Abstract
SummaryThis paper describes a novel method for mapping between basis representation of a field variable over a domain in the context of numerical modelling and inverse problems. In the numerical solution of inverse problems, a continuous scalar or vector field over a domain may be represented in different finite‐dimensional basis approximations, such as an unstructured mesh basis for the numerical solution of the forward problem, and a regular grid basis for the representation of the solution of the inverse problem. Mapping between the basis representations is generally lossy, and the objective of the mapping procedure is to minimise the errors incurred. We present in this paper a novel mapping mechanism that is based on a minimisation of the L2 or H1 norm of the difference between the two basis representations. We provide examples of mapping in 2D and 3D problems, between an unstructured mesh basis representative of an FEM approximation, and different types of structured basis including piecewise constant and linear pixel basis, and blob basis as a representation of the inverse basis. A comparison with results from a simple sampling‐based mapping algorithm shows the superior performance of the method proposed here. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.
Highlights
Inverse problems arise in almost all areas of science whenever the parameters of a model need to be inferred from observed data
Row 3 shows the result of mapping the basis coefficients for each of the basis expansions back to the source basis representation (V ! W ), and the bottom row shows the differences between the mapped images in row 3 and the source image
We have presented in this paper a novel method for mapping a field variable x.r/, defined in a bounded domain, from one finite-dimensional basis representation xU .r/ to a different representation xV .r/
Summary
Inverse problems arise in almost all areas of science whenever the parameters of a model need to be inferred from observed data. Evaluation of f requires a numerical solution strategy that discretises the continuous into a finite-dimensional problem and represents the parameters x.r/ and field variables by a finite-dimensional basis expansion U. The choice of basis for the forward problem is governed by considerations of computational stability and accuracy of the numerical model This is often not an optimal choice for the inverse problem, where it may be more appropriate to adapt the basis expansion to physical limits of resolution of the reconstruction and to enforce smoothness constraints in the admissible solutions. [4] we have presented a simple mapping method that relied on U being of higher resolution than V such that basis functions of V can be adequately represented by an expansion in U, reducing the mapping problem to operations in U that can be performed with standard FEM tools. The integrals are evaluated with a quadrature rule that is either exact, where V is piecewise polynomial, or approximate in the general case
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