Authors:Anna Bertram; Ralf Zimmermann Pages: 1693 - 1716 Abstract: Cokriging is a variable-fidelity surrogate modeling technique which emulates a target process based on the spatial correlation of sampled data of different levels of fidelity. In this work, we address two theoretical questions associated with the so-called new Cokriging method for variable-fidelity modeling: A mandatory requirement for the well-posedness of the Cokriging emulator is the positive definiteness of the associated Cokriging correlation matrix. Spatial correlations are usually modeled by positive definite correlation kernels, which are guaranteed to yield positive definite correlation matrices for mutually distinct sample points. However, in applications, low-fidelity information is often available at high-fidelity sample points and the Cokriging predictor may benefit from the additional information provided by such an inclusive sampling. We investigate the positive definiteness of the Cokriging covariance matrix in both of the aforementioned cases and derive sufficient conditions for the well-posedness of the Cokriging predictor. The approximation quality of the Cokriging predictor is highly dependent on a number of model- and hyper-parameters. These parameters are determined by the method of maximum likelihood estimation. For standard Kriging, closed-form optima of the model parameters along hyper-parameter profile lines are known. Yet, these do not readily transfer to the setting of Cokriging, since additional parameters arise, which exhibit a mutual dependence. In previous work, this obstacle was tackled via a numerical optimization. Here, we derive closed-form optima for all Cokriging model parameters along hyper-parameter profile lines. The findings are illustrated by numerical experiments. PubDate: 2018-12-01 DOI: 10.1007/s10444-017-9585-1 Issue No:Vol. 44, No. 6 (2018)

Authors:Omer San; Romit Maulik Pages: 1717 - 1750 Abstract: Many reduced-order models are neither robust with respect to parameter changes nor cost-effective enough for handling the nonlinear dependence of complex dynamical systems. In this study, we put forth a robust machine learning framework for projection-based reduced-order modeling of such nonlinear and nonstationary systems. As a demonstration, we focus on a nonlinear advection-diffusion system given by the viscous Burgers equation, which is a prototypical setting of more realistic fluid dynamics applications due to its quadratic nonlinearity. In our proposed methodology the effects of truncated modes are modeled using a single layer feed-forward neural network architecture. The neural network architecture is trained by utilizing both the Bayesian regularization and extreme learning machine approaches, where the latter one is found to be more computationally efficient. A significant emphasis is laid on the selection of basis functions through the use of both Fourier bases and proper orthogonal decomposition. It is shown that the proposed model yields significant improvements in accuracy over the standard Galerkin projection methodology with a negligibly small computational overhead and provide reliable predictions with respect to parameter changes. PubDate: 2018-12-01 DOI: 10.1007/s10444-018-9590-z Issue No:Vol. 44, No. 6 (2018)

Authors:Peter Benner; Christian Himpe; Tim Mitchell Pages: 1751 - 1768 Abstract: The identification of reduced-order models from high-dimensional data is a challenging task, and even more so if the identified system should not only be suitable for a certain data set, but generally approximate the input-output behavior of the data source. In this work, we consider the input-output dynamic mode decomposition method for system identification. We compare excitation approaches for the data-driven identification process and describe an optimization-based stabilization strategy for the identified systems. PubDate: 2018-12-01 DOI: 10.1007/s10444-018-9592-x Issue No:Vol. 44, No. 6 (2018)

Authors:Bülent Karasözen; Murat Uzunca Pages: 1769 - 1796 Abstract: An energy preserving reduced order model is developed for two dimensional nonlinear Schrödinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD. PubDate: 2018-12-01 DOI: 10.1007/s10444-018-9593-9 Issue No:Vol. 44, No. 6 (2018)

Authors:Zoran Tomljanović; Christopher Beattie; Serkan Gugercin Pages: 1797 - 1820 Abstract: We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the \(\mathcal {H}_{2}\) system norm. The objective function is non-convex and the associated optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates ‘interpolatory’ reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the \(\mathcal {H}_{2}\) system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments. PubDate: 2018-12-01 DOI: 10.1007/s10444-018-9605-9 Issue No:Vol. 44, No. 6 (2018)

