35 research outputs found
Symplectic Model Order Reduction in RBmatlab
No full textAn add-on to the open-source library RBmatlab (v1.16.09) which includes symplectic model reduction techniques that are discussed in the preprint
Buchfink, P. & Bhatt, A. & Haasdonk, B.: Symplectic Model Order Reduction with Non-Orthonormal Bases (arXiv:1902.10523)
In the preprint MATLAB revision 2017b was used.</p
Symplectic Model Order Reduction in RBmatlab
No full text<p>An add-on to the open-source library <a href="https://www.morepas.org/software/rbmatlab/">RBmatlab</a> (v1.16.09) which includes symplectic model reduction techniques that are discussed in the preprint</p>
<p>Buchfink, P. & Bhatt, A. & Haasdonk, B.: Symplectic Model Order Reduction with Non-Orthonormal Bases (<a href="https://arxiv.org/abs/1902.10523">arXiv:1902.10523</a>)</p>
<p>In the preprint MATLAB revision 2017b was used.</p>The new version is more performant since it computes the different decompositions (for the reduced-order basis construction) only once and reuses them
Structure-preserving model reduction on subspaces and manifolds
No full textMathematical models are a key enabler to understand complex processes across all branches of research and development since such models allow us to simulate the behavior of the process without physically realizing it. However, detailed models are computationally demanding and, thus, are frequently prohibited from being evaluated (a) multiple times for different parameters, (b) in real time or (c) on hardware with low computational power. The field of model (order) reduction (MOR) aims to approximate such detailed models with more efficient surrogate models that are suitable for the tasks (a-c). In classical MOR, the solutions of the detailed model are approximated in a problem-specific, low-dimensional subspace, which is why we refer to it as MOR on subspaces. The subspace is characterized by a reduced basis that can be computed from given data with a so-called basis generation technique.
The two key aspects in this thesis are: (i) structure-preserving MOR techniques and (ii) MOR on manifolds. Preserving given structures throughout the reduction is important to obtain physically consistent reduced models. We demonstrate this for Lagrangian and Hamiltonian systems, which are dynamical systems that guarantee preservation of energy over time. MOR on manifolds, on the other hand, broadens the applicability of MOR to problems that cannot be treated efficiently with MOR on subspaces
Experimental Comparison of Symplectic and Non-symplectic Model Order Reduction on an Uncertainty Quantification Problem
No full textSoftwarepackage CCMOR2
No full text The dataset entails the code of the CCMOR2 package developed in Matlab.
This project aims to model and perform certified model order reduction on multi-physical systems. The systems are formulated in the port-Hamiltonian framework which incorporates useful system theoretic properties such as stability and passivity.
The approach is subvided into three parts:
Modeling of the multi-physical system in the port-Hamiltonian framework
Reduce the high-dimensional system in a structure-preserving manner
Certify the reduction by performing a-posteriori error analysis
We refer to the README for further information on how to use this software.</p
Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds and Approximation with Weakly Symplectic Autoencoder
No full textClassical model reduction techniques project the governing equations onto linear subspaces of the high-dimensional state-space. For problems with slowly decaying Kolmogorov-nwidths such as certain transport-dominated problems, however, classical linear-subspace reduced-order models (ROMs) of low dimension might yield inaccurate results. Thus, the concept of classical linear-subspace ROMs has to be extended to more general concepts, like model order teduction (MOR) on manifolds. Moreover, as we are dealing with Hamiltonian systems, it is crucial that the underlying symplectic structure is preserved in the reduced model, as otherwise it could become unphysical in the sense that the energy is not conserved or stability properties are lost. To the best of our knowledge, existing literature addresses either MOR on manifolds or symplectic model reduction for Hamiltonian systems, but not their combination. In this work, we bridge the two aforementioned approaches by providing a novel projection technique called symplectic manifold Galerkin (SMG), which projects the Hamiltonian system onto a nonlinear symplectic trial manifold such that the reduced model is again a Hamiltonian system. We derive analytical results such as stability, energy-preservation, and a rigorous a posteriori error bound. Moreover, we construct a weakly symplectic deep convolutional autoencoder as a computationally practical approach to approximate a nonlinear symplectic trial manifold. Finally, we numerically demonstrate the ability of the method to achieve higher accuracy than (non-)structure-preserving linear-subspace ROMs and non-structure-preserving MOR on manifold techniques.</p
Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds
No full textFor projection-based linear-subspace model order reduction (MOR), it is well known that the Kolmogorov n-width describes the best-possible error for a reduced order model (ROM) of size n. In this paper, we provide approximation bounds for ROMs on polynomially mapped manifolds. Inparticular, we showt hat the approximation bounds depend on the polynomial degree p of the mapping function as well as on the linear Kolmogorov n-width for the underlying problem. This results in a Kolmogorov (n,p)-width, which describes a lower bound for the best-possible error for a ROM on polynomially mapped manifolds of polynomial degree p and reduced size n
Approximation Bounds for Model Reduction on Polynomially Mapped Manifolds
No full textFor projection-based linear-subspace model order reduction (MOR), it is well
known that the Kolmogorov n-width describes the best-possible error for a
reduced order model (ROM) of size n. In this paper, we provide approximation
bounds for ROMs on polynomially mapped manifolds. In particular, we show that
the approximation bounds depend on the polynomial degree p of the mapping
function as well as on the linear Kolmogorov n-width for the underlying
problem. This results in a Kolmogorov (n, p)-width, which describes a lower
bound for the best-possible error for a ROM on polynomially mapped manifolds of
polynomial degree p and reduced size n.Comment: 11 pages, 1 figur
Symplectic model order reduction with non-orthonormal bases
No full textParametric high-fidelity simulations are of interest for a wide range of applications. But the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g. structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such a ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction
