Iteration penalty method

We consider penalized operator equations approximating variational inequalities. For equations with strongly monotonous operators we construct an iterative method, prove convergence of solutions, and obtain error estimates. 

Let K he a closed convex subset in a reflexive Banach space V let an operator A act from V into V and let f G V he given. Consider the variational inequality 

The imbeddings are continuous and dense, and there is a constant c 0 such that 

Given a small parameter c 0, we write down the penalized equation 

This property obviously implies coercivity and strict monotonicity of A. The right-hand side of (1.105) belongs to V since H c V. Then, by Theorem 1.14, there exists a unique solution V,n = 0,1,2. to 

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Kovtunenko V.A. (1994b) An iteration penalty method for variational inequalities with strongly monotonous operators. Siberian Math. J. 35 (4), 735-738. 

Kovtunenko V.A. (1994c) Iteration penalty method for the contact elasto-plastic problem. Control and Cybernetics 23 (4), 803-808. 

Kovtunenko V.A. (1996b) An iterative penalty method for a problem with constraints on the inner boundary. Siberian Math. J. 37 (3), 508-512. 

Kovtunenko V.A. (1997b) Iterative penalty method for plate with a crack. Adv. Math. Sci. Appl. 7 (2), 667-674. 

Kovtunenko V.A. (1993) An iterative methods for solving variational inequalities of the contact elastoplastic problem by the penalty method. Comp. Maths. Math. Phys. 33 (9), 1245-1249. 

Some of the conclusions drawn are that, for our specific finite element models (non-symmetric, using penalty method for some connections, possible softening behavior), direct solvers outperform the iterative solver significantly. As expected direct solver were not as scalable as iterative solvers, however, specifics of our finite element models (dealing with soil-structure interaction) resulted in poor initial performance of iterative solvers, that, even with excellent performance scaling, could not catch up with the efficiency of direct solvers. IT is also important to note that parallel direct solvers, such as MUMPS and SPOOLES provided the best performance and would be recommended for use with finite element models that, as ours did, feature non-symmetry, are poorly conditioned (they are ill-posed due to use of penalty method) and can be negative definite (for softening materials). 

In this section we deal with the simplified nonpenetration condition of the crack faces considered in the previous section. We formulate the model of a plate with a crack accounting for only horizontal displacements and construct approximate equations using penalty and iterative methods. The convergence of these solutions is proved and its application to the onedimensional problem is discussed. Analytical solutions for the model of a bar with a cut are obtained. The results of this section can be found in (Kovtunenko, 1996c, 1996d). 

An alternate method for introducing pulse restrictions has been introduced by one of us (AB) recently within an iterative scheme for solving the optimal control equations [70], The idea is that a new reference field e(t) is constructed based on the field from the previous iteration after the application of a filter function F to ensure the fulfilment of some predesigned temporal and spectral properties. Therefore, a penalty term of the form... 

To improve the convergence of the gradient-type method, Tannor et al. [81, 93] suggested employing the Krotov iteration method [102]. In formulating their method, they utilize a penalty function of the form /[e(f)] = pe (f). In Tannor s Krotov method, the fcth iteration step of the solution process is given by... 

The user supplied weighting constant, (>0), should have a large value during the early iterations of the Gauss-Newton method when the parameters are away from their optimal values. As the parameters approach the optimum, should be reduced so that the contribution of the penalty function is essentially negligible (so that no bias is introduced in the parameter estimates). 

Figure 7.7 shows the convergence of the penalty function method. At each outer iteration r, the plotted objective functional value is the optimal with corresponding constraint violation quantified as q. It was 17.2 at the very beginning (corresponding to the initial controls) and dropped finally to 5.9 x 10 at the convergence, which was attained in 11 outer iterations. 

The classical i penalty Junction method starts with an assigned value for the parameter ko, solves the unconstrained problem, and verifies that the constraints are fulfilled satisfactorily. If unsatisfied, we impose k- +i > k and the procedure is iterated. 

As above, the SQP method is not iterated without controls, but a merit function of Chapter 12 is usually adopted (for instance, i or the augmented Lagrangian penalty Junction alpf) to deem whether the iterations converge to the solution. 

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