Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems

Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model. Here, we present an approach for reduced model construction, that preserves the symplectic symmetry of dissipative Hamiltonian systems. The method constructs a closed reduced Hamiltonian system by coupling the full model with a canonical heat bath. This allows the reduced system to be integrated with a symplectic integrator, resulting in a correct dissipation of energy, preservation of the total energy and, ultimately, in the stability of the solution. Accuracy and stability of the method are illustrated through the numerical simulation of the dissipative wave equation and a port-Hamiltonian model of an electric circuit

Structure Preserving Model Reduction of Parametric Hamiltonian Systems

While reduced-order models (ROMs) have been popular for efficiently solving large systems of differential equations, the stability of reduced models over long-time integration is of present challenges. We present a greedy approach for ROM generation of parametric Hamiltonian systems that captures the symplectic structure of Hamiltonian systems to ensure stability of the reduced model. Through the greedy selection of basis vectors, two new vectors are added at each iteration to the linear vector space to increase the accuracy of the reduced basis. We use the error in the Hamiltonian due to model reduction as an error indicator to search the parameter space and identify the next best basis vectors. Under natural assumptions on the set of all solutions of the Hamiltonian system under variation of the parameters, we show that the greedy algorithm converges with exponential rate. Moreover, we demonstrate that combining the greedy basis with the discrete empirical interpolation method also preserves the symplectic structure. This enables the reduction of the computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy, and stability of this model reduction technique is illustrated through simulations of the parametric wave equation and the parametric Schrodinger equation

Geometric Model Order Reduction

During the past decade, model order reduction (MOR) has been successfully applied to reduce the computational complexity of elliptic and parabolic systems of partial differential equations (PDEs). However, MOR of hyperbolic equations remains a challenge. Symmetries and conservation laws, which are a distinctive feature of such systems, are often destroyed by conventional MOR techniques, resulting in a perturbed and often unstable reduced system. The goal of this thesis is to study and develop model order reduction techniques that can preserve nonlinear invariants, symmetries, and conservation laws and to understand the stability properties of these methods compared to conventional techniques. Hamiltonian systems, as systems that are driven by symmetries, are studied intensively from the point of view of MOR. Furthermore, a conservative model reduction of fluid flow is presented. It is illustrated that conserving invariants, conservation laws, and symmetries not only result in a physically meaningful reduced system but also result in an accurate and robust reduced system with enhanced stability

Goal-oriented Uncertainty Quantification for Inverse Problems via Variational Encoder-Decoder Networks

In this work, we describe a new approach that uses variational encoder-decoder (VED) networks for efficient goal-oriented uncertainty quantification for inverse problems. Contrary to standard inverse problems, these approaches are \emph{goal-oriented} in that the goal is to estimate some quantities of interest (QoI) that are functions of the solution of an inverse problem, rather than the solution itself. Moreover, we are interested in computing uncertainty metrics associated with the QoI, thus utilizing a Bayesian approach for inverse problems that incorporates the prediction operator and techniques for exploring the posterior. This may be particularly challenging, especially for nonlinear, possibly unknown, operators and nonstandard prior assumptions. We harness recent advances in machine learning, i.e., VED networks, to describe a data-driven approach to large-scale inverse problems. This enables a real-time goal-oriented uncertainty quantification for the QoI. One of the advantages of our approach is that we avoid the need to solve challenging inversion problems by training a network to approximate the mapping from observations to QoI. Another main benefit is that we enable uncertainty quantification for the QoI by leveraging probability distributions in the latent space. This allows us to efficiently generate QoI samples and circumvent complicated or even unknown forward models and prediction operators. Numerical results from medical tomography reconstruction and nonlinear hydraulic tomography demonstrate the potential and broad applicability of the approach.Comment: 28 pages, 13 figure

