Abstract
The mathematics of inverse problems has relevance across numerous estimation problems in science and engineering. L1 regularization has attracted recent attention in reconstructing the system properties in the case of sparse inverse problems; i.e., when the true property sought is not adequately described by a continuous distribution, in particular in Compressed Sensing image reconstruction. In this work, we focus on the application of L1 regularization to a class of inverse problems; relaxation-relaxation, T1–T2, and diffusion-relaxation, D–T2, correlation experiments in NMR, which have found widespread applications in a number of areas including probing surface interactions in catalysis and characterizing fluid composition and pore structures in rocks. We introduce a robust algorithm for solving the L1 regularization problem and provide a guide to implementing it, including the choice of the amount of regularization used and the assignment of error estimates. We then show experimentally that L1 regularization has significant advantages over both the Non-Negative Least Squares (NNLS) algorithm and Tikhonov regularization. It is shown that the L1 regularization algorithm stably recovers a distribution at a signal to noise ratio<20 and that it resolves relaxation time constants and diffusion coefficients differing by as little as 10%. The enhanced resolving capability is used to measure the inter and intra particle concentrations of a mixture of hexane and dodecane present within porous silica beads immersed within a bulk liquid phase; neither NNLS nor Tikhonov regularization are able to provide this resolution. This experimental study shows that the approach enables discrimination between different chemical species when direct spectroscopic discrimination is impossible, and hence measurement of chemical composition within porous media, such as catalysts or rocks, is possible while still being stable to high levels of noise.
Highlights
In many applications, the properties of the system to be measured, referred to as the ‘distribution’, are distorted by a physical process or the measuring instrument itself
The L1 regularization method is used to process 2D Nuclear Magnetic Resonance (NMR) correlation experiments of mixtures of hexane and dodecane in different physical environments where the Non-Negative Least Squares (NNLS) and Tikhonov regularization methods fail to distinguish between the different components
The L1 regularization method was applied to a class of ill–conditioned inverse problems: spin–lattice relaxation – spin– spin relaxation, T1–T2, and diffusion – spin–spin relaxation, D–T2, correlation experiments in NMR
Summary
The properties of the system to be measured, referred to as the ‘distribution’, are distorted by a physical process or the measuring instrument itself. Unlike forward problems, inverse problems are typically ill– conditioned This means that small differences in the acquired signal from an experiment, caused by the random nature of noise or experimental error, can lead to significantly different reconstructed distributions. In many inverse problems, the signal and the distribution are related by transforms which are harder to invert, such as exponential decay functions in the case of 2D NMR correlation experiments In this manuscript, an algorithm is introduced which is robust in reconstructing distributions from ill-conditioned, L1 regularized inverse problems. The similarity of relaxation time constants and diffusion coefficients of hexane and dodecane make it impossible to resolve these two components using the NNLS algorithm or Tikhonov regularization in 2D NMR correlation experiments. Resolving the individual diffusion coefficients and relaxation time constants in both the bulk liquid phase and within the porous medium, as well as quantifying the relative amount of each component in both phases, within the same experiment, is invaluable in understanding the behaviour of multiple chemical components inside catalysts [47] and rocks [48] in situ
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