Introduction to Inverse Problems for Differential Equations by Alemdar Hasanov Hasanoğlu & Vladimir G. Romanov

Author:Alemdar Hasanov Hasanoğlu & Vladimir G. Romanov
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Moreover, for the rate of convergence of the sequence following estimate holds:

(3.4.32)

where is the Lipschitz constant and , is the diameter of the set .

Proof

It is known that the minimum problem

for a continuous convex functional in a bounded closed and convex set has a solution. Hence each minimizing sequence weakly converges to an element of the solution set , that is , as .

To prove the rate of the convergence, we introduce the numerical sequence defined as

(3.4.33)

For a functional with Lipschitz continuous Fréchet gradient convexity is equivalent to the following inequality:

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