced approximation spaces are constructed by the (weak) greedy algorithm. We propose and analyze a hierarchical a posteriori error estimator which evaluates the difference of two reduced approximations of different accuracy. Based on the a priori error analysis of the (weak) greedy algorithm, it is expected that the hierarchical error estimator is sharp with efficiency index close to one, if the Kolmogorov N-with decays fast for the underlying problem and if a suitable saturation assumption for the reduced approximation is satisfied. We investigate the tightness of the hierarchical a posteriori estimator both from a theoretical and numerical perspective. For the respective approximation with higher accuracy, we study and compare basis enrichment of Lagrange- and Taylor-type reduced bases. Numerical experiments indicate the efficiency for both, the construction of a reduced basis using the hierarchical error estimator in a greedy algorithm, and for tight online certification of reduced approximations. This is particularly relevant in cases where the \(\inf \)-\(\sup \) constant may become small depending on the parameter. In such cases, a standard residual-based error estimator—complemented by the successive constrained method to compute a lower bound of the parameter dependent \(\inf \)-\(\sup \) constant—may become infeasible. PubDate: 2019-11-14

Abstract: Abstract We consider model order reduction for a free boundary problem of an osmotic cell that is parameterized by material parameters as well as the initial shape of the cell. Our approach is based on an Arbitrary-Lagrangian-Eulerian description of the model that is discretized by a mass-conservative finite element scheme. Using reduced basis techniques and empirical interpolation, we construct a parameterized reduced order model in which the mass conservation property of the full-order model is exactly preserved. Numerical experiments are provided that highlight the performance of the resulting reduced order model. PubDate: 2019-11-13

Abstract: Abstract A statistical learning approach for high-dimensional parametric PDEs related to uncertainty quantification is derived. The method is based on the minimization of an empirical risk on a selected model class, and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors. PubDate: 2019-10-05

Abstract: Abstract We present an offline/online computational procedure for computing the dual norm of parameterized linear functionals. The approach is motivated by the need to efficiently compute residual dual norms, which are used in model reduction to estimate the error of a given reduced solution. The key elements of the approach are (i) an empirical test space for the manifold of Riesz elements associated with the parameterized functional and (ii) an empirical quadrature procedure to efficiently deal with parametrically non-affine terms. We present a number of theoretical and numerical results to identify the different sources of error and to motivate the proposed technique, and we compare the approach with other state-of-the-art techniques. Finally, we investigate the effectiveness of our approach to reduce both offline and online costs associated with the computation of the time-averaged residual indicator proposed in Fick et al. (J. Comput. Phys. 371, 214–243 2018). PubDate: 2019-09-11