Theoretical aspects of ill-posed problems in statistics

Ill-posed problems arise in a wide variety of practical statistical situations, ranging from biased sampling and Wicksell's problem in stereology to regression, errors-in-variables and empirical Bayes models. The common mathematics behind many of these problems is operator inversion. When this inverse is not continuous a regularization of the inverse is needed to construct approximate solutions. In the statistical literature, however, ill-posed problems are rather often solved in an ad hoc manner which obccures these common features. It is our purpose to place the concept of regularization within a general and unifying framework and to illustrate its power in a number of interesting statistical examples. We will focus on regularization in Hilbert spaces, using spectral theory and reduction to multiplication operators. A partial extension to a Banach function space is briefly considered.Research supported by the Air Force Office of Scientific Research.     

Theoretical analysis of discrete contact problems with Coulomb friction

A discrete model of the two-dimensional Signorini problem with Coulomb friction and a coefficient of friction F depending on the spatial variable is analysed. It is shown that a solution exists for any F and is globally unique if F is sufficiently small. The Lipschitz continuity of this unique solution as a function of F as well as a function of the load vector f is obtained. Furthermore, local uniqueness of solutions for arbitrary F > 0 is studied. The question of existence of locally Lipschitz-continuous branches of solutions with respect to the coefficient F is converted to the question of existence of locally Lipschitz-continuous branches of solutions with respect to the load vector f. A condition guaranteeing the existence of locally Lipschitz-continuous branches of solutions in the latter case and results for determining their directional derivatives are given. Finally, the general approach is illustrated on an elementary example, whose solutions are calculated exactly.     

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.     

Family of numerical methods based on non-polynomial splines for solution of contact problems

In this paper a new technique based on quartic non-polynomial spline functions connecting spline functions values at mid knots and their corresponding values of the fourth-order derivatives is developed. This approach leads to a family of numerical methods for computing approximations to the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is shown that the present family of methods gives better approximations. Existing second and fourth-order finite-difference and spline functions based methods developed at mid knots become special cases of the new approach. Numerical examples are given to illustrate applicability and efficiency of the new methods.     

Theoretical and numerical aspects in the multiphasic modelling of human brain tissue

A surgical intervention is often required if the functionality of the sensitive human brain tissue is seriously compromised, e. g., due to the occurrence of malignant brain tumours. A promising method for an effective tumour-treatment procedure is given by the so-called convection-enhanced drug delivery (CED), cf. [1]. In this regard, the aim of this contribution is to simulate the expected effects as well as coupled impacts of a (scheduled) CED-procedure with the help of numerical computations, which base on a sophisticated multiphasic and multi-physical modelling strategy applied to human brain tissue. In particular, a quaternary porous-media model, cf. [3–5], is used for the discussion of selected numerical examples and demonstrates the applicability of the model. In detail, the optimal catheter placement and the application of multiple infusion catheters are studied in terms of the occurring anisotropic therapeutic spreading of the infused drug. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extremal problems for numerical series

In this paper, we consider extremal problems for numerical positive series. The terms of these series are pairwise products of the elements of two sequences, one of which is fixed and the other varies within a given set of sequences. We obtain exact solutions for a number of such problems. As one of the possible applications of the results obtained, we find solutions of some extremal problems related to best n-term approximations of periodic functions.     

Some optimal and inverse problems for orthotropic noncircular cylindrical shells

In this paper, we consider problems of optimal control involving stressed or strained states of orthotropic, noncircular cylindrical shells. It is assumed that the thickness of the shell is variable. The thickness and the radius of curvature of the directrix of the shell are assumed to be the controls. Existence of solutions for the optimal control problems considered is shown. In particular, existence of solutions for the problem of the minimal weight shell and the problem of nearest-to-equal-strength shell is shown. We present results on the approximation of the optimal control problems by a sequence of finite-dimensional problems, which may be reduced to nonlinear programming problems.     

Comparative analysis of two algorithms for numerical solution of nonlinearstatic problems for multilayered anisotropic shells of revolution 2. Account of transverse compression

Two algorithms for numerical solution of static problems for multilayer anisotropic shells of revolution are discussed. The first algorithm is based on a differential approach using the method of discrete orthogonalization, and the second one—on the finite element method with linear local approximation in the meridional direction. It is assumed that the layers of the shell are made of linearly elastic, anisotropic materials. As the unknown functions, six displacements of the shell are chosen, which often simplifies the definition of static problems for multilayer shells. The calculation of a cross-ply cylindrical shell stretched in the axial direction is considered. It is shown that taking account of the transverse compression, anisotropy, and geometrical nonlinearity is important for the given class of problems.Tambov State Technical University, Tambov, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 435–446, May–June, 1999.     

Modelling of topological derivatives for contact problems

The problem of topology optimization is considered for free boundary problems of thin obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis. Such results lead to the topological derivatives of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimization. Partially supported by the grant 4 T11A 01524 of the State Committee for the Scientific Research of the Republic of Poland     

Duality for implicit variational problems and numerical applications

The present paper deals with a dual characterization of the solutions of implicit variational problems. Some general results relating the solutions of the dual problem with those of the primal one are applied to variational and quasi-variational inequalities, Nash equilibria, saddle points and fixed points.

Finally a dual method for the numerical solutions of some quasi-variational inequalities is developed.     

Asymptotic solution of heat-conduction problems for shells of variable thickness

The temperature field in thin cylindrical shells of variable thickness on heating by a plane-parallel pulsed heat flux in the presence of convective heat transfer with the surrounding medium are approximately determined. By introducting well-founded practical constraints on the thermophysical parameters, the nonlinear problem is reduced to linear form.Translated frm Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 94–97, 1987.     

Mixed boundary-value problems of thermoelasticity for anisotropic-in-plan inhomogeneous toroidal shells

Problems of thermoelasticity for an anisotropic-in-plan inhomogeneous thin toroidal shell are solved by asymptotic integration of the equations of the three-dimensional problem of the theory of an anisotropic inhomogeneous solid for various boundary conditions. Recurrence formulae are derived for the components of the asymmetric stress tensor and the displacement vector. An example is given.     

Thermoelastic rolling contact problems for multilayer materials

The paper deals with the existence of solutions to the thermoelastic rolling contact problems for nonhomogeneous materials. One of the contacting surfaces is assumed to be covered with a graded material coating. The thermal and mechanical features of the coating material depend on its depth. The thermoelastic contact problem is governed by the system of mildly coupled evolutionary boundary value problems with discontinuous coefficients. Quasistatic approach is employed. This approach is based on the assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The Faedo–Galerkin approach combined with the penalization and smoothing approach are used to show the existence of solutions to this contact problem. The operator splitting method is used to solve the problem numerically. Numerical results indicating the reduction of mechanically and/or thermally induced stresses are provided.