Sentence examples for multilevel solution from inspiring English sources

end{aligned} The approximate solution (u_{n}) obtained in this way is called the nth multilevel solution of (1).

In general, an approximation (u_{n, i+1}) for ((n+i+1))st multilevel solution is defined by setting u_{n, 0} = u_{n}.

(5) The ((n+i th multilevel solution (u_{n+i}in S_{n+i}) satisfies the equation mathbf{A}_{n+i}mathbf{u}_{n+i}= mathbf{f}_{n+i}.

Let (u^) be the isolated solution of (2), (u_{n}in S_{n}) be the nth (standard) multilevel solution obtained from the wavelets of order p and (u_{n, i}) be the ((n+i th multilevel augmentation solution of (2).

First, we consider the distance between the weak solution u of (1) and the nth multilevel solution (u_{n}in S_{n}) obtained by the wavelets of order p. Theorem 4.1 in [13] stated that, for (uin H_{0}^{1}(varOmega ) cap H^{s} varOmega ) ), there exists a positive constant C such that |u-u_{n}| + 2^{-n}|u-u_{n}|_{1} leq C bigl(2^{-n} bigr)^{s}|u|_{s},quad 1leq sleq p+1.

Therefore, careful analysis, deliberate testing, and a phased approach to the implementation of innovative technologies are necessary to achieve the multilevel solutions of the smart ICU.

First, to explore multilevel solutions, the algorithm recursively tries to add extra index codebooks both at coarser and finer levels.

We have generalized our search algorithm for the two-level map equation to recursively search for multilevel solutions.

By the hypotheses on (mathcal{A}_{n}^{-1}) and (mathcal {D}_{n}^{-1}), we see that the ((n+i th multilevel augmentation solution (u_{n+i}) and the ((n+i th multilevel augmentation solution (u_{n, i}) exist for all (ngeq N) and (iinmathbb{N}cup{0}).

We call (u_{n, 1}) the ((n+1)) st multilevel augmentation solution of (1).

The ((n+i+1))th multilevel augmentation solution, (u_{n,i+1}), can be solved inductively.