Conservative Model Order Reduction for Fluid Flow

In the past decade, model order reduction (MOR) has been successful in reducing the computational complexity of elliptic and parabolic systems of partial differential equations (PDEs). However, MOR of hyperbolic equations remains a challenge. Symmetries and conservation laws, which are a distinctive feature of such systems, are often destroyed by conventional MOR techniques which result in a perturbed, and often unstable reduced system. The importance of conservation of energy is well-known for a correct numerical integration of fluid flow. In this paper, we discuss model reduction, that exploits skew-symmetry of conservative and centered discretization schemes, to recover conservation of energy at the level of the reduced system. Moreover, we argue that the reduced system, constructed with the new method, can be identified by a reduced energy that mimics the energy of the high-fidelity system. Therefore, the loss in energy, associated with the model reduction, remains constant in time. This results in an, overall, correct evolution of the fluid that ensures robustness of the reduced system. We evaluate the performance of the proposed method through numerical simulation of various fluid flows, and through a numerical simulation of a continuous variable resonance combustor model

Symplectic Model-Reduction with a Weighted Inner Product

In the recent years, considerable attention has been paid to preserving structures and invariants in reduced basis methods, in order to enhance the stability and robustness of the reduced system. In the context of Hamiltonian systems, symplectic model reduction seeks to construct a reduced system that preserves the symplectic symmetry of Hamiltonian systems. However, symplectic methods are based on the standard Euclidean inner products and are not suitable for problems equipped with a more general inner product. In this paper we generalize symplectic model reduction to allow for the norms and inner products that are most appropriate to the problem while preserving the symplectic symmetry of the Hamiltonian systems. To construct a reduced basis and accelerate the evaluation of nonlinear terms, a greedy generation of a symplectic basis is proposed. Furthermore, it is shown that the greedy approach yields a norm bounded reduced basis. The accuracy and the stability of this model reduction technique is illustrated through the development of reduced models for a vibrating elastic beam and the sine-Gordon equation

Learning regularization parameters of inverse problems via deep neural networks:Paper

In this work, we describe a new approach that uses deep neural networks (DNN) to obtain regularization parameters for solving inverse problems. We consider a supervised learning approach, where a network is trained to approximate the mapping from observation data to regularization parameters. Once the network is trained, regularization parameters for newly obtained data can be computed by efficient forward propagation of the DNN. We show that a wide variety of regularization functionals, forward models, and noise models may be considered. The network-obtained regularization parameters can be computed more efficiently and may even lead to more accurate solutions compared to existing regularization parameter selection methods. We emphasize that the key advantage of using DNNs for learning regularization parameters, compared to previous works on learning via optimal experimental design or empirical Bayes risk minimization, is greater generalizability. That is, rather than computing one set of parameters that is optimal with respect to one particular design objective, DNN-computed regularization parameters are tailored to the specific features or properties of the newly observed data. Thus, our approach may better handle cases where the observation is not a close representation of the training set. Furthermore, we avoid the need for expensive and challenging bilevel optimization methods as utilized in other existing training approaches. Numerical results demonstrate the potential of using DNNs to learn regularization parameters.Comment: 27 pages, 16 figure

Uncertainty Quantification of Inclusion Boundaries in the Context of X-Ray Tomography

In this work, we describe a Bayesian framework for the X-ray computed tomography (CT) problem in an infinite-dimensional setting. We consider reconstructing piecewise smooth fields with discontinuities where the interface between regions is not known. Furthermore, we quantify the uncertainty in the prediction. Directly detecting the discontinuities, instead of reconstructing the entire image, drastically reduces the dimension of the problem. Therefore, the posterior distribution can be approximated with a relatively small number of samples. We show that our method provides an excellent platform for challenging X-ray CT scenarios (e.g. in case of noisy data, limited angle, or sparse angle imaging). We investigate the accuracy and the efficiency of our method on synthetic data. Furthermore, we apply the method to the real-world data, tomographic X-ray data of a lotus root filled with attenuating objects. The numerical results indicate that our method provides an accurate method in detecting boundaries between piecewise smooth regions and quantifies the uncertainty in the prediction, in the context of X-ray CT