{
  "id": "1212.4132",
  "version": "v1",
  "published": "2012-12-17T20:41:35.000Z",
  "updated": "2012-12-17T20:41:35.000Z",
  "title": "Sparse Dynamics for Partial Differential Equations",
  "authors": [
    "Hayden Schaeffer",
    "Stanley Osher",
    "Russel Caflisch",
    "Cory Hauck"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-17T20:41:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partial differential equations",
      "sparse dynamics",
      "high frequency source terms",
      "basis approximation",
      "essential dynamics"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1073/pnas.1302752110",
      "journal": "Proceedings of the National Academy of Science",
      "year": 2013,
      "month": "Apr",
      "volume": 110,
      "number": 17,
      "pages": 6634
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013PNAS..110.6634S"
    }
  }
}