The symbol approximation method: a numerical approach to the approximation of the symbol of self-adjoint operators
Résumé
We propose a simple numerical procedure to approach the symbol of a self-adjoint linear operator by using trace estimates of a corresponding discretization matrix A, with numerical data. The Symbol Approximation Method (SAM) is based on an adaptation of the matrix trace estimator to successive distinct numerical spectral bands in order to build a piece-wise constant function as an approximation of the symbol of . The decomposition of the spectral interval into band of frequencies is proposed with several approaches, from the formal spectral to the multi-grid one. We apply the new method to different operators when discretized in finite differences or in finite elements. The SAM is also proposed as a tool for the modeling of waves equations, and is presented a means to capture an additional linear damping term (hidden operator) in hydrodynamics models such as Korteweig–de Vries or Benjamin Ono equations.