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A general framework for numerically stable reconstructions in Hilbert spaces

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Many computational problems require the reconstruction of a function-a signal or image, for example-from a collection of its samples. In abstract terms, we seek to recover an element of a Hilbert space in a particular basis (or frame), given its measurements (inner products) with respect to another, fixed basis. Whilst this problem has been studied extensively in the last several decades, existing methods are prone to be numerically unstable, and may require rather restrictive conditions on the type of element to be reconstructed in order to ensure convergence.

The purpose of this talk is to describe a new approach for this problem. It transpires that the abstract reconstruction problem has a straightforward and well-posed infinite-dimensional formulation. Using certain operator-theoretic considerations, we can derive a finite-dimensional analogue, suitable for computations, that retains such structure. This yields a convergent numerical method that is both numerically stable and, as it turns out, near-optimal. Moreover, techniques from compressed sensing can also be incorporated to accurately recover sparse signals whilst significantly undersampling.

One example of this framework, with application to spectral methods for hyperbolic PDEs, is the accurate recovery of a piecewise smooth function from its (discrete or continuous) Fourier or orthogonal polynomial coefficients. We shall describe this example in detail, and discuss a number of advantages over more common techniques.

This is joint work with Anders Hansen (Cambridge)

This talk is part of the Numerical Analysis series.

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