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$${\phi }_{{\rm{opt}}}=\mathop{{\rm{argmin}}}\limits_{\phi }{ {\mathcal{L}}}_{P}[I(\phi ),{I}_{{\rm{obj}}}]$$ (1) \({ {\mathcal{L}}}_{P}\)is also referred as a loss function, which usually describes the difference between the reconstructed intensityI(ϕ) ...
(7) We want to know under what circumstances ui approaches a limiting value of u, call the fixed point in mathematics, as i approaches infinity. For this analysis, we'll define converging as |ui+1 − ui| < |ui − ui−1|, (8) for all i > I, where I is some unknown, ...
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It is worth highlighting that as \(N\) approaches infinity (\(N\to \infty \)), the solution becomes complete, offering a theoretically analytical solution. In practical applications, the numerical solution for the tubular systems holds significant importance, usually obtained with recurrence ...
{N}\). Note that the control interval is not the same as the integration interval, and on every control interval\([t_k,t_{k+1}]\)one can apply a FESD method with multiple variable integration steps. The state approximations are denoted by\(x_k \approx x(t_k)\), the control ...
$${\phi }_{{\rm{opt}}}=\mathop{{\rm{argmin}}}\limits_{\phi }{ {\mathcal{L}}}_{P}[I(\phi ),{I}_{{\rm{obj}}}]$$ (1) \({ {\mathcal{L}}}_{P}\)is also referred as a loss function, which usually describes the difference between the reconstructed intensityI(ϕ) ...
In Fig. 2, we display the endpoints h_\pm (x) as the integration variable x runs from the threshold to infinity. Because \kappa (M^2,M_\pi ^2,x) is holomorphic in the cut x plane, the endpoints h_\pm move – for \delta > 0 –along curves that are infinitely often differentiab...
As t approaches infinity, this smooth error function globally converges to 0. Consequently, the solution obtained through the BAFARNN model (4) also globally converges to the theoretical solution of DQRF (1). Proof. The representation of the BAFARNN model (4) is divided into two parts as ...
This procedure guarantees that the parameters of the penalty function are increasing enough because it is a necessary condition for non-stationary penalty functions [28], i.e., when k → ∞ , parameters must also go to infinity. The second point is generated by x k ′ + 1 = x a c ...