Irregularity Of Shape Optimization Problems AndAn Improvement Technique

For solution of dissipation energy minimization problems in a steady-state viscous flow field domain we applied the traction method that was proposed as a solution to domain optimization problems including a variation-constrained boundary in elliptic boundary value problems. Comparing this method to the mathematical programming method directly using the shape gradient function we can easily apply this method to domain optimization problems including the variation-constrained natural boundary. Also, we can easily implement this method with general-purpose FEM programs or BEM programs. We show the successful results of two-dimensional Stokes flow problems of a bending channel and a channel in which an isolated body exists.

New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces

Local convergence theorems for gradient methods in hilbert space

We present a numerical analysis method and results using the traction method for optimization problems of domains in which linear elastic problems are defined. In this paper we consider the application of the traction method which was proposed as a solution to domain optimization problems in elliptic boundary value problems. The minimization problems of the mean compliance were treated. Using the Lagrange multiplier method, we obtain the shape gradient functions for these domain optimization problems from the optimality criteria. In this drawing process we consider the variation of the surface force acting on the boundary and the variation of the stiffness function and the body force distributed in the domain. We show successful solutions of the problem of an infinite plate with a hole and of the problems of a rectangular plate clamped at both ends.

In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.

In this paper, we consider the Newton–Schur method in Hilbert space and obtain quadratic convergence. For the symmetric elliptic eigenvalue problem discretized by the standard finite element method and nonoverlapping domain decomposition method, we use the Steklov–Poincaré operator to reduce the eigenvalue problem on the domain into the nonlinear eigenvalue subproblem on , which is the union of subdomain boundaries. We prove that the convergence rate for the Newton–Schur method is , where the constant is independent of the fine mesh size and coarse mesh size , and and are errors after and before one iteration step, respectively. For one specific inner product on , a sharper convergence rate is obtained, and we can prove that . Numerical experiments confirm our theoretical analysis.

Kernel-based methods have had hefty success in a wide range of fields over the past decade, they are founded on the robust mathematical framework of reproducing kernel Hilbert spaces (RKHS), this space provides an interesting framework for the development of adaptive nonlinear filters. In this paper, we present a comparative study between the kernel method in Hilbert space with a reproducing kernel, and linear adaptive algorithms that is least mean square (LMS), normalized least mean square (NLMS) and recursive least square (RLS) algorithms. Simulation results show excellent performance of the kernel algorithm for identification of single-input single-output (SISO) systems, compared to the linear adaptive algorithms, this by adopting the very fast fading channels called Broadband Radio Access Network (BRAN A and BRAN C).

An adaptive multilevel approach to parabolic equations I. General theory and 1D implementation

In this article we adjust and correct some proofs that appeared in the article (El-Gebeily, M. and O'Regan, D., 2006, Existence and quasilinearization methods in Hilbert spaces, Journal of Mathematical Analysis and Applications, 324, 344–357) in connection with the abstract operator problem in a Hilbert space . Both existence results and abstract quasilinearization are adjusted and better streamlined.

Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods in Hilbert spaces. Besides simplifying computations it allows to provide a good understanding of properties of the forward problem compared to the prior information introduced by the regularization methods. In the last decade nonlinear variational approaches such as ` or total variation regularizations became quite prominent regularization techniques with certain properties being superior to standard methods. In the analysis of those, singular values and vectors did not play any role so far, for the obvious reason that these problems are nonlinear, together with the issue of defining singular values and singular vectors in the first place. In this paper however we want to start a study of singular values and vectors for nonlinear variational regularization of linear inverse problems, with particular focus on singular onehomogeneous regularization functionals. A major role is played by the smallest singular value, which we define as the ground state of an appropriate functional combining the (semi)norm introduced by the forward operator and the regularization functional. The optimality condition for the ground state further yields a natural generalization to higher singular values and vectors involving the subdifferential of the regularization functional, although we shall see that the Rayleigh principle may fail for higher singular values. Using those definitions of singular values and vectors, we shall carry over two main properties from the world of linear regularization. The first one is gaining information about scale, respectively the behavior of regularization techniques at different scales. This also leads to novel estimates at different scales, generalizing the estimates for the coefficients in the linear singular value expansion. The second one is to provide classes of exact solutions for variational regularization methods. We will show that all singular vectors can be reconstructed up to a scalar factor by the standard Tikhonov-type regularization approach even in the presence of (small) noise. Moreover, we will show that they can even be reconstructed without any bias by the recently popularized inverse scale space method.

The principal object of the investigation is a family of scalar products defined on a Hilbert space ℋ. For this general case of the theory of linear statistical inference, we obtain a counterpart of the Neyman-Fisher factorization theorem, and we introduce and study counterparts of the Fisher information and of the information matrix that preserve all the important properties of their classical prototypes. By additionally introducing the concept of observation interpreted as a linear functional on ℋ that satisfies a certain additional condition, we can construct a natural counterpart of the maximum likelihood method.

An open-loop two-person zero-sum linear quadratic (LQ for short) stochastic differential game is considered. The controls for both players are allowed to appear in both the drift and diffusion of the state equation, the weighting matrices in the payoff/cost functional are not assumed to be definite/non-singular, and the cross-terms between two controls are allowed to appear. A forward-backward stochastic differential equation (FBSDE, for short) and a generalized differential Riccati equation are introduced, whose solvability leads to the existence of the open-loop saddle points for the corresponding two-person zero-sum LQ stochastic differential game, under some additional mild conditions. The main idea is a thorough study of general two-person zero-sum LQ games in Hilbert spaces.

The theory of focal points and conjugate points is an important part of the study of problems in the calculus of variations and control theory. Hestenes has shown that for many problems this theory may be studied by Hilbert space methods. In a previous paper the author has extended the theory of Hestenes to elliptic quadratic forms $J(x;\sigma )$ defined on $\mathfrak {A}(\sigma )$ where $\sigma$ is a member of the metric space $(\Sigma ,\rho )$ and $\mathfrak {A}(\sigma )$ denotes a closed subspace of $\mathfrak {A}$. A fundamental part of this extension is concerned with inequalities dealing with the signature $s(\sigma )$ and nullity $n(\sigma )$ of $J(x;\sigma )$ on $\mathfrak {A}(\sigma )$ where $\sigma$ is in a $\rho$ neighborhood of a fixed point ${\sigma _0}$ in $\Sigma$. The purpose of this paper is threefold. The first purpose is to show that the extended theory includes the focal point hypotheses of Hestenes. The second purpose is to obtain a focal point theory much like that of Hestenes. It is interesting to note that our theory is based only on the nonnegative integers $s(\lambda )$ and $n(\lambda )$. This will facilitate later work on numerical calculations of focal points. Our final purpose is to obtain an abstract focal interval theory in which the usual focal points are (degenerate) focal intervals. While previous authors have considered specific problems, no general results for the focal interval case seem to be contained in the literature. An expression for the number of focal intervals on a subinterval $(\lambda ’,\lambda '')$ of $[a,b]$ is given. This expression is a key result for our work on approximation of focal intervals (to be published). For completeness we give comparison theorems for focal point problems. In addition an example is given for problems in optimal control theory. The correspondence between our focal intervals and solutions to the differential equations of the example is given.

In this paper we show that the author’s Two Nonzero Lemma (TNCL) can be applied to present a simple proof for a very useful equality which was first proved by Karl Gustafson in 1968. Gustafson used Hilbert space methods, including convexity of the Hilbert space norm, to prove this identity which was the basis of his matrix trigonometry. By applying TNCL, we will reduce the problem to a simple problem of ordinary calculus.

Chapter 3 - Numerical search methods for unconstrained optimization problems