论文标题
高斯连贯状态对高频Helmholtz解决方案有效近似
Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states
论文作者
论文摘要
我们介绍了专门设计的新有限维空间，以近似于高频Helmholtz问题的解决方案，并在尺寸$ d $中具有光滑可变系数。这些离散空间由高斯连贯状态跨越，这些状态具有要位于相空间中的关键特性。我们通过利用溶液的（已知）微定量特性来仔细选择跨越近似空间的高斯相干状态。对于大量的源术语（包括平面波散射问题），此选择会导致离散的空间，这些空间为所有WaveNumber $ k $提供了均匀的近似错误，并具有许多自由度缩放为$ k^{d-1/2} $，我们严格地建立了这些误差。相比之下，对于基于（分段）多项式的离散空间，自由度的数量至少必须扩展为$ k^d $才能实现相同的属性。这些理论结果由一维数值示例说明，其中提议的离散空间与最小二乘的变化公式结合在一起。
We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension $d$. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber $k$ with a number of degrees of freedom scaling as $k^{d-1/2}$, which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as $k^d$ to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation.
