Volume 18, Issue 5

In this paper, the criterion of resolvability of the inverse problem of heat conduction is established.

We derive an iterative algorithm for an Inverse Problem of Option Pricing. The aim is to determine the local volatility such that the corresponding solutions of the Black–Scholes equation match the quoted market prices. Market data are given as option prices with different strikes and maturities. This leads to a parameter estimation problem for parabolic differential equations. This inverse problem is nonlinear and ill-posed. To overcome these difficulties, we apply an inexact Newton method. Thus, we derive a regularizing iterative algorithm of Levenberg–Marquardt type. As an underlying operator, we introduce the propagation operator. It maps, for given initial data, the unknown volatility to the final data. Then, we determine its Gâteaux-differential operator via a parabolic equation of second order. Moreover, its corresponding adjoint, called backpropagation operator, is obtained via a backward parabolic equation. Finally, to illustrate the efficiency of the method, we present some numerical results for simulated data and for observations extracted from the market. Besides, we discuss different strategies for choosing the regularizing parameters.

This paper is devoted to regularization in the process of derivative's approximation of integrated semigroups in time variables. We consider the direct method and also the method based on A. N. Tikchonov's approach. It is shown that discrete derivative converges in strong sense and the order of convergence in general Banach space is obtained. The presentation is given in the abstract framework of discrete approximation scheme, which includes finite element methods, finite difference schemes and projection methods.

We study the problem of reconstructing a wavefront from measurements by Shack–Hartmann-type sensors. Mathematically, this leads to the problem of reconstructing a function from a discrete set of averages of the gradient. We investigate this problem in a scale of Hilbert spaces and show that the best approximate solutions are asymptotically ill-posed depending on the scale of the Hilbert spaces. Moreover, we prove convergence of the best approximate solutions and of approximations obtained by the truncated singular value decomposition for increasing dimension of measurements.

Let the steady-state deflection u ( x ) of a beam of length l be governed by the Euler–Bernoulli fourth-order ordinary differential beam equation ( a ( x ) u ″( x ))″ = ƒ( x ) for x ∈ (0, l ), subject to Dirichlet (clamped beams) boundary copnditions. We consider the identification of the spacewise dependent flexural rigidity a ( x ) of the beam, given the load ƒ( x ) and the measured deflection u ( x ) by the comparison model method – a direct method which has been known and applied for sometime in diffusion processes of transport in porous media, but not in the elasticity beam theory, as developed in this study.