arXiv subjects

Peter Benner

Publications and source records attributed to Peter Benner.

At least 127 records · Page 7Linked to original sources

arXiv2020

A Non-Intrusive Method to Inferring Linear Port-Hamiltonian Realizations using Time-Domain Data

Port-Hamiltonian systems have gained a lot of attention in recent years due to their inherent valuable properties in modeling and control. In this paper, we are interested in constructing linear port-Hamiltonian systems from time-domain input-output data. We discuss a non-intrusive methodology that is comprised of two main ingredients -- (a) inferring frequency response data from time-domain data and (b) constructing an underlying port-Hamiltonian realization using the inferred frequency response data. We illustrate the proposed methodology by means of two numerical examples and also compare it with two other system identification methods to infer the frequency response from the input-output data.

eess.SY↗

arXiv2020

Relative Perturbation Theory for Quadratic Hermitian Eigenvalue Problem

In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $λ^2 M x + λC x + K x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices and $C$ is a general Hermitian matrix. We base our findings on new results for an equivalent regular Hermitian matrix pair $A-λB$. The new bounds can be applied to many interesting quadratic eigenvalue problems appearing in applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments.

math.NA↗

arXiv2020

Machine Learning for Material Characterization with an Application for Predicting Mechanical Properties

Currently, the growth of material data from experiments and simulations is expanding beyond processable amounts. This makes the development of new data-driven methods for the discovery of patterns among multiple lengthscales and time-scales and structure-property relationships essential. These data-driven approaches show enormous promise within materials science. The following review covers machine learning applications for metallic material characterization. Many parameters associated with the processing and the structure of materials affect the properties and the performance of manufactured components. Thus, this study is an attempt to investigate the usefulness of machine learning methods for material property prediction. Material characteristics such as strength, toughness, hardness, brittleness or ductility are relevant to categorize a material or component according to their quality. In industry, material tests like tensile tests, compression tests or creep tests are often time consuming and expensive to perform. Therefore, the application of machine learning approaches is considered helpful for an easier generation of material property information. This study also gives an application of machine learning methods on small punch test data for the determination of the property ultimate tensile strength for various materials. A strong correlation between small punch test data and tensile test data was found which ultimately allows to replace more costly tests by simple and fast tests in combination with machine learning.

cs.LG↗

arXiv2020

Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms

This work presents a non-intrusive model reduction method to learn low-dimensional models of dynamical systems with non-polynomial nonlinear terms that are spatially local and that are given in analytic form. In contrast to state-of-the-art model reduction methods that are intrusive and thus require full knowledge of the governing equations and the operators of a full model of the discretized dynamical system, the proposed approach requires only the non-polynomial terms in analytic form and learns the rest of the dynamics from snapshots computed with a potentially black-box full-model solver. The proposed method learns operators for the linear and polynomially nonlinear dynamics via a least-squares problem, where the given non-polynomial terms are incorporated in the right-hand side. The least-squares problem is linear and thus can be solved efficiently in practice. The proposed method is demonstrated on three problems governed by partial differential equations, namely the diffusion-reaction Chafee-Infante model, a tubular reactor model for reactive flows, and a batch-chromatography model that describes a chemical separation process. The numerical results provide evidence that the proposed approach learns reduced models that achieve comparable accuracy as models constructed with state-of-the-art intrusive model reduction methods that require full knowledge of the governing equations.

math.NA↗

arXiv2020

Data-Driven Snapshot Calibration via Monotonic Feature Matching

Snapshot matrices of hyperbolic equations have a slow singular value decay, resulting in inefficient reduced-order models. We develop on the idea of inducing a faster singular value decay by computing snapshots on a transformed spatial domain, or the so-called snapshot calibration/transformation. We are particularly interested in problems involving shock collision, shock rarefaction-fan collision, shock formation, etc. For such problems, we propose a realizable algorithm to compute the spatial transform using monotonic feature matching. We consider discontinuities and kinks as features, and by carefully partitioning the parameter domain, we ensure that the spatial transform has properties that are desirable both from a theoretical and an implementation standpoint. We use these properties to prove that our method results in a fast m-width decay of a so-called calibrated manifold. A crucial observation we make is that due to calibration, the m-width does not only depend on m but also on the accuracy of the full order model, which is in contrast to elliptic and parabolic problems that do not need calibration. The method we propose only requires the solution snapshots and not the underlying partial differential equation (PDE) and is therefore, data-driven. We perform several numerical experiments to demonstrate the effectiveness of our method.

math.NA↗

arXiv2020

A rational Even-IRA algorithm for the solution of T-even polynomial eigenvalue problems

In this work we present a rational Krylov subspace method for solving real large-scale polynomial eigenvalue problems with T-even (that is, symmetric/skew-symmetric) structure. Our method is based on the Even-IRA algorithm. To preserve the structure, a sparse T-even linearization from the class of block minimal bases pencils is applied. Due to this linearization, the Krylov basis vectors can be computed in a cheap way. A rational decomposition is derived so that our method explicitly allows for changes of the shift during the iteration. This leads to a method that is able to compute parts of the spectrum of a T-even matrix polynomial in a fast and reliable way.

