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Register a new userOptimal reduced model algorithms for data-based state estimation
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Added by
James Nichols
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Reduced model spaces, such as reduced basis and polynomial chaos, are linear spaces $V_n$ of finite dimension $n$ which are designed for the efficient approximation of families parametrized PDEs in a Hilbert space $V$. The manifold $mathcal{M}$ that gathers the solutions of the PDE for all admissible parameter values is globally approximated by the space $V_n$ with some controlled accuracy $epsilon_n$, which is typically much smaller than when using standard approximation spaces of the same dimension such as finite elements. Reduced model spaces have also been proposed in [13] as a vehicle to design a simple linear recovery algorithm of the state $uinmathcal{M}$ corresponding to a particular solution when the values of parameters are unknown but a set of data is given by $m$ linear measurements of the state. The measurements are of the form $ell_j(u)$, $j=1,dots,m$, where the $ell_j$ are linear functionals on $V$. The analysis of this approach in [2] shows that the recovery error is bounded by $mu_nepsilon_n$, where $mu_n=mu(V_n,W)$ is the inverse of an inf-sup constant that describe the angle between $V_n$ and the space $W$ spanned by the Riesz representers of $(ell_1,dots,ell_m)$. A reduced model space which is efficient for approximation might thus be ineffective for recovery if $mu_n$ is large or infinite. In this paper, we discuss the existence and construction of an optimal reduced model space for this recovery method, and we extend our search to affine spaces. Our basic observation is that this problem is equivalent to the search of an optimal affine algorithm for the recovery of $mathcal{M}$ in the worst case error sense. This allows us to perform our search by a convex optimization procedure. Numerical tests illustrate that the reduced model spaces constructed from our approach perform better than the classical reduced basis spaces.
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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.
The exploration of complex physical or technological processes usually requires exploiting available information from different sources: (i) physical laws often represented as a family of parameter dependent partial differential equations and (ii) data provided by measurement devices or sensors. The amount of sensors is typically limited and data acquisition may be expensive and in some cases even harmful. This article reviews some recent developments for this small-data scenario where inversion is strongly aggravated by the typically large parametric dimensionality. The proposed concepts may be viewed as exploring alternatives to Bayesian inversion in favor of more deterministic accuracy quantification related to the required computational complexity. We discuss optimality criteria which delineate intrinsic information limits, and highlight the role of reduced models for developing efficient computational strategies. In particular, the need to adapt the reduced models -- not to a specific (possibly noisy) data set but rather to the sensor system -- is a central theme. This, in turn, is facilitated by exploiting geometric perspectives based on proper stable variational formulations of the continuous model.
In this paper, we investigate the randomized algorithms for block matrix multiplication from random sampling perspective. Based on the A-optimal design criterion, the optimal sampling probabilities and sampling block sizes are obtained. To improve the practicability of the block sizes, two modified ones with less computation cost are provided. With respect to the second one, a two step algorithm is also devised. Moreover, the probability error bounds for the proposed algorithms are given. Extensive numerical results show that our methods outperform the existing one in the literature.
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.
We introduce a novel, computationally inexpensive approach for imaging with an active array of sensors, which probe an unknown medium with a pulse and measure the resulting waves. The imaging function uses a data driven estimate of the internal wave originating from the vicinity of the imaging point and propagating to the sensors through the unknown medium. We explain how this estimate can be obtained using a reduced order model (ROM) for the wave propagation. We analyze the imaging function, connect it to the time reversal process and describe how its resolution depends on the aperture of the array, the bandwidth of the probing pulse and the medium through which the waves propagate. We also show how the internal wave can be used for selective focusing of waves at points in the imaging region. This can be implemented experimentally and can be used for pixel scanning imaging. We assess the performance of the imaging methods with numerical simulations and compare them to the conventional reverse-time migration method and the backprojection method introduced recently as an application of the same ROM.
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