Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients

One of the basic inverse problems in an anisotropic media is the determination of coefficients in a bounded domain with a single measurement. We consider the problem of finding the coefficient of the second derivatives in a second-order hyperbolic equation with variable coefficients.

Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement. Moreover we show that our uniqueness results yield the Lipschitz stability estimate in L ² space for solution to the inverse problem under consideration.     

An optimal control model of oil discovery and extraction

Starting with Hotelling (1931) [7] the stock of non-renewable resources have been treated as fixed. Along the line of Pindyck (1978) [8] and Greiner and Semmler (in press) [5] we treat the stock of oil resources as time varying, depending on new discoveries. The resource is finite and only a part of the resource is known while the rest has not yet been discovered. The discovery leads to a rise of known oil resource which can then be optimally exploited. The optimal control model has two state variables, the known stock of the resource and the cumulated past extraction. The control variable is the optimal extraction rate. The optimal control model assumes a monopolistic resource owner who maximizes intertemporal profits from exploiting the resource where the price of the resource depends on the extraction rate, the known stock of the resource, and the cumulated past extraction. The model is solved for a finite time horizon using NUDOCCCS, a numerical solution method to solve finite horizon optimal control problems. Various parameter constellations are explored. For certain parameter constellations the price path becomes U-Shaped as some empirical research, see Greiner and Semmler (in press) [5], have found to hold for actual price data. This holds if the stock of the initially known resource is small.     

波动方程系数反问题的求解方法

讨论了一维波动方程系数反演的一种求解方法,将解进行一阶渐进展开,得到相应的反问题,将其转化为第二类Volttera型积分方程组,证明了反问题解的存在唯一性.     

Feature extraction using two-dimensional local graph embedding based on maximum margin criterion

In this paper, we propose a novel method for image feature extraction, namely the two-dimensional local graph embedding, which is based on maximum margin criterion and thus not necessary to convert the image matrix into high-dimensional image vector and directly avoid computing the inverse matrix in the discriminant criterion. This method directly learns the optimal projective vectors from 2D image matrices by simultaneously considering local graph embedding and maximum margin criterion. The proposed method avoids huge feature matrix problem in Eigenfaces, Fisherfaces, Laplacianfaces, maximum margin criterion (MMC) and inverse matrix in 2D Fisherfaces, 2D Laplacianfaces and 2D Local Graph Embedding Discriminant Analysis (2DLGEDA) so that computational time would be saved for feature extraction. Experimental results on the Yale and the USPS databases show the effectiveness of the proposed method under various experimental conditions.     

On inverse crack problems in elastostatics

In solid mechanics, nondestructive testing has been an important technique in gathering information about unknown cracks, or defects in material. From a mathematical point of view, this is described as an inverse problem of partial differential equations, that is, the problem is to extract information about the location and shape of an unknown crack from the surface displacement field and traction on the boundary of the elastic material. By using the enclosure method introduced by Prof. Ikehata we can derive the extraction formula of an unknown linear crack from a single set of measured boundary data. Then, we need to have precise properties of a solution of the corresponding boundary value problem; for instance, an expansion formula around the crack tip. In this paper we consider the inverse problem concentrating on this point. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On dimension of inverse limits with upper semicontinuous set-valued bonding functions

We give results about the dimension of continua, obtained by combining inverse limits of inverse sequences of metric spaces and one-valued bonding maps with inverse limits of inverse sequences of metric spaces and upper semicontinuous set-valued bonding functions, by standard procedure introduced in [I. Bani?, Continua with kernels, Houston J. Math. (2006), in press].     

A problem of data mining in E-commerce

Data mining is generally defined as the science of nontrivial extraction of implicit, previously unknown, and potentially useful information from datasets. There are many websites on the Internet that provide extensive information about products and allow users post comments on various products and rate the product on a scale of 1 to 5. During the past decade, the need for intelligent algorithms for calculating and organizing extremely large sets of data has grown exponentially. In this article we investigate the extent to which a product’s average user rating can be predicted, using a manageable subset of a data set. For this we use a linearization-algorithm based prediction model and sketch how an inverse problem can be formulated to yield a smooth local volatility function of user ratings. The MAPLE programs that implement the proposed algorithm show that the method is reasonably accurate for the reconstruction of volatility of user ratings, which is useful both in accurate user predictions as well as computing sensitivity.     

Closure of the squared Zakharov-Shabat eigenstates

By solving the inverse scattering problem for a third-order (degenerate) eigenvalue problem, we can find the closure of the squared eigenfunctions of the Zakharov-Shabat equations. The question of the completeness of squared eigenstates occurs in many aspects of “inverse scattering transforms” (solving nonlinear evolution equations exactly by inverse scattering techniques) as well as in various aspects of the inverse scattering problem. The method we use is quite suggestive as to how one might find the closure of the squared eigenfunctions of other eigenvalue equations, and we point the strong analogy between our results and the problem of finding the closure of the eigenvectors of a nonself-adjoint matrix.     

On the Main Aspects of the Inverse Conductivity Problem

We consider a nonlinear inverse problem for an elliptic partial differential equation known as the Calder{\''o}n problem or the inverse conductivity problem. Based on several results, we briefly summarize them to motivate this research field. We give a general view of the problem by reviewing the available results for $C^2$ conductivities. After reducing the original problem to the inverse problem for a Schr\"odinger equation, we apply complex geometrical optics solutions to show its uniqueness. After extending the ideas of the uniqueness proof result, we establish a stable dependence between the conductivity and the boundary measurements. By using the Carleman estimate, we discuss the partial data problem, which deals with measurements that are taken only in a part of the boundary.     