math.NA↗

arXiv2020

Space and Chaos-Expansion Galerkin POD Low-order Discretization of PDEs for Uncertainty Quantification

The quantification of multivariate uncertainties in partial differential equations can easily exceed any computing capacity unless proper measures are taken to reduce the complexity of the model. In this work, we propose a multidimensional Galerkin Proper Orthogonal Decomposition that optimally reduces each dimension of a tensorized product space. We provide the analytical framework and results that define and quantify the low-dimensional approximation. We illustrate its application for uncertainty modeling with Polynomial Chaos Expansions and show its efficiency in a numerical example.

math.DS↗

arXiv2020

Efficient and Accurate Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem for Crystalline Systems

Optical properties of materials related to light absorption and scattering are explained by the excitation of electrons. The Bethe-Salpeter equation is the state-of-the-art approach to describe these processes from first principles (ab initio), i.e. without the need for empirical data in the model. To harness the predictive power of the equation, it is mapped to an eigenvalue problem via an appropriate discretization scheme. The eigenpairs of the resulting large, dense, structured matrix can be used to compute dielectric properties of the considered crystalline or molecular system. The matrix always shows a $2\times 2$ block structure. Additionally, certain definiteness properties typically hold. One form can be acquired for crystalline systems, another one is more general and can for example be used to study molecules. In this work, we present new theoretical results characterizing the structure of the two forms in the language of non-standard scalar products. These results enable us to develop a new perspective on the state-of-the-art solution approach for crystalline systems. This new viewpoint is used to develop two new methods for solving the eigenvalue problem. One requires less computational effort while providing the same degree of accuracy. The other one improves the expected accuracy, compared to methods currently in use, with a comparable performance. Both methods are well suited for high performance environments and only rely on basic numerical linear algebra building blocks.

math.NA↗

arXiv2020

An Adaptive Sampling Approach for the Reduced Basis Method

The offline time of the reduced basis method can be very long given a large training set of parameter samples. This usually happens when the system has more than two independent parameters. On the other hand, if the training set includes fewer parameter samples, the greedy algorithm might produce a reduced-order model with large errors at the samples outside of the training set. We introduce a method based on a surrogate error model to efficiently sample the parameter domain such that the training set is adaptively updated starting from a coarse set with a small number of parameter samples. A sharp a posteriori error estimator is evaluated on a coarse training set. Radial Basis Functions are used to interpolate the error estimator over a separate fine training set. Points from the fine training set are added into the coarse training set at every iteration based on a user defined criterion. In parallel, parameter samples satisfying a defined tolerance are adaptively removed from the coarse training set. The approach is shown to avoid high computational costs by using a small training set and to provide a reduced-order model with guaranteed accuracy over a fine training set. Further, we show numerical evidence that the reduced-order model meets the defined tolerance over an independently sampled test set from the parameter domain.

math.NA↗

arXiv2020

Data-Driven Learning of Reduced-order Dynamics for a Parametrized Shallow Water Equation

This paper discusses a non-intrusive data-driven model order reduction method that learns low-dimensional dynamical models for a parametrized shallow water equation. We consider the shallow water equation in non-traditional form (NTSWE). We focus on learning low-dimensional models in a non-intrusive way. That means, we assume not to have access to a discretized form of the NTSWE in any form. Instead, we have snapshots that are obtained using a black-box solver. Consequently, we aim at learning reduced-order models only from the snapshots. Precisely, a reduced-order model is learnt by solving an appropriate least-squares optimization problem in a low-dimensional subspace. Furthermore, we discuss computational challenges that particularly arise from the optimization problem being ill-conditioned. Moreover, we extend the non-intrusive model order reduction framework to a parametric case where we make use of the parameter dependency at the level of the partial differential equation. We illustrate the efficiency of the proposed non-intrusive method to construct reduced-order models for NTSWE and compare it with an intrusive method (proper orthogonal decomposition). We furthermore discuss the predictive capabilities of both models outside the range of the training data.

math.NA↗

arXiv2020

Structure-Preserving Interpolation for Model Reduction of Parametric Bilinear Systems

In this paper, we present an interpolation framework for structure-preserving model order reduction of parametric bilinear dynamical systems. We introduce a general setting, covering a broad variety of different structures for parametric bilinear systems, and then provide conditions on projection spaces for the interpolation of structured subsystem transfer functions such that the system structure and parameter dependencies are preserved in the reduced-order model. Two benchmark examples with different parameter dependencies are used to demonstrate the theoretical analysis.

