Globally Constructed Adaptive Local Basis Set for Spectral Projectors of Second Order Differential Operators
Globally Constructed Adaptive Local Basis Set for Spectral Projectors of Second Order Differential Operators
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DOI：
10.1137/17m1140236
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发表时间：
2017-07
期刊：
影响因子：
--
通讯作者：
Yingzhou Li;Lin Lin-Lin
中科院分区：
文献类型：
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作者：
Yingzhou Li;Lin Lin-Lin
二阶微分算子的谱投影在量子物理和其他科学工程应用中起着重要的作用。为了解决局部特征并获得收敛的结果，通常所需的自由度数量远远大于光谱投影仪的秩。这将导致计算和存储方面的巨大成本。本文提出了一种构造对给定微分算子自适应的基集的方法。对基集进行系统改进，并将投影机的局部特征嵌入基集。因此，所需的自由度的数量只是一个小常数倍的投影仪的秩。基集的构造采用随机化过程，只需要对全局域上的少量向量应用微分算子，而每个基函数本身在严格局部域上得到支持，并且在全局域上不连续。利用不连续伽辽金（DG）方法从这样的基集系统地逼近了全局域上的光谱投影。全局构造过程非常灵活，即使运算符包含非局部潜在项，也可以一致地构造局部基集。我们利用具有局部势的一维、二维和三维线性问题，以及类似于量子物理中Hartree-Fock问题的具有非局部势的一维非线性问题，验证了全局构造自适应局部基集的有效性。
Spectral projectors of second order differential operators play an important role in quantum physics and other scientific and engineering applications. In order to resolve local features and to obtain converged results, typically the number of degrees of freedom needed is much larger than the rank of the spectral projector. This leads to significant cost in terms of both computation and storage. In this paper, we develop a method to construct a basis set that is adaptive to the given differential operator. The basis set is systematically improvable, and the local features of the projector is built into the basis set. As a result the required number of degrees of freedom is only a small constant times the rank of the projector. The construction of the basis set uses a randomized procedure, and only requires applying the differential operator to a small number of vectors on the global domain, while each basis function itself is supported on strictly local domains and is discontinuous across the global domain. The spectral projector on the global domain is systematically approximated from such a basis set using the discontinuous Galerkin (DG) method. The global construction procedure is very flexible, and allows a local basis set to be consistently constructed even if the operator contains a nonlocal potential term. We verify the effectiveness of the globally constructed adaptive local basis set using one-, two- and three-dimensional linear problems with local potentials, as well as a one dimensional nonlinear problem with nonlocal potentials resembling the Hartree-Fock problem in quantum physics.