Authors:Patrick Kürschner Pages: 1821 - 1844 Abstract: In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on the efficient numerical realization of this model reduction approach in case of large system dimensions. We discuss numerical methods to deal with the resulting matrix exponential functions and Lyapunov equations which are solved for low-rank approximations. Our main tool for this purpose are rational Krylov subspace methods. We also discuss the eigenvalue decay and numerical rank of the solutions of the Lyapunov equations. These results, and also numerical experiments, will show that depending on the final time horizon, the numerical rank of the Lyapunov solutions in time-limited balanced truncation can be smaller compared to standard balanced truncation. In numerical experiments we test the approaches for computing low-rank factors of the involved Lyapunov solutions and illustrate that time-limited balanced truncation can generate reduced order models having a higher accuracy in the considered time region. PubDate: 2018-12-01 DOI: 10.1007/s10444-018-9608-6 Issue No:Vol. 44, No. 6 (2018)

Authors:Ion Victor Gosea; Mihaly Petreczky; Athanasios C. Antoulas; Christophe Fiter Pages: 1845 - 1886 Abstract: We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians. PubDate: 2018-12-01 DOI: 10.1007/s10444-018-9610-z Issue No:Vol. 44, No. 6 (2018)

Authors:Andrea Carracedo Rodriguez; Serkan Gugercin; Jeff Borggaard Pages: 1887 - 1916 Abstract: Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been successfully extended to nonparametric bilinear dynamical systems. However, this has not yet occurred for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce necessary conditions that the projection subspaces must satisfy to obtain parametric tangential interpolation of each subsystem transfer function. These conditions also guarantee that the parameter sensitivities (Jacobian) of each subsystem transfer function are matched tangentially by those of the corresponding reduced-order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including additional vectors in the projection subspaces. As in the parametric linear case, the basis construction for two-sided projections does not require computing the Jacobian or the Hessian. PubDate: 2018-12-01 DOI: 10.1007/s10444-018-9611-y Issue No:Vol. 44, No. 6 (2018)

Authors:Xiaodong Cheng; Jacquelien M. A. Scherpen Pages: 1917 - 1939 Abstract: This paper considers the network structure preserving model reduction of power networks with distributed controllers. The studied system and controller are modeled as second-order and first-order ordinary differential equations, which are coupled to a closed-loop model for analyzing the dissimilarities of the power units. By transfer functions, we characterize the behavior of each node (generator or load) in the power network and define a novel notion of dissimilarity between two nodes by the \(\mathcal {H}_{2}\) -norm of the transfer function deviation. Then, the reduction methodology is developed based on separately clustering the generators and loads according to their behavior dissimilarities. The characteristic matrix of the resulting clustering is adopted for the Galerkin projection to derive explicit reduced-order power models and controllers. Finally, we illustrate the proposed method by the IEEE 30-bus system example. PubDate: 2018-12-01 DOI: 10.1007/s10444-018-9617-5 Issue No:Vol. 44, No. 6 (2018)

Authors:Carmen Gräßle; Michael Hinze Pages: 1941 - 1978 Abstract: The main focus of the present work is the inclusion of spatial adaptivity for the snapshot computation in the offline phase of model order reduction utilizing proper orthogonal decomposition (POD-MOR) for nonlinear parabolic evolution problems. We consider snapshots which live in different finite element spaces, which means in a fully discrete setting that the snapshots are vectors of different length. From a numerical point of view, this leads to the problem that the usual POD procedure which utilizes a singular value decomposition of the snapshot matrix, cannot be carried out. In order to overcome this problem, we here construct the POD model/basis using the eigensystem of the correlation matrix (snapshot Gramian), which is motivated from a continuous perspective and is set up explicitly, e.g., without the necessity of interpolating snapshots into a common finite element space. It is an advantage of this approach that the assembly of the matrix only requires the evaluation of inner products of snapshots in a common Hilbert space. This allows a great flexibility concerning the spatial discretization of the snapshots. The analysis for the error between the resulting POD solution and the true solution reveals that the accuracy of the reduced-order solution can be estimated by the spatial and temporal discretization error as well as the POD error. Finally, to illustrate the feasibility of our approach, we present a test case of the Cahn–Hilliard system utilizing h-adapted hierarchical meshes and two settings of a linear heat equation using nested and non-nested grids. PubDate: 2018-12-01 DOI: 10.1007/s10444-018-9620-x Issue No:Vol. 44, No. 6 (2018)

