Narrow Results

This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000] and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it.

We consider the problem of recovering surface vibrations from acoustic pressure measurements taken in the interior or the exterior of a region. We give two formulations of the problem. One is based on a representation of the pressure as layer potentials and the other is based on approximation by a class of specific solutions to the Helmholtz equation. Boundary element methods are developed to approximate the integral operators. A conjugate gradient algorithm based on Lanczos bidiagonalization is applied to compute regularized solutions to the ill-conditioned equations in the presence of measurement errors. Numerical examples which compare the two formulations are presented.

We present a novel parameter choice strategy for the conjugate gradient regularization algorithm which does not assume a priori information about the magnitude of the measurement error. Our approach is to regularize within the Krylov subspaces associated with the normal equations. We implement conjugate gradient via the Lanczos bidiagonalization process with reorthogonalization, and then we construct regularized solutions using the SVD of a bidiagonal projection constructed by the Lanczos process. We compare our method with the one proposed by Hanke and Raus and illustrate its performance with numerical experiments, including detection of acoustic boundary vibrations.

We study the problem of nonparametric estimation of the risk-neutral densities from options data. The underlying statistical problem is known to be ill-posed and needs to be regularized. We propose a novel regularized empirical sieve approach for the estimation of the risk-neutral densities which relies on the notion of the minimal martingale entropy measure. The proposed approach can be used to estimate the so-called pricing kernels which play an important role in assessing the risk aversion over equity returns. The asymptotic properties of the resulting estimate are analyzed and its empirical performance is illustrated.

The simple and efficient procedure of trend identification in economic time series using the FFT method has been proposed. The effect of trend identification on the behavior of agents on stock markets has been considered. The spectral decomposition of economic data as ill-posed problems has been studied. Connection of trend identification with data smoothing and regularization for numerical differentiation of empirical data has been discussed. Relations among data subjective complexity, spectral regularization parameter and temporal preferences of agent have been shown. The selection of the cut-off frequency for data smoothing should correspond to the investment horizon of the economic agent.

We apply a wavelet dual least squares method to a backward heat equation. Connecting Shannon wavelet bases with a special project method, the dual least squares method, we can obtain a regularized solution. Meanwhile, a new error estimate between the approximate solution and exact solution is proved.

We consider a Cauchy problem for a modified Helmholtz equation, especially when we give the optimal error bound for this problem. Some spectral regularization methods and a revised Tikhonov regularization method are used to stabilize the problem from the viewpoint of general regularization theory. Hölder-type stability error estimates are provided for these regularization methods. According to the optimality theory of regularization, the error estimates are order optimal.

In this note we prove a stability estimate for an inverse heat source problem. Based on the obtained stability estimate, we present a generalized Tikhonov regularization method and obtain the error estimate. Numerical experiment shows that the generalized Tikhonov regularization works well.

Numerical fractional differentiation is a classical ill-posed problem in the sense that a small perturbation in the data can cause a large change in the fractional derivative. In this paper, we consider a wavelet regularization method for solving a reconstruction problem for numerical fractional derivative with noise. A Meyer wavelet projection regularization method is given, and the Hölder-type stability estimates under both apriori and aposteriori regularization parameter choice rules are obtained. Some numerical examples show that the method works well.

Range Restricted GMRES (RRGMRES) can be considered as regularizing iterative method and its iterates as regularized solutions. In this paper, we use two preconditioned versions of this method for solving ill-posed inverse problems. Our proposed preconditioner matrix is appropriate to use directly for linear systems of the form $Ax=b$ with square ill-conditioned coefficient matrix. We show numerically that the Preconditioned RRGMRES method has more stable solution compared to RRGMRES.

In this paper, we report on a strategy for computing the numerical approximate solution for a class of ill-posed operator equations in Hilbert spaces: $K:E\to F,Kf=g$. This approach is a combination of Tikhonov regularization method and the finite rank approximation of $K^{\ast }K$. Finally, numerical results are given to show the effectiveness of this method.

