Abstract:
This will be an informal talk where the content can be adapted to the interests of the audience. Below is a selection of topics.
Developing new methods for nano-holotomography offers a broad range of interesting mathematical challenges, of which I would like to share a few pitfalls and advances. The standard forward operator is the so-called Fresnel transform, a convolution operator with a highly oscillatory kernel that does not decay at infinity. What sort of discrepancies arise when implementing such an object on a finite dimensional grid? Is the discrete operator at all a good surrogate for the real-world counterpart? I will show some perhaps unexpected behaviours and also explain them mathematically.
In near-field imaging with a conic beam, a key classical result is the Fresnel scaling theorem, which states that an object in a conic beam will look, on a detector, exactly as the object would look in a parallel beam, albeith with a different propagation distance and different scaling. How does this theorem look if multiple objects are placed in the beam? How does it change if an object is placed before the focal spot?
Finally, maybe of more general interest, most applied mathematical problems nowadays are solved with iterative solvers, such as Gradient or Conjugate Gradient Descent. A key obstacle for many users is parameter-choises, such as step-length, where there is little concrete guidiance in the litterature and poor or ad hoc choices lead to suboptimal performance. I will discuss new, generally applicable schemes for optimal parameter choices, which has emerged recently in collaboration with ANL, and show some promising results.
Bio: Marcus Carlsson received his PhD in Mathematics from Lund University in 2007 and has since then worked at Purdue and in Chile. Since 2016 he is an associate professor at Lund University. While originally working in complex analysis and operator theory, he got interested in applied problems during his years at Purdue, working with seismic imaging in close collaboration with industry. He has a very broad interest in mathematics, but his core contributions to science are in the fields of multidimensional frequency estimation and compressive sensing, with a focus on exploring applications in various imaging setups. He is currently working with V. Nikitin and D. Gursoy, among others, on developing new methods for nano-holotomography.
Location:
In person: 401/A1100
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