Dynamical low-rank approximation for radiation transport with random input
Final Report Abstract
Computer simulations are of significant importance for problems in radiation transport ranging from nuclear engineering to cancer therapy. However, conventional simulation techniques exhibit a large memory footprint and prohibitive computational costs. This stems mainly from the large number of variables the solution depends on which are time, energy, spatial position, direction of travel and uncertainties. In this work we have tackled these challenged by employing dynamical low-rank approximation (DLRA) which is a computational method that recently gained a remarkable amount of attention due to its reduced memory and computational costs required to perform accurate numerical simulations. The objectives of this project were to 1) construct novel DLRA methods for radiation transport and especially radiation treatment planning, 2) explore the stability of the proposed methods, and 3) investigate how developed methods can be used on high performance computing architectures. A notable finding of this project is that DLRA is especially well suited for proton radiation therapy simulations as the dynamics of proton radiation can be captured very accurately at significantly reduced computational costs and memory requirements. Moreover, the reduced memory footprint makes DLRA very well suited for GPU architectures which are common in clinical usage. The applicability of DLRA in proton therapy simulations has been made possible by several achievements in understanding the mathematical properties of DLRA which has led to novel discretizations which are provably stable and capture essential properties of the problem. Furthermore, this mathematical understanding has enabled us to construct a new time integrator (that is, a numerical method which evolves the solution in time) which enhances the efficiency of DLRA while extending the parallelism of the method. This new integrator can impact a large number of different applications and improves their efficiency for high performance computing systems.
Publications
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“Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equations”. In: Advances in Neural Information Processing Systems 35 (2022), pp. 20051–20063.
Steffen Schotthöfer et al.
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A Parallel Rank-Adaptive Integrator for Dynamical Low-Rank Approximation. SIAM Journal on Scientific Computing, 46(3), B205-B228.
Ceruti, Gianluca; Kusch, Jonas & Lubich, Christian
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Asymptotic-Preserving and Energy Stable Dynamical Low-Rank Approximation. SIAM Journal on Numerical Analysis, 62(1), 73-92.
Einkemmer, Lukas; Hu, Jingwei & Kusch, Jonas
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Energy Stable and Conservative Dynamical Low-Rank Approximation for the Su–Olson Problem. SIAM Journal on Scientific Computing, 46(2), B137-B158.
Baumann, Lena; Einkemmer, Lukas; Klingenberg, Christian & Kusch, Jonas
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Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: a study using POD-Galerkin and dynamical low-rank approximation. Advances in Computational Mathematics, 50(4).
Koellermeier, Julian; Krah, Philipp & Kusch, Jonas
