A robust error estimator and a residual-free error indicator for reduced basis methods
A robust error estimator and a residual-free error indicator for reduced basis methods
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DOI：
10.1016/j.camwa.2018.11.032
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发表时间：
2017-10
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影响因子：
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通讯作者：
Yanlai Chen;Jiahua Jiang;A. Narayan
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文献类型：
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作者：
Yanlai Chen;Jiahua Jiang;A. Narayan
缩减基方法（RBM）是求解参数化偏微分方程的一种严格的模型缩减方法.它确定了一个低维子空间的近似参数的解决方案是嵌入在高维空间的流形。随后在该子空间中构造降阶模型。RBM依赖于残差为基础的误差指标orposteriorerror界，以指导建设的减少解决方案的子空间，作为一个停止标准，并证明产生的代理解决方案。不幸的是，这是众所周知的，标准算法的剩余范数计算遭受过早停滞的水平上的平方根的机器precision.In本文中，我们开发了两个替代品的标准离线阶段的减少基础算法。首先，我们设计了一个强大的策略计算的残差指标，允许RBM算法丰富的解决方案子空间的精度超过根机精度。其次，我们基于插值理论中的勒贝格函数提出了一种新的误差指标。该误差指标不需要计算残差范数，而是仅需要计算RBM解的能力。这个无残差的指标是严格的，因为它限制了RBM近似所犯的错误，但直到一个不可计算的乘法常数。因此，无残差指示器在离线RBM阶段期间选择快照时是有效的，但目前不能用于证明近似所犯的错误。然而，它规避了需要fora后验分析的数值方法，因此可以有效的问题，这样一个严格的估计是很难得到的。
The Reduced Basis Method (RBM) is a rigorous model reduction approach for solving parameterizedpartial differential equations. It identifies a low-dimensional subspace for approximation of the parametric solution manifold that is embedded in high-dimensional space. A reduced order model is subsequently constructed in this subspace. RBM relies on residual-based error indicators ora posteriorierror bounds to guide construction of the reduced solution subspace, to serve as a stopping criteria, and to certify the resulting surrogate solutions. Unfortunately, it is well-known that the standard algorithm for residual norm computation suffers from premature stagnation at the level of the square root of machine precision.In this paper, we develop two alternatives to the standard offline phase of reduced basis algorithms. First, we design a robust strategy for computation of residual error indicators that allows RBM algorithms to enrich the solution subspace with accuracy beyond root machine precision. Secondly, we propose a new error indicator based on the Lebesgue function in interpolation theory. This error indicator does not require computation of residual norms, and instead only requires the ability to compute the RBM solution. This residual-free indicator is rigorous in that it bounds the error committed by the RBM approximation, but up to an uncomputable multiplicative constant. Because of this, the residual-free indicator is effective in choosing snapshots during the offline RBM phase, but cannot currently be used to certify error that the approximation commits. However, it circumvents the need fora posteriorianalysis of numerical methods, and therefore can be effective on problems where such a rigorous estimate is hard to derive.