  Multidimensional NMR inversion using Kronecker products poses several
challenges. First, kernel compression is only possible when the kernel matrices
are separable, and in recent years, there has been an increasing interest in
NMR sequences with non-separable kernels. Second, in three or more dimensions,
the singular value decomposition is not unique; therefore kernel compression is
not well-defined for higher dimensions. Without kernel compression, the
Kronecker product yields matrices that require large amounts of memory, making
the inversion intractable for personal computers. Finally, incorporating
arbitrary regularization terms is not possible using the Lawson-Hanson (LH) or
the Butler-Reeds-Dawson (BRD) algorithms. We develop a minimization-based
inversion method that circumvents the above problems by using multilinear forms
to perform multidimensional NMR inversion without using kernel compression or
Kronecker products. The new method is memory efficient, requiring less than
0.1% of the memory required by the LH or BRD methods. It can also be extended
to arbitrary dimensions and adapted to include non-separable kernels, linear
constraints, and arbitrary regularization terms. Additionally, it is easy to
implement because only a cost function and its first derivative are required to
perform the inversion.