math.NA↗

arXiv2020

Clustering-Based Model Order Reduction for Nonlinear Network Systems

Clustering by projection has been proposed as a way to preserve network structure in linear multi-agent systems. Here, we extend this approach to a class of nonlinear network systems. Additionally, we generalize our clustering method which restores the network structure in an arbitrary reduced-order model obtained by projection. We demonstrate this method on a number of examples.

math.NA↗

arXiv2020

Adaptive Basis Construction and Improved Error Estimation for Parametric Nonlinear Dynamical Systems

An adaptive scheme to generate reduced-order models for parametric nonlinear dynamical systems is proposed. It aims to automatize the POD-Greedy algorithm combined with empirical interpolation. At each iteration, it is able to adaptively determine the number of the reduced basis vectors and the number of the interpolation basis vectors for basis construction. The proposed technique is able to derive a suitable match between the reduced basis and the interpolation basis vectors, making the generation of a stable, compact and reliable reduced-order model possible. This is achieved by adaptively adding new basis vectors or removing unnecessary ones, at each iteration of the greedy algorithm. An efficient output error indicator plays a key role in the adaptive scheme. We also propose an improved output error indicator based on previous work. Upon convergence of the POD-Greedy algorithm, the new error indicator is shown to be sharper than the existing ones, implicating that a more reliable reduced-order model can be constructed. The proposed method is tested on several nonlinear dynamical systems, namely, the viscous Burgers' equation and two other models from chemical engineering.

math.NA↗

arXiv2020

GR decompositions and their relations to Cholesky-like factorizations

For a given matrix, we are interested in computing GR decompositions $A=GR$, where $G$ is an isometry with respect to given scalar products. The orthogonal QR decomposition is the representative for the Euclidian scalar product. For a signature matrix, a respective factorization is given as the hyperbolic QR decomposition. Considering a skew-symmetric matrix leads to the symplectic QR decomposition. The standard approach for computing GR decompositions is based on the successive elimination of subdiagonal matrix entries. For the hyperbolic and symplectic case, this approach does in general not lead to a satisfying numerical accuracy. An alternative approach computes the QR decomposition via a Cholesky factorization, but also has bad stability. It is improved by repeating the procedure a second time. In the same way, the hyperbolic and the symplectic QR decomposition are related to the $LDL^T$ and a skew-symmetric Cholesky-like factorization. We show that methods exploiting this connection can provide better numerical stability than elimination-based approaches.

math.NA↗

arXiv2020

A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation

We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with non-homogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the temporal discretization. Due to the low regularity induced by the boundary noise, convergence orders above 1/2 in space and 1/4 in time cannot be expected. We prove such optimal convergence orders for our full discretization when the distributed noise and the initial condition are sufficiently smooth. Under less smooth conditions, the convergence order is further decreased. Our results only assume that the related (deterministic) differential Riccati equation can be approximated with a certain convergence order, which is easy to achieve in practice. We confirm these theoretical results through a numerical experiment in a two dimensional domain.

math.NA↗

arXiv2020

On the Eigenvalue Distribution for a Beam with Attached Masses

We study a mathematical model of a hinged flexible beam with piezoelectric actuators and electromagnetic shaker in this paper. The shaker is modelled as a mass and spring system attached to the beam. To analyze free vibrations of this mechanical system, we consider the corresponding spectral problem for a fourth-order differential operator with interface conditions that characterize the shaker dynamics. The characteristic equation is studied analytically, and asymptotic estimates of eigenvalues are obtained. The eigenvalue distribution is also illustrated by numerical simulations under a realistic choice of mechanical parameters.

math.OC↗

arXiv2020

Stabilization of Crystallization Models Governed by Hyperbolic Systems

This paper deals with mathematical models of continuous crystallization described by hyperbolic systems of partial differential equations coupled with ordinary and integro-differential equations. The considered systems admit nonzero steady-state solutions with constant inputs. To stabilize these solutions, we present an approach for constructing control Lyapunov functionals based on quadratic forms in weighted L2-spaces. It is shown that the proposed control design scheme guarantees exponential stability of the closed-loop system.

math.OC↗

arXiv2020

Matrix Equations, Sparse Solvers: M-M.E.S.S.-2.0.1 -- Philosophy, Features and Application for (Parametric) Model

Matrix equations are omnipresent in (numerical) linear algebra and systems theory. Especially in model order reduction (MOR) they play a key role in many balancing based reduction methods for linear dynamical systems. When these systems arise from spatial discretizations of evolutionary partial differential equations, their coefficient matrices are typically large and sparse. Moreover, the numbers of inputs and outputs of these systems are typically far smaller than the number of spatial degrees of freedom. Then, in many situations the solutions of the corresponding large-scale matrix equations are observed to have low (numerical) rank. This feature is exploited by M-M.E.S.S. to find successively larger low-rank factorizations approximating the solutions. This contribution describes the basic philosophy behind the implementation and the features of the package, as well as its application in the model order reduction of large-scale linear time-invariant (LTI) systems and parametric LTI systems.

cs.MS↗