On the determination of the potential function from given orbits

The paper deals with the problem of finding the field of force that generates a given (N ? 1)-parametric family of orbits for a mechanical system with N degrees of freedom. This problem is usually referred to as the inverse problem of dynamics. We study this problem in relation to the problems of celestial mechanics. We state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov’s problem. We apply the obtained results to generalize the theorem enunciated by Joukovski in 1890, solve the inverse Stäckel problem and solve the problem of constructing the potential-energy function U that is capable of generating a bi-parametric family of orbits for a particle in space. We determine the equations for the sought-for function U and show that on the basis of these equations we can define a system of two linear partial differential equations with respect to U which contains as a particular case the Szebehely equation. We solve completely a special case of the inverse dynamics problem of constructing U that generates a given family of conics known as Bertrand’s problem. At the end we establish the relation between Bertrand’s problem and the solutions to the Heun differential equation. We illustrate our results by several examples.     

Ben's consolidation of knowledge structures about infinite sets

According to a recently proposed model for processes of abstraction in context, the construction of a new structure is to be followed by a consolidation phase. In this paper, we develop an empirically based, theoretical analysis of consolidation that emerges from a sequence of interviews about the comparison of infinite sets with a talented student. We take for granted that construction has occurred in the first interview and analyze the second one. Our analysis shows that consolidation can be identified by means of the psychological and cognitive characteristics of self-evidence, confidence, immediacy, flexibility and awareness. We also found three modes of thinking conducive to consolidation, one related to problem solving, one to reflective activity and an intermediate one.     

除环上左线性方程组的反问题

推广并改进了实数域上线性方程组的反问题及其一系列结果，解决了除环上左线性方程组更具广泛性的一类反问题，给出了此类反问题有（斜）自共轭解及（半）正定自共轭解的充要条件及其解集结构．     

具有两种流体的轴对称渗流的一个自由边界问题

本文处理带有两种流体的轴对称的一个自由边界问题，其中在渗流区域的上部都是油，下部是水，这是同时取油注水的一个数学模型。下面，我们将用复分析方法求出此自由边界问题的一个解，并证明其解的唯一性。     

Solving Inverse Problems for Hyperbolic Equations via the Regularization Method

In the paper, we first deduce an optimization problem from an inverse problem for a general operator equation and prove that the optimization problem possesses a unique, stable solution that converges to the solution of the original inverse problem, if it exists, as a regularization factor goes to zero. Secondly, we apply the above results to an inverse problem determining the spatially varying coefficients of a second order hyperbolic equation and obtain a necessary condition, which can be used to get an approximate solution to the inverse problem.     

解Biot固结方程的有限元方法

饱和土固结的Biot理论[1]将固结过程作为一个弹性体应力和孔隙流流动的耦合问题,和Terzhigi理论[2]相比,它更能确切地反映固结机理.本文用经典变分原理导得固结问题一般的Biot有限元方程,具有明确的物理意义.这一结果已用来分析巴家咀土坝的固结过程,计算结果和工程实践一致.     

Barndorff-Nielsen and Shephard model: oil hedging with variance swap and option

In this paper the Barndorff-Nielsen and Shephard (BN-S) model is implemented to find an optimal hedging strategy for the oil commodity from the Bakken, a new region of oil extraction that is benefiting from fracking technology. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The results indicate that oil can be optimally hedged with the use of a combination of options and variance swaps. Theoretical results related to the variance process are established and implemented for the analysis of the variance swap. In this paper we also determined the optimal amount of the underlying oil commodity that has to be held for minimizing the hedging error. The model and analysis are used to numerically analyze hedging decisions for managing price risk in Bakken oil commodities. From the numerical results, a number of important features of the usefulness of the Barndorff-Nielsen and Shephard model are illustrated.

     

Non-linear analysis of the consolidation of an elastic saturated soil with incompressible fluid and variable permeability by FEM

Research interest in the mechanical behaviour of soils is growing as a result of an increasing number of geomechanical problems involving consolidation effects. The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid and variable permeability. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the convergence of the method using a technique based on the proof of solution’s existence. Finally, we then solved this constitutive model by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data. Therefore, the model can be used for quantitative predictions of consolidation behaviour of soils with permeability dependent on the settlement.     

Asymptotic analysis of the growth of cake layers in filters

The problem of fluid flow in a two-dimensional pleated filteris considered. Of particular interest is the change in the flowdue to cake build-up on the surface of the filter material.The flow is taken to be Darcy flow in the cake and the filtermaterial, with Stokes' flow outside the cake. The particlesin the flow are taken to be transported with the flow and tostick to the cake without slippage or resuspension, and thecake is taken to be incompressible. The flow is considered invarious geometries, particularly long thin filters and corners.The main parameter in the problem is the ratio of the filter-materialresistance to the cake resistance, and limiting cases are considered.Travelling waves of cake build-up are found for arbitrary time-dependentvariations in the inflow conditions. The time taken for thefilter to become clogged by the cake is also considered.     

A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.     

A global in time existence and uniqueness result for an integrodifferential hyperbolic inverse problem with memory effect

We consider an inverse problem for a one-dimensional integrodifferential hyperbolic system, which comes from a simplified model of thermoelasticity. This inverse problem aims to identify the displacement u, the temperature η and the memory kernel k simultaneously from the weighted measurement data of temperature. By using the fixed point theorem in suitable Sobolev spaces, the global in time existence and uniqueness results of this inverse problem are obtained. Moreover, we prove that the solution to this inverse problem depends continuously on the noisy data in suitable Sobolev spaces. For this nonlinear inverse problem, our theoretical results guarantee the solvability for the proposed physical model and the well-posedness for small measurement time τ, which is quite different from general inverse problems.