"Regularized approximation problems from a computational point of view"Herremans, AstridFor many computational problems in science and engineering, a representation for the solution naturally follows from the given problem and expert knowledge thereof. For example, one may add singular functions to a basis in order to represent a solution with known singularities. However, even though such representations may be linearly independent from an analytic point of view, they are often linearly dependent from a numerical point of view, leading to ill-conditioned approximation problems. This is a known phenomenon in the analysis of subsequences of frames and, indeed, many of these practical approximation sets can be identified as truncated frames. Accurate approximations can still be obtained using regularization, yet a precise analysis of these methods is often lacking. We present recent results regarding the convergence of regularized approximations, along with an efficient algorithm designed to exploit structural properties of the infinite-dimensional problem. |
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