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首页> 外文期刊>SIAM Journal on Scientific Computing >A DIMENSIONAL REDUCTION APPROACH BASED ON THE APPLICATION OF REDUCED BASIS METHODS IN THE FRAMEWORK OF HIERARCHICAL MODEL REDUCTION
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A DIMENSIONAL REDUCTION APPROACH BASED ON THE APPLICATION OF REDUCED BASIS METHODS IN THE FRAMEWORK OF HIERARCHICAL MODEL REDUCTION

机译：基于降基方法的分层降阶方法在层次模型降阶框架中的应用

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摘要

In this article we introduce a new dimensional reduction approach which is based on the application of reduced basis (RB) techniques in the hierarchical model reduction (HMR) framework. Considering problems that exhibit a dominant spatial direction, the idea of HMR is to perform a Galerkin projection onto a reduced space, which combines the full solution space in the dominant direction with a reduction space in the transverse direction. The latter is spanned by modal orthonormal basis functions. While so far the basis functions in the HMR approach have been chosen a priori [S. Perotto, A. Ern, and A. Veneziani, Multiscale Model. Simul., 8 (2010), pp. 1102-1127], for instance, as Legendre or trigonometric polynomials, in this work a highly nonlinear approximation is employed for the construction of the reduction space. To this end we first derive a lower dimensional parametrized problem in the transverse direction from the full problem where the parameters reflect the influence from the unknown solution in the dominant direction. Exploiting the good approximation properties of RB methods, we then construct a reduction space by applying a proper orthogonal decomposition to a set of snapshots of the parametrized partial differential equation. For an efficient construction of the snapshot set we apply adaptive refinement in parameter space based on an a posteriori error estimate that is also derived in this article. We introduce our method for general elliptic problems such as advection-diffusion equations in two space dimensions. Numerical experiments demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full dimensional problem and the computational efficiency of our new adaptive approach.
机译：在本文中，我们介绍了一种新的降维方法，该方法基于在基础模型简化（HMR）框架中应用了缩减基础（RB）技术。考虑到表现出主要空间方向的问题，HMR的想法是在缩小的空间上执行Galerkin投影，该空间将主要方向上的完整解空间与横向上的缩小空间结合在一起。后者由模态正交基函数跨越。到目前为止，HMR方法的基本功能已被优先选择。 Perotto，A。Ern和A.Venezanii，多尺度模型。 Simul。，8（2010），pp。1102-1127]，例如勒让德（Legendre）或三角多项式，在这项工作中，高度非线性近似被用于构造还原空间。为此，我们首先从整个问题中得出一个横向的低维参数化问题，其中参数反映了主导方向上未知解的影响。利用RB方法的良好近似特性，然后通过对参数化偏微分方程的一组快照应用适当的正交分解，构造出约简空间。为了有效地构建快照集，我们基于后验误差估计在参数空间中应用自适应细化，该估计也从本文中得出。我们介绍了解决一般椭圆问题的方法，例如二维空间中的对流扩散方程。数值实验表明，所提出的降维近似值可以快速收敛到全尺寸问题的解决方案，并且可以提高我们新的自适应方法的计算效率。

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