Authors:Raúl M. Falcón; Víctor Álvarez; Félix Gudiel Abstract: Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = (lij) of order n over a field \(\mathbb {K}\) as the set of zeros of the binomial ideal \(\langle x_{i}x_{j}-x_{l_{ij}}\colon (i,j) \text { is a non-empty cell in} L \rangle \subseteq \mathbb {K}[x_{1},\ldots ,x_{n}]\) . Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic. PubDate: 2018-12-12 DOI: 10.1007/s10444-018-9654-0

Authors:Xuanxuan Zhou; Tingchun Wang; Luming Zhang Abstract: Two numerical methods are presented for the approximation of the Zakharov-Rubenchik equations (ZRE). The first one is the finite difference integrator Fourier pseudospectral method (FFP), which is implicit and of the optimal convergent rate at the order of O(N−r + τ2) in the discrete L2 norm without any restrictions on the grid ratio. The second one is to use the Fourier pseudospectral approach for spatial discretization and exponential wave integrator for temporal integration. Fast Fourier transform is applied to the discrete nonlinear system to speed up the numerical computation. Numerical examples are given to show the efficiency and accuracy of the new methods. PubDate: 2018-12-11 DOI: 10.1007/s10444-018-9651-3

Authors:Xiangyun Meng; Martin Stynes Abstract: We consider the singularly perturbed fourth-order boundary value problem ε2Δ2u −Δu = f on the unit square \({\Omega }\subset \mathbb {R}^{2}\) , with boundary conditions u = ∂u/∂n = 0 on ∂Ω. Here, ε ∈ (0,1) is a small parameter. The problem is solved numerically by means of Adini finite elements—a simple nonconforming finite element method for this problem. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with N2 elements is constructed and convergence of the method is proved in a ‘broken’ version of the Sobolev norm \(v\mapsto \left (\varepsilon ^{2} v _{2}^{2} + v _{1}^{2} \right )^{1/2}\) . For a particular choice of the mesh, the error in the computed solution is at most C [ε1/2(N− 1 lnN)2 + min {ε1/2,ε− 3/2N− 2} + N− 3], where the constant C is independent of ε and N. Numerical results support our theoretical convergence rates, even for an example where not all the hypotheses of our theory are satisfied. PubDate: 2018-11-30 DOI: 10.1007/s10444-018-9646-0

Authors:Martin Ambroz; Martin Balažovjech; Matej Medl’a; Karol Mikula Abstract: We introduce a new approach to wildland fire spread modeling. We evolve a 3-D surface curve, which represents the fire perimeter on the topography, as a projection to a horizontal plane. Our mathematical model is based on the empirical laws of the fire spread influenced by the fuel, wind, terrain slope, and shape of the fire perimeter with respect to the topography (geodesic and normal curvatures). To obtain the numerical solution, we discretize the arising intrinsic partial differential equation by a semi-implicit scheme with respect to the curvature term. For the advection term discretization, we use the so-called inflow-implicit/outflow-explicit approach and an implicit upwind technique which guarantee the solvability of the corresponding linear systems by an efficient tridiagonal solver without any time step restriction and also the robustness with respect to singularities. A fast treatment of topological changes (splitting and merging of the curves) is described and shown on examples as well. We show the experimental order of convergence of the numerical scheme, we demonstrate the influence of the fire spread model parameters on a testing and real topography, and we reconstruct a simulated grassland fire as well. PubDate: 2018-11-26 DOI: 10.1007/s10444-018-9650-4

Authors:Nikolaos Rekatsinas; Rob Stevenson Abstract: In this work, we construct a well-posed first-order system least squares (FOSLS) simultaneously space-time formulation of parabolic PDEs. Using an adaptive wavelet solver, this problem is solved with the best possible rate in linear complexity. Thanks to the use of a basis that consists of tensor products of wavelets in space and time, this rate is equal to that when solving the corresponding stationary problem. Our findings are illustrated by numerical results. PubDate: 2018-11-23 DOI: 10.1007/s10444-018-9644-2

