Sparse dynamics for partial differential equations
Sparse dynamics for partial differential equations
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DOI：
10.1073/pnas.1302752110
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发表时间：
2013-04-23
影响因子：
11.1
通讯作者：
Osher, Stanley
中科院分区：
文献类型：
--
作者：
Schaeffer, Hayden;Caflisch, Russel;Osher, Stanley
当解被限制在给定基的一个稀疏子集内时，我们研究了几个微分方程的近似动态。通过简单地对基近似的系数应用软阈值处理，在每个时间步都强制实施这种限制。通过减少或压缩在每一步表示解所需的信息，只有基本动态被表示出来。在许多情况下，存在从微分方程导出的自然基，它们促进了稀疏性。我们发现我们的方法成功地简化了具有高频源项的对流方程、扩散方程、弱激波和涡度方程的动态。
We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms.