quadmbox{(PDL1P~[14])} (2.2) where (nabla_{h}) denotes the numerical approximation of the gradient ∇, associated with the mesh size h, (r_{1}) and (r_{2}) are constants to be determined, (p^{n}) is the dual variable representing the gradient of the primal variable ((u^{n})), and (u^) is an intermediate solution.
When the objective functional contains the gradient of the state variable, mixed finite element methods should be used for discretization of the state equation with which both the scalar variable and its flux variable can be approximated in the same accuracy.
This relation involves the conditional filtered average of the magnitude of the gradient of the progress variable.
The damage model is based on a continuum model including the gradient of the damage variable.
For the adiabatic boundary condition, the gradient of the dependent variable normal to the boundary should be zero, i.e., ∂ φ/∂ y = 0.
The gradient of the control variable corrects the convective flux in the nonlinear state equation so as to enforce mass conservation while minimizing deviations from the target state.
In many control problems, the objective functional contains the gradient of the state variables.
As we have seen, problem (P) includes parabolic equation which is nonlinear with respect to the gradient of the solution, and with variable exponents of nonlinearity.
This regression coefficient corresponds to the gradient of the rate of change of the dependent variable (y) per unit change in the independent variable (x).
where X k = I + p ∑ j = 1 K H ̄ kj C j H ̄ kj C j H, and Y k = I + p ∑ j ≠ k K H ̄ kj C j H ̄ kj C j H (29). Since the sum-rate is real valued function and C k ∀k are complex variables, the gradient of the sum-rate can be calculated using the differential with respect to C k.
The first operator performs moves along the axes while the second simultaneously perturbs all the variables to follow the gradient of the fitness landscape.
