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Register a new userL1-based reduced over collocation and hyper reduction for steady state and time-dependent nonlinear equations
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Added by
Yanlai Chen
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The task of repeatedly solving parametrized partial differential equations (pPDEs) in, e.g. optimization or interactive applications, makes it imperative to design highly efficient and equally accurate surrogate models. The reduced basis method (RBM) presents as such an option. Enabled by a mathematically rigorous error estimator, RBM constructs a low-dimensional subspace of the parameter-induced high fidelity solution manifold from which an approximate solution is computed. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However, this decomposition, usually through the empirical interpolation method (EIM) when the PDE is nonlinear or its parameter dependence nonaffine, is either challenging to implement, or severely degrades online efficiency. In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear pPDEs on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding the efficiency degradation inherent to a traditional application of EIM. Two critical ingredients of the scheme are collocation at about twice as many locations as the dimension of the reduced solution space, and an efficient L1-norm-based error indicator for the strategic selection of the parameter values to build the reduced solution space. Together, these two ingredients render the proposed L1-ROC scheme both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in alternative RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of L1-ROC and its superior stability performance.
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The onerous task of repeatedly resolving certain parametrized partial differential equations (pPDEs) in, e.g. the optimization context, makes it imperative to design vastly more efficient numerical solvers without sacrificing any accuracy. The reduced basis method (RBM) presents itself as such an option. With a mathematically rigorous error estimator, RBM seeks a surrogate solution in a carefully-built subspace of the parameter-induced high fidelity solution manifold. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However, this decomposition, usually through the empirical interpolation method (EIM) when the PDE is nonlinear or its parameter dependence nonaffine, is either challenging to implement, or severely degrading to the online efficiency. In this paper, we augment and extend the EIM approach in the context of solving pPDEs in two different ways, resulting in the Reduced Over-Collocation methods (ROC). These are stable and capable of avoiding the efficiency degradation inherent to a direct application of EIM. There are two ingredients of these methods. First is a strategy to collocate at about twice as many locations as the number of bases for the surrogate space. The second is an efficient approach for the strategic selection of the parameter values to build the reduced solution space for which we study two choices, a recent empirical L1 approach and a new indicator based on the reduced residual. Together, these two ingredients render the schemes, L1-ROC and R2-ROC, online efficient and immune from the efficiency degradation of EIM for nonlinear and nonaffine problems offline and online. Numerical tests on three different families of nonlinear problems demonstrate the high efficiency and accuracy of these new algorithms and their superior stability performance.
The need for multiple interactive, real-time simulations using different parameter values has driven the design of fast numerical algorithms with certifiable accuracies. The reduced basis method (RBM) presents itself as such an option. RBM features a mathematically rigorous error estimator which drives the construction of a low-dimensional subspace. A surrogate solution is then sought in this low-dimensional space approximating the parameter-induced high fidelity solution manifold. However when the system is nonlinear or its parameter dependence nonaffine, this efficiency gain degrades tremendously, an inherent drawback of the application of the empirical interpolation method (EIM). In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear partial differential equations on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding the efficiency degradation. Two critical ingredients of the scheme are collocation at about twice as many locations as the number of basis elements for the reduced approximation space, and an efficient error indicator for the strategic building of the reduced solution space. The latter, the main contribution of this paper, results from an adaptive hyper reduction of the residuals for the reduced solution. Together, these two ingredients render the proposed R2-ROC scheme both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in traditional RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of our R2-ROC and its superior stability performance.
State estimation aims at approximately reconstructing the solution $u$ to a parametrized partial differential equation from $m$ linear measurements, when the parameter vector $y$ is unknown. Fast numerical recovery methods have been proposed based on reduced models which are linear spaces of moderate dimension $n$ which are tailored to approximate the solution manifold $mathcal{M}$ where the solution sits. These methods can be viewed as deterministic counterparts to Bayesian estimation approaches, and are proved to be optimal when the prior is expressed by approximability of the solution with respect to the reduced model. However, they are inherently limited by their linear nature, which bounds from below their best possible performance by the Kolmogorov width $d_m(mathcal{M})$ of the solution manifold. In this paper we propose to break this barrier by using simple nonlinear reduced models that consist of a finite union of linear spaces $V_k$, each having dimension at most $m$ and leading to different estimators $u_k^*$. A model selection mechanism based on minimizing the PDE residual over the parameter space is used to select from this collection the final estimator $u^*$. Our analysis shows that $u^*$ meets optimal recovery benchmarks that are inherent to the solution manifold and not tied to its Kolmogorov width. The residual minimization procedure is computationally simple in the relevant case of affine parameter dependence in the PDE. In addition, it results in an estimator $y^*$ for the unknown parameter vector. In this setting, we also discuss an alternating minimization (coordinate descent) algorithm for joint state and parameter estimation, that potentially improves the quality of both estimators.
In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and ghost point diffusion maps (GPDM), to solve the time-dependent advection-diffusion PDE on unknown smooth manifolds without and with boundaries. The core idea is to directly approximate the spatial components of the differential operator on the manifold with a local integral operator and combine it with the standard implicit time difference scheme. When the manifold has a boundary, a simplified version of the GPDM approach is used to overcome the bias of the integral approximation near the boundary. The Monte-Carlo discretization of the integral operator over the point cloud data gives rise to a mesh-free formulation that is natural for randomly distributed points, even when the manifold is embedded in high-dimensional ambient space. Here, we establish the convergence of the proposed solver on appropriate topologies, depending on the distribution of point cloud data and boundary type. We provide numerical results to validate the convergence results on various examples that involve simple geometry and an unknown manifold. Additionally, we also found positive results in solving the one-dimensional viscous Burgers equation where GPDM is adopted with a pseudo-spectral Galerkin framework to approximate nonlinear advection term.
In this paper, we propose a coupled Discrete Empirical Interpolation Method (DEIM) and Generalized Multiscale Finite element method (GMsFEM) to solve nonlinear parabolic equations with application to the Allen-Cahn equation. The Allen-Cahn equation is a model for nonlinear reaction-diffusion process. It is often used to model interface motion in time, e.g. phase separation in alloys. The GMsFEM allows solving multiscale problems at a reduced computational cost by constructing a reduced-order representation of the solution on a coarse grid. In arXiv:1301.2866, it was shown that the GMsFEM provides a flexible tool to solve multiscale problems by constructing appropriate snapshot, offline and online spaces. In this paper, we solve a time dependent problem, where online enrichment is used. The main contribution is comparing different online enrichment methods. More specifically, we compare uniform online enrichment and adaptive methods. We also compare two kinds of adaptive methods. Furthermore, we use DEIM, a dimension reduction method to reduce the complexity when we evaluate the nonlinear terms. Our results show that DEIM can approximate the nonlinear term without significantly increasing the error. Finally, we apply our proposed method to the Allen Cahn equation.
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