Multidimensional NMR Inversion without Kronecker Products: Multilinear Inversion

摘要

Multidimensional NMR inversion using Kronecker products poses severalchallenges. First, kernel compression is only possible when the kernel matricesare separable, and in recent years, there has been an increasing interest inNMR sequences with non-separable kernels. Second, in three or more dimensions,the singular value decomposition is not unique; therefore kernel compression isnot well-defined for higher dimensions. Without kernel compression, theKronecker product yields matrices that require large amounts of memory, makingthe inversion intractable for personal computers. Finally, incorporatingarbitrary regularization terms is not possible using the Lawson-Hanson (LH) orthe Butler-Reeds-Dawson (BRD) algorithms. We develop a minimization-basedinversion method that circumvents the above problems by using multilinear formsto perform multidimensional NMR inversion without using kernel compression orKronecker products. The new method is memory efficient, requiring less than0.1% of the memory required by the LH or BRD methods. It can also be extendedto arbitrary dimensions and adapted to include non-separable kernels, linearconstraints, and arbitrary regularization terms. Additionally, it is easy toimplement because only a cost function and its first derivative are required toperform the inversion.