Computing the orientational-average of diffusion-weighted MRI signals: a comparison of different techniques

Numerous applications in diffusion MRI involve computing the orientationally-averaged diffusion-weighted signal. Most approaches implicitly assume, for a given b-value, that the gradient sampling vectors are uniformly distributed on a sphere (or ‘shell’), computing the orientationally-averaged signal through simple arithmetic averaging. One challenge with this approach is that not all acquisition schemes have gradient sampling vectors distributed over perfect spheres. To ameliorate this challenge, alternative averaging methods include: weighted signal averaging; spherical harmonic representation of the signal in each shell; and using Mean Apparent Propagator MRI (MAP-MRI) to derive a three-dimensional signal representation and estimate its ‘isotropic part’. Here, these different methods are simulated and compared under different signal-to-noise (SNR) realizations. With sufficiently dense sampling points (61 orientations per shell), and isotropically-distributed sampling vectors, all averaging methods give comparable results, (MAP-MRI-based estimates give slightly higher accuracy, albeit with slightly elevated bias as b-value increases). As the SNR and number of data points per shell are reduced, MAP-MRI-based approaches give significantly higher accuracy compared with the other methods. We also apply these approaches to in vivo data where the results are broadly consistent with our simulations. A statistical analysis of the simulated data shows that the orientationally-averaged signals at each b-value are largely Gaussian distributed.

Finding the Optimal Multimodel Averaging Method for Global Hydrological Simulations

Global gridded precipitations have been extensively considered as the input of hydrological models for runoff simulations around the world. However, the limitations of hydrologic models and the inaccuracies of the precipitation datasets could result in large uncertainty in hydrological forecasts and water resource estimations. Therefore, it is of great importance to investigate the hydrological value of a weighted combination of hydrological models driven by different precipitation datasets. In addition, due to the diversities of combination members and climate conditions, hydrological simulation for watersheds under different climate conditions may show various sensitivities to the weighted combinations. This study undertakes a comprehensive analysis of various multimodel averaging methods and schemes (i.e., the combination of the members in averaging) to identify the most skillful and reliable multimodel averaging application. To achieve this, four hydrological models driven by six precipitation datasets were used as averaging members. The behaviors of 9 averaging methods and 11 averaging schemes in hydrological simulations were tested over 2277 watersheds distributed in different climate regions in the world. The results show the following: (1) The multi-input averaging schemes (i.e., members consist of one model driven by multiple precipitation datasets) generally perform better than the multimodel averaging schemes (i.e., members consist of multiple models driven by the same precipitation dataset) for each averaging method; (2) The use of multiple members can improve the averaging performances. Six averaging members are found to be necessary and advisable, since using more than six members only imrpoves the estimation results slightly, as compared with using all 24 members; (3) The advantage of using averaging methods for hydrological modeling is region dependent. The averaging methods, in general, produced the best results in the warm temperate region, followed by the snow and equatorial regions, while a large difference among various averaging methods is found in arid and arctic regions. This is mainly due to the different averaging methods being affected to a different extent by the poorly performed members in the arid and arctic regions; (4) the multimodel superensemble method (MMSE) is recommended for its robust and outstanding performance among various climatic regions.

String-averaging methods for best approximation to common fixed point sets of operators: the finite and infinite cases

String-averaging is an algorithmic structure used when handling a family of operators in situations where the algorithm in hand requires to employ the operators in a specific order. Sequential orderings are well known, and a simultaneous order means that all operators are used simultaneously (in parallel). String-averaging allows to use strings of indices, constructed by subsets of the index set of all operators, to apply the operators along these strings, and then to combine their end-points in some agreed manner to yield the next iterate of the algorithm. String-averaging methods were discussed and used for solving the common fixed point problem or its important special case of the convex feasibility problem. In this paper we propose and investigate string-averaging methods for the problem of best approximation to the common fixed point set of a family of operators. This problem involves finding a point in the common fixed point set of a family of operators that is closest to a given point, called an anchor point, in contrast with the common fixed point problem that seeks any point in the common fixed point set.We construct string-averaging methods for solving the best approximation problem to the common fixed points set of either finite or infinite families of firmly nonexpansive operators in a real Hilbert space. We show that the simultaneous Halpern–Lions–Wittman–Bauschke algorithm, the Halpern–Wittman algorithm, and the Combettes algorithm, which were not labeled as string-averaging methods, are actually special cases of these methods. Some of our string-averaging methods are labeled as “static” because they use a fixed pre-determined set of strings. Others are labeled as “quasi-dynamic” because they allow the choices of strings to vary, between iterations, in a specific manner and belong to a finite fixed pre-determined set of applicable strings. For the problem of best approximation to the common fixed point set of a family of operators, the full dynamic case that would allow strings to unconditionally vary between iterations remains unsolved, although it exists and is validated in the literature for the convex feasibility problem where it is called “dynamic string-averaging”.

Prediction of Mean Responses of RC Bridges Considering the Incident Angle of Ground Motions and Displacement Directions

Inelastic dynamic analyses were carried out using 3D and 2D models to predict the mean seismic response of four-span reinforced concrete (RC) bridges considering directionality effects. Two averaging methods, including an advanced method considering displacement direction, were used for the prediction of the mean responses to account for different incident angles of ground motion records. A method was developed to predict the variability of the mean displacement predictions due to variability in the incident angles of the records for different averaging methods. When the concepts of averaging in different directions were used, significantly different predictions were obtained for the directionality effects. The accuracy of the results obtained using 2D and 3D analyses with and without the application of the combination rules for the prediction of the mean seismic demands considering the incident angle of the records was investigated. The predictions from different methods to account for the records incident angles were evaluated probabilistically. Recommendations were made for the use of the combination rules to account for the directivity effects of the records and to predict the actual maximum displacement, referred to as the maximum radial displacement.

Response of Cantilever Model with Inertia Nonlinearity under Transverse Basal Gaussian Colored Noise Excitation

Considering the curvature nonlinearity and longitudinal inertia nonlinearity caused by geometrical deformations, a slender inextensible cantilever beam model under transverse pedestal motion in the form of Gaussian colored noise excitation was studied. Present stochastic averaging methods cannot solve the equations of random excited oscillators that included both inertia nonlinearity and curvature nonlinearity. In order to solve this kind of equations, a modified stochastic averaging method was proposed. This method can simplify the equation to an Itô differential equation about amplitude and energy. Based on the Itô differential equation, the stationary probability density function (PDF) of the amplitude and energy and the joint PDF of the displacement and velocity were studied. The effectiveness of the proposed method was verified by numerical simulation.