This is achieved by computing the gradient (vec{g}) and the Hessian H (or its approximation) at each iteration of the nonlinear regression.
Further, the traditional way to learn the parameters of a deep architecture is to minimize an objective function by computing the gradient over all the parameters using the backpropagation algorithm [14] with a nonlinear optimizer.
By rewriting the matrix quadratic form in the log-likelihood term of (34) as a vector quadratic form in terms of h ̲ ( f ) and by computing the gradient of Q G and equating it to zero, we obtain h ̲ ( f ) = γ Σ h ̲ − 1 ( f ) + 1 σ b 2 ( f ) ∑ n = 1 N ( R ̂ s ( n, f ) ⊗ I I ) T − 1 × γ Σ h ̲ − 1 ( f ) μ h ̲ ( f ) + 1 σ b 2 ( f ) ∑ n = 1 N vec ( R ̂ xs ( n, f ) ) (35).
Often it is helpful to interpret MEG fields measured by magnetometers (and axial gradiometers, e.g., Gross et al. 2012) after transforming the data to a planar gradient configuration, that is, by computing the gradient tangential to the scalp.
A voxel-wise approximation of the surface normal was obtained by computing the gradient direction for each voxel within a scalar field derived by a combination of these distance functions (ФWM– ФPBS).
Ruiz et al. [26] presented a texture orientation detection algorithm by computing the dominant gradient and reducing the unnecessary directional candidate modes in the RDO process, saving average time of 30.1% compared with HM 14.0.
Finally, we investigate two different approaches to compute the gradient by the adjoint state method and show the significant advantages of using OpenCL for FWI.
Working on the horizontal direction, we first compute the gradient sign map by quantizing the gradient magnitude as follows (the same process is extended to the vertical direction): SignMap x ( x, y ) = 2 if G x ( x, y ) < 0 1 if G x ( x, y ) > 0 0 if G x ( x, y ) = 0 (9).
Use the adjoint method to compute the gradient of the log likelihood function from a Kalman filter.
Although this objective is not differentiable, we can compute the gradient of a smoothed version.