Authors:Mariam Al-Maskari; Samir Karaa Abstract: We consider the numerical approximation of a generalized fractional Oldroyd-B fluid problem involving two Riemann-Liouville fractional derivatives in time. We establish regularity results for the exact solution which play an important role in the error analysis. A semidiscrete scheme based on the piecewise linear Galerkin finite element method in space is analyzed, and optimal with respect to the data regularity error estimates are established. Further, two fully discrete schemes based on convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are investigated and related error estimates for smooth and nonsmooth data are derived. Numerical experiments are performed with different values of the problem parameters to illustrate the efficiency of the method and confirm the theoretical results. PubDate: 2018-11-15 DOI: 10.1007/s10444-018-9649-x

Authors:Daniele C. R. Gomes; Mauro A. Rincon; Maria Darci G. da Silva; Gladson O. Antunes Abstract: In this paper, we investigate some mathematical and numerical aspects of a one-dimensional nonlinear Schrödinger problem defined in a noncylindrical domain. By a change of variable, we transform the original problem into an equivalent one defined in a cylindrical domain. To obtain the existence and uniqueness of the solution, we apply the Faedo-Galerkin method and results of compactness. The numerical simulation is performed by means of the finite element method in the associated space and the finite difference method in the temporal part, to get an approximate numerical solution. In addition, we will make an analysis of the rate of convergence of the applied methods. Finally, we will show that the results of the numerical simulation are in agreement with the theoretical analysis. PubDate: 2018-11-13 DOI: 10.1007/s10444-018-9643-3

Authors:Xianyi Zeng Abstract: We propose a general hybrid-variable (HV) framework to solve linear advection equations by utilizing both cell-average approximations and nodal approximations. The construction is carried out for 1D problems, where the spatial discretization for cell averages is obtained from the integral form of the governing equation whereas that for nodal values is constructed using hybrid-variable discrete differential operators (HV-DDO); explicit Runge-Kutta methods are employed for marching the solutions in time. We demonstrate the connection between the HV-DDO and Hermite interpolation polynomials, and show that it can be constructed to arbitrary order of accuracy. In particular, we derive explicit formula for the coefficients to achieve the optimal order of accuracy given any compact stencil of the HV-DDO. The superconvergence of the proposed HV methods is then proved: these methods have one-order higher spatial accuracy than the designed order of the HV-DDO; in contrast, for conventional methods that only utilize one type of variables, the two orders are the same. Hence, the proposed method can potentially achieve higher-order accuracy given the same computational cost, comparing to existing finite difference methods. We then prove the linear stability of sample HV methods with up to fifth-order accuracy in the case of Cauchy problems. Next, we demonstrate how the HV methods can be extended to 2D problems as well as nonlinear conservation laws with smooth solutions. The performance of the sample HV methods are assessed by extensive 1D and 2D benchmark tests of linear advection equations, the nonlinear Euler equations, and the nonlinear Buckely-Leverett equation. PubDate: 2018-11-10 DOI: 10.1007/s10444-018-9647-z

Authors:Hui Liang; Martin Stynes Abstract: General Riemann-Liouville linear two-point boundary value problems of order αp, where n − 1 < αp < n for some positive integer n, are investigated on the interval [0,b]. It is shown first that the natural degree of regularity to impose on the solution y of the problem is \(y\in C^{n-2}[0,b]\) and \(D^{\alpha _{p}-1}y\in C[0,b]\) , with further restrictions on the behavior of the derivatives of y(n− 2) (these regularity conditions differ significantly from the natural regularity conditions in the corresponding Caputo problem). From this regularity, it is deduced that the most general choice of boundary conditions possible is \(y(0) = y^{\prime }(0) = {\dots } = y^{(n-2)}(0) = 0\) and \({\sum }_{j = 0}^{n_{1}}\beta _{j}y^{(j)}(b_{1}) =\gamma \) for some constants βj and γ, with b1 ∈ (0,b] and \(n_{1}\in \{0, 1, \dots , n-1\}\) . A wide class of transformations of the problem into weakly singular Volterra integral equations (VIEs) is then investigated; the aim is to choose the transformation that will yield the most accurate results when the VIE is solved using a collocation method with piecewise polynomials. Error estimates are derived for this method and for its iterated variant. Numerical results are given to support the theoretical conclusions. PubDate: 2018-11-08 DOI: 10.1007/s10444-018-9645-1