Sentence examples for by the implicit method from inspiring English sources

Proposition 3.1 The net { x t } generated by the implicit method (3.2) is bounded.

Then, as α m → 0, the sequence {x m } generated by the implicit method (3.1) converges to x ̃ ∈ F i x ( T ).

Theorem 3.3 The net { x t } generated by the implicit method (3.2) converges in norm, as t → 0 +, to the unique solution x ∗ of the following variational inequality: x ∗ ∈ Γ, 〈 ( I − f ) x ∗, x − x ∗ 〉 ≥ 0, x ∈ Γ. (3.13).

Theorem 3.1 { x t } generated by the implicit method (3.1) converges in norm as t → 0 + to the unique solution x ∗ ∈ Fix ( T ) of the variational inequality 〈 ( μ F ¯ − f ¯ ) x ∗, j ( x ∗ − u ) 〉 ≤ 0, u ∈ Fix ( T ).

We next show that limsup n → ∞ x ̃, x ̃ - x n ≤ 0, where x ̃ = lim m → ∞ y m and {y m } be the sequence defined by the implicit method (3.1).

Theorem 3.1 Suppose that F ( T ) ∩ ( A + B ) − 1 0. Then the net { x t } defined by the implicit method (3.1) converges strongly, as t → 0, to a point x ˜ ∈ F ( T ) ∩ ( A + B ) − 1 0, which is the unique solution of the variational inequality (3.2).

In Section 3, by discretizing the implicit method of the regularization obtained in Section 2, we introduce an iteration process and prove its strong convergence.

The additional computational costs are raised by 10%% once the implicit method is activated, but only in this case Coulomb collisions are properly treated.

The results show that though the implicit integration enables larger time steps to be used compared to the common explicit methods, the overall speed up is overruled by higher computational costs of the implicit method.

The efficiency of the implicit method proposed by He (1999) depends on the parameter heavily; while it varies for individual problem, that is, different problem has different "suitable" parameter, which is difficult to find.

We apply the same asymptotic analysis to a discrete version of the problem, in which the frequency variable is discretized by the multigroup method, the direction variable by the discrete-ordinates method, the time variable by the fully implicit method, and the spatial variable by a subcell-balance method.