A review of the authors' results is given. Several methods are discussed for solving nonlinear equations F(u) = f, where F is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy principle for solving the equation is formulated and justified. Various versions of the Dynamical Systems Method (DSM) for solving the equation are formulated. These versions of the DSM include a regularized Newton-type method, a gradient-type method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation F(u) = f is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to equation F(u) = f is justified. New nonlinear differential inequalities are derived and applied to a study of large-time behavior of solutions to evolution equations. Discrete versions of these inequalities are established.

We consider an inverse problem for the Poisson equation $-\mathrm{\Delta }u=f$ in the square $\mathrm{\Omega }=(0,1)\times (0,1)$ which consists of determining the source $f$ from boundary measurements. Such problem is ill-posed. We restrict ourselves to a class of functions $f(x_1,x_2)={\phi }_1(x_2)g_1(x_1)+{\phi }_2(x_2)g_2(x_1)$. To illustrate our method, we first assume that $g_1$ and $g_2$ are known functions with partial data at the boundary. For the reconstruction, we consider approximations by the Fourier series, therefore we obtain an ill-posed linear system which requires a regularization strategy. In the general case, we propose an iterative algorithm based on the full data at the boundary. Finally, some numerical results are presented to show the effectiveness of the proposed reconstruction algorithms.

The simple and efficient procedure of trend identification in economic time series using the FFT method has been proposed. The effect of trend identification on the behavior of agents on stock markets has been considered. The spectral decomposition of economic data as ill-posed problems has been studied. Connection of trend identification with data smoothing and regularization for numerical differentiation of empirical data has been discussed. Relations among data subjective complexity, spectral regularization parameter and temporal preferences of agent have been shown. The selection of the cut-off frequency for data smoothing should correspond to the investment horizon of the economic agent.

We give a survey of the image normalization technique for certain classes of Wiener-Hopf operators (WHOs) associated to ill-posed boundary-transmission value problems. We briefly describe the method of normalization for the positive half-plane and apply it to the diffraction problem with generalized Dirichlet boundary conditions. Then, we consider a WHO equivalent to a convolution type operator associated to a boundary value problem on a strip. Finally, we give an example of a generalization of the method for a junction of two infinite half-planes.

A multitude of important problems can be cast as nonsmooth variational problems in function spaces, and hence in an infinite-dimensional, setting. Traditionally numerical approaches to such problems are based on first order methods. Only more recently Newton-type methods are systematically investigated and their numerical efficiency is explored. The notion of Newton differentiability combined with path following is of central importance. It will be demonstrated how these techniques are applicable to problems in mathematical imaging, and variational inequalities. Special attention is paid to optimal control with partial differential equations as constraints.

Inverse problems is a vibrant and expanding branch of mathematics that has found numerous applications. In this work, we will survey some of the main techniques available in the literature to solve the inverse problem of identifying variable parameters in partial differential equations. Besides carefully defining the problem, we give simple examples to depict some of the difficulties associated with the study of this inverse problem. We will discuss in sufficient detail twelve different methods available to solve this particular class of inverse problems. We also analyze some of their most exciting applications. We will also point out some of the research directions which can be pursued in this fascinating branch of applied and industrial mathematics.

A new dynamical systems method is studied for solving nonlinear operator equation . This method consists of the construction of a nonlinear dynamical systems, that is, a Cauchy problem , where Φ is a suitable operator, which has the following properties: i) :∃u(t) ∀t > 0, ii) :∃u(∞), and iii) . Conditions on are given which allow one to chosen Φ such that i), ii), and iii) hold. This method yields also a convergent iterative method for finding true solution y. Stable approximation to a solution of the equation is constructed by this DSM when f is unknown but fδ is known, when ||f − fδ|| ≤ δ.

In this paper, we proposed a piecewise Tikhonov regularization scheme to overcome the disadvantage of the classical Tikhonov regularization for solving the Radon integral equation whose solution is piecewise continues arising in CT imaging. In our new scheme, we firstly detect the location of discontinuous interface by applying the classical Tikhonov regularization. Then a so-called piecewise Tikhonov regularization method is applied to reconstruction the smooth solution in each piece. Some numerical examples has also be presented to demonstrate the effectiveness of this two-steps scheme.