Tikhonov regularization of nonlinear III-posed problems in hilbert scales

In this paper we consider nonlinear ill-posed problems F(x) = y ₀, where x and y ₀ are elements of Hilbert spaces X and Y, respectively. We solve these problems by Tikhonov regularization in a Hilbert scale. This means that the regularizing norm is stronger than the norm in X. Smoothness conditions are given that guarantee convergence rates with respect to the data noise in the original norm in X. We also propose a variant of Tikhonov regularization that yields these rates without needing the knowledge of the smoothness conditions. In this variant F is allowed to be known only approximately and X can be approximated by a finite-dimensional subspace. Finally, we illustrate the required conditions for a simple parameter estimation problem for regularization in Sobolev spaces.     

Dynamical Systems Method (DSM) for general nonlinear equations

If F:H→H is a map in a Hilbert space H, , and there exists y such that F(y)=0, F^′(y)≠0, then equation F(u)=0 can be solved by a DSM (dynamical systems method). This method yields also a convergent iterative method for finding y, and this method converges at the rate of a geometric series. It is not assumed that y is the only solution to F(u)=0. A stable approximation to a solution of the equation F(u)=f is constructed by a DSM when f is unknown but f_δ is known, where f_δ−f≤δ.     

An implicit Landweber method for nonlinear ill-posed operator equations

In this paper, we are interested in the solution of nonlinear inverse problems of the form F(x)=y. We propose an implicit Landweber method, which is similar to the third-order midpoint Newton method in form, and consider the convergence behavior of the implicit Landweber method. Using the discrepancy principle as a stopping criterion, we obtain a regularization method for ill-posed problems. We conclude with numerical examples confirming the theoretical results, including comparisons with the classical Landweber iteration and presented modified Landweber methods.     

A New Gradient Method for Ill-Posed Problems

In this paper, we present a new gradient method for linear and nonlinear ill-posed problems F(x) = y. Combined with the discrepancy principle as stopping rule it is a regularization method that yields convergence to an exact solution if the operator F satisfies a tangential cone condition. If the exact solution satisfies smoothness conditions, then even convergence rates can be proven. Numerical results show that the new method in most cases needs less iteration steps than Landweber iteration, the steepest descent or minimal error method.     

Approximate Controllability for the Semilinear Heat Equation Involving Gradient Terms

This paper deals with the approximate controllability of the semilinear heat equation, when the nonlinear term depends on both the state y and its spatial gradient y and the control acts on any nonempty open subset of the domain. Our proof relies on the fact that the nonlinearity is globally Lipschitz with respect to (y, y). The approximate controllability is viewed as the limit of a sequence of optimal control problems. Another key ingredient is a unique continuation property proved by Fabre (Ref. 1) in the context of linear heat equations. Finally, we prove that approximate controllability can be obtained simultaneously with exact controllability over finite-dimensional subspaces.     

Hereditary optimal control problems: Numerical method based upon a padé approximation

In this paper, we consider a particular approximation scheme which can be used to solve hereditary optimal control problems. These problems are characterized by variables with a time-delayed argumentx(t – ). In our approximation scheme, we first replace the variable with an augmented statey(t) x(t - ). The two-sided Laplace transform ofy(t) is a product of the Laplace transform ofx(t) and an exponential factor. This factor is approximated by a first-order Padé approximation, and a differential relation fory(t) can be found. The transformed problem, without any time-delayed argument, can then be solved using a gradient algorithm in the usual way. Four problems are solved to illustrate the validity and usefulness of this technique.This research was supported in part by the National Aeronautics and Space Administration under NASA Grant NCC-2-106.     

Ill-posedness and local ill-posedness concepts in hilbert spaces *

In this paper, we study ill-posedness concepts of nonlinear and linear operator equations in a Hilbert space setting. Such ill-posedness information may help to select appropriate optimization approaches for the stable approximate solution of inverse problems, which are formulated by the operator equations. We define local ill-posedness of a nonlinear operator equation F(x) = y ₀ in a solution point x ₀:and consider the interplay between the nonlinear problem and its linearization using the Fréchet derivative F′(x ₀). To find a corresponding ill-posedness concept for the linearized equation we define intrinsic ill-posedness for linear operator equations A x = y and compare this approach with the ill-posedness definitions due to Hadamard and Nashed     

On the cost of controlling unstable systems: The case of boundary controls

We discuss the cost of controlling parabolic equations of the formy _t + δ² y +kδy = 0 in a bounded smooth domain Ώ ofℝ ^d by means of a boundary control. More precisely, we are interested in the cost of controlling from zero initial state to a given final state (in a suitable approximate sense) at timeT > 0 and in the behavior of this cost ask → ∞. We introduce finite-dimensional Galerkin approximations and prove that they are exactly controllable. Moreover, we also prove that the cost of controlling converges exponentially to zero ask → ∞. This proves, roughly speaking, that when the system becomes more unstable it is easier to control. The system under consideration does not admit a variational formulation. Thus, in order to introduce its Galerkin approximation, we first approximate it by means of a singular perturbation. We also develop a method for the construction of special Galerkin bases well adapted to the control problem. Dedicated to John E. Lagnese on his 60th Birthday. Supported by project PB93-1203 of the DGICYT (Spain) and grant CHRX-CT94-0471 of the European Union.     

