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Why discretize?

Inverse problems are common across the geosciences: imaging in geophysics, history matching, parameter estimation, and many of these require constrained optimization using partial differential equations (PDEs) where the derivative of mesh variables are sought. Finite difference, finite element and finite volume techniques allow subdivision of continuous differential equations into discrete domains. The knowledge and appropriate application of these methods is fundamental to simulating physical processes. Many inverse problems in the geosciences are solved using stochastic techniques or external finite difference based tools (e.g. PEST); these are robust to local minima and the programmatic implementation, respectively, however these methods do not scale to millions of parameters to be estimated. This sort of scale is necessary for solving many of the inverse problems in geophysics and increasingly hydrogeology (e.g. electromagnetics, gravity, and fluid flow problems).

In the context of the inverse problem, when the physical properties, the domain, and the boundary conditions are not necessarily known, the simplicity and efficiency in mesh generation are important criteria. Complex mesh geometries, such as body fitted grids, commonly used when the domain is explicitly given, are less appropriate. Additionally, when considering the inverse problem, it is important that operators and their derivatives are accessible to interrogation and extension. The goal of this work is to provide a high level background to finite volume techniques abstracted across four mesh types:

  1. tensor product mesh :class:`discretize.TensorMesh`
  2. cylindrically symmetric mesh :class:`discretize.CylMesh`
  3. curvilinear mesh :class:`discretize.CurvilinearMesh`
  4. octree and quadtree meshes :class:`discretize.TreeMesh`

discretize contributes an overview of finite volume techniques in the context of geoscience inverse problems, which are treated in a consistent way across various mesh types, highlighting similarities and differences.