The Gaussian phase distribution approximation enables analysis of restricted diffusion encoded by general gradient waveforms but fails to account for the diffraction-like features that may occur for simple pore geomet...The Gaussian phase distribution approximation enables analysis of restricted diffusion encoded by general gradient waveforms but fails to account for the diffraction-like features that may occur for simple pore geometries.We investigate the range of validity of the approximation by random walk simulations of restricted diffusion in a cylinder using isotropic diffusion encoding sequences as well as conventional single gradient pulse pairs and oscillating gradient waveforms.The results show that clear deviations from the approximation may be observed at relative signal attenuations below 0.1 for onedimensional sequences with few oscillation periods.Increasing the encoding dimensionality and/or number of oscillations while extending the total duration of the waveform diminishes the non-Gaussian effects while preserving the low apparent diffusivities characteristic of restriction.展开更多
In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant n...In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means,e.g.,arithmetic mean,geometric mean,harmonic mean,and general mean.By associating DTI with perturbation formula,i.e.,a formula to relate the tensor-valued function difference with respect the difference of the function input tensors,the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case,respectively.We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean,and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.展开更多
基金financially supported by the Swedish Research Council(2022-04422_VR)。
文摘The Gaussian phase distribution approximation enables analysis of restricted diffusion encoded by general gradient waveforms but fails to account for the diffraction-like features that may occur for simple pore geometries.We investigate the range of validity of the approximation by random walk simulations of restricted diffusion in a cylinder using isotropic diffusion encoding sequences as well as conventional single gradient pulse pairs and oscillating gradient waveforms.The results show that clear deviations from the approximation may be observed at relative signal attenuations below 0.1 for onedimensional sequences with few oscillation periods.Increasing the encoding dimensionality and/or number of oscillations while extending the total duration of the waveform diminishes the non-Gaussian effects while preserving the low apparent diffusivities characteristic of restriction.
基金supported by the National Natural Science Foundation of China under grant No.12271108Shanghai Municipal Science and Technology Commission under grant No.22WZ2501900Innovation Program of Shanghai Municipal Education Commission
文摘In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means,e.g.,arithmetic mean,geometric mean,harmonic mean,and general mean.By associating DTI with perturbation formula,i.e.,a formula to relate the tensor-valued function difference with respect the difference of the function input tensors,the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case,respectively.We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean,and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.