Exploring multivariate Padé approximants for multiple hypergeometric series

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions F _i , i = 1,...,4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the F _i onto ϰ=0 or y=0, namely, the Gauss function ₂ F ₁(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Padé approximants. In section 3 we prove that the table of homogeneous multivariate Padé approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F ₁(a, b, b′; c; x; y) with a = b = b′ = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Padé approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Padé approximants in this context and discussing directions for future work. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Riemann surfaces of analytic eigenvalue functions

If y(t)=y(t;μ ,λ) is the solution of the differential equation y”+(λ +μ w(t))y=0, a≤ t≤ b, determined by the initial conditions y(a)=0, y’(a)=1, then F(μ ,λ) =y(b;μ ,λ) is an entire function of two variables. The zeros of F are the eigenpairs of a two-parameter Sturm–Liouville problem. The multi-valued solution λ (μ) of F(μ ,λ) =0 is introduced, and the connectedness of its underlying Riemann surface is investigated. The results are extensions of those previously obtained by Schäfke. As an example, the Lamé equation is considered.     

Universal bases and greedy algorithms for anisotropic function classes

We suggest a three-step strategy to find a good basis (dictionary) for non-linear m-term approximation. The first step consists of solving an optimization problem of finding a near best basis for a given function class F, when we optimize over a collection D of bases (dictionaries). The second step is devoted to finding a universal basis (dictionary) D _u ∈ D for a given pair (F, D) of collections: F of function classes and D of bases (dictionaries). This means that D_u provides near optimal approximation for each class F from a collection F. The third step deals with constructing a theoretical algorithm that realizes near best m-term approximation with regard to D _u for function classes from F. In this paper we work this strategy out in the model case of anisotropic function classes and the set of orthogonal bases. The results are positive. We construct a natural tensor-product-wavelet-type basis and prove that it is universal. Moreover, we prove that a greedy algorithm realizes near best m-term approximation with regard to this basis for all anisotropic function classes.     

An Operation Connected to a Young-Type Inequality

Given two φ-functions F and G we consider the largest φ-function H = F ⊕ G such that the Young-type inequality H(xy) ? F(x) + G(y) holds for all x, y > 0. We prove an equivalence theorem for F ⊕ G with the best constants and, for the special case when F and G are log-convex and satisfy a certain growth condition, a representation formula for F G. Moreover, further properties and examples are presented and the relations to similar results are discussed.     

On First Range Times of Linear Diffusions

In this paper we consider first range times (with randomised range level) of a linear diffusion on R. Inspired by the observation that the exponentially randomised range time has the same law as a similarly randomised first exit time from an interval, we study a large family of non-negative 2-dimensional random variables (X,X′) with this property. The defining feature of the family is F^c(x,y)=F^c(x+y,0), ∀ x ≥ 0, y ≥ 0, where F^c(x,y):=P (X > x, X′ > y) We also explain the Markovian structure of the Brownian local time process when stopped at an exponentially randomised first range time. It is seen that squared Bessel processes with drift are serving hereby as a Markovian element.     

A Smoothing Newton Method for General Nonlinear Complementarity Problems

Smoothing Newton methods for nonlinear complementarity problems NCP(F) often require F to be at least a P ₀-function in order to guarantee that the underlying Newton equation is solvable. Based on a special equation reformulation of NCP(F), we propose a new smoothing Newton method for general nonlinear complementarity problems. The introduction of Kanzow and Pieper's gradient step makes our algorithm to be globally convergent. Under certain conditions, our method achieves fast local convergence rate. Extensive numerical results are also reported for all complementarity problems in MCPLIB and GAMSLIB libraries with all available starting points.     

Best m-term one-sided trigonometric approximation of some function classes defined by a kind of multipliers

In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the norm Lp （1 ≤ p ≤ ∞） of multiplier function classes and the corresponding m-term Greedy-liked one-sided trigonometric approximation results.     

Approximation Methods for Solving the Cauchy Problem

In this paper we give some new results concerning solvability of the 1-dimensional differential equation y′ = f(x, y) with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if f is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.     

F[x,y] as a Dialgebra and a Leibniz Algebra

In this article, we define a new associative dialgebra over a polynomial algebra F[x, y] with two indeterminates x and y. Left derivations, right derivations, derivations, and automorphisms of F[x, y] as associative dialgebra are determined. Meanwhile, we also determine all homogeneous derivations of F[x, y] as ?-graded Leibniz algebra, and automorphisms of Leibniz algebra F[x, y] preserving the standard filtration.     

Best Approximation and Cyclic Variation Diminishing Kernels

We study best uniform approximation of periodic functions from

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where the kernelK(x, y) is strictly cyclic variation diminishing, and related problems including periodic generalized perfect splines. For various approximation problems of this type, we show the uniqueness of the best approximation and characterize the best approximation by extremal properties of the error function. The results are proved by using a characterization of best approximants from quasi-Chebyshev spaces and certain perturbation results.     

Best-possible bounds on sets of bivariate distribution functions

The fundamental best-possible bounds inequality for bivariate distribution functions with given margins is the Fréchet–Hoeffding inequality: If H denotes the joint distribution function of random variables X and Y whose margins are F and G, respectively, then max(0,F(x)+G(y)−1)H(x,y)min(F(x),G(y)) for all x,y in [−∞,∞]. In this paper we employ copulas and quasi-copulas to find similar best-possible bounds on arbitrary sets of bivariate distribution functions with given margins. As an application, we discuss bounds for a bivariate distribution function H with given margins F and G when the values of H are known at quartiles of X and Y.     

Number of solutions for cubic Thue equations with automorphisms

Let be an irreducible cubic form with positive discriminant, and with non-trivial automorphisms. We show that the Thue equation F(x,y) = 1 has at most three integer solutions except for a few known cases. For the proof, we use an explicitly expressed cubic form which is equivalent to F. To obtain an upper bound for the size of solutions, we use the Padé approximation method developed in our former work. To obtain a lower bound for the size of solutions, we use a result of R. Okazaki on gaps between solutions, which is obtained by geometric consideration. 2000 Mathematics Subject Classification Primary—11D25, 11D59