超声层析成像的迭代重建

利用变形Born迭代方法,建立了超声衍射重建算法。在迭代过程中,为了解决超声逆散射问题中的非线性性,需要反复地求解前向散射方程和逆散射方程,以达到全场和未知函数的近似,较好地重建物体内部的断层图象。由于逆散射方程是一个不适定性的方程组,要用正则化方法处理方程的不适定性问题,使迭代方法收敛于问题的真实解,才能成功地应用于较高对比度物体的图象重建问题。用Picard准则对不适定问题进行了分析,给出了通过简单图形．确定模型受噪声污染情况以及正则化方法适用范围的方法。在重建实验中。对建立的图像重建算法进行了实验仿真。达到了较好的效果。     

Asymptotic stability of a stochastic delay equation

The investigation of a nonlinear stochastic delay equation with structural tool and regenerative time delays is presented. The conditions of Hopf bifurcation are computed in order to describe the regions of stability and instability. Explicit expressions characterizing the influence of nonlinear and stochastic perturbations, valid in the first order centre manifold approximation, are derived. In addition to this, we describe the underlying mathematical ideas of the centre manifold reduction of delay differential equations to ordinary differential equations for fixed time delays.     

Ill-posed and well-posed problems in inverse elastodynamic scattering for nondestructive evaluation

The practical significance of ill-posedness in a data reduction problem is reviewed. Inverse elastodynamic scattering is shown to be ill-posed in general, although suitably restricted problems may be well-posed. These results underscore the need to analyze carefully the errors of data reduction problems in NDE, and to focus attention on final results of an NDE exercise, rather than on intermediate steps.     

一类非线性反应扩散方程的有限维反馈控制

本文研究了一类反应扩散方程的有限维反馈控制问题,用惯性流形理论,在按连续范数逼近的意义下,把一个受控系统控制到一个预先给定的有限维目标系统。     

Dynamics of a Modified Predator-Prey System to Allow for a Functional Response and Time Delay

A modified predator-prey system described by two differential equations and an algebraic equation is discussed. Formulae for determining the direction of a Hopf bifurcation and the stability of the bifurcating periodic solutions are derived differential-algebraic system theory, bifurcation theory and centre manifold theory. Numerical simulations illustrate the results, which includes quite complex dynamical behaviour.     

A comparison of boundary element method formulations for steady state anisotropic heat conduction problems

In this paper the boundary element method (BEM) is numerically implemented in order to solve steady state anisotropic heat conduction problems. Various types of elements, namely, constant elements, continuous and discontinuous linear elements and continuous and discontinuous quadratic elements are used. The performances of these various BEM formulations are compared for both the direct well-posed Dirichlet problem and the inverse ill-posed Cauchy problem, revealing several features of the BEM. Furthermore, previously undetermined analytical solutions for the integrals associated with linear and quadratic elements are presented.     

不适定热传导反问题的两种求解方法及其比较

本文以不适定热传导反问题为对象，采用两种方法进行了求解。一种方法基于对具有测量误差的边界条件进行适当的微扰，使之化为适定问题；另一种方法基于Ｔｉｋｈｏｎｏｖ的正则平滑思想，对反问题中的输入数据进行平滑处理，以便使函数及其一阶导数均实现一致逼近。通过计算与求解表明，两种方法均能得到具有一定精度与稳定性的结果，其中以正则化法更为理想     

数值求解Symm积分方程的正则化MINRES方法

Symm积分方程在位势理论中具有重要应用,它是Hadamard意义下的不适定问题。离散该方程将产生对称线性不适定系统。基于GCV准则,并应用截断奇异值分解,本文提出数值求解Symm积分方程的正则化MINRES方法。与Tikhonov正则化方法相比,在数据出现噪声的情况下,新方法能有效地求得Symm积分方程的数值解。     

Investigation of the ABC flow instability with application of centre manifold reduction

We demonstrate that application of a generalized centre manifold is advantageous when low-dimensional reductions of continuous dynamical systems of hydrodynamic type are considered. The centre manifold theorem that we use differs from the standard one in the following way. We define a centre manifold as an invariant manifold, tangent to the eigenspace of the linearization of the mapping defining a dynamical system, where the eigenspace is associated with eigenvalues with small but not necessarily zero real parts (usually only eigenvalues with zero real parts are employed). As a result, several bifurcations following the onset of instability of the trivial steady state are reproduced in the eight-dimensional system that we obtain, whilst in the six-dimensional system constructed with application of the conventional centre manifold theorem only the first bifurcation of the steady state is reproduced correctly.     

Solution of the Retrospective Inverse Heat Conduction Problem with Parametric Optimization

The retrospective inverse heat conduction problem has been solved as a problem of optimal control of an object with distributed parameters. The initial ill-posed statement of the inverse problem is transformed into a conditionally well-posed one when the limitations imposed on the second derivative of the desired control, which corresponds to a reduction of the set of control actions to the class of continuous and continuously differentiable functions, are taken into account. Preliminary parameterization of the control actions makes it possible to formulate a mathematical programming problem, which can be solved based on the analytical method of minimax optimization with alternance specific features of the desired optimal temperature deviations.     

Group preserving scheme for the Cauchy problem of the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation by group preserving scheme (GPS) which is an ill-posed problem, because the solution does not depend continuously on the data. For this, the Laplace equation, by using a semi-discretization method namely method of line, is converted to an ODEs system and then obtained ODEs system is considered by GPS. Stability of GPS for ill-posed Laplace equation is shown. The problem numerical results show the efficiency and power of this method.     

Rate-dependent control strategies stabilize limb forces during human locomotion

A spring-mass model accurately predicts centre of mass dynamics for hopping and running animals and is pervasive throughout experimental and theoretical studies of legged locomotion. Given the neuromechanical complexity of the leg, it remains unclear how joint dynamics are selected to achieve such simple centre of mass movements consistently from step to step and across changing conditions. Human hopping is a tractable experimental model to study how net muscle moments, or joint torques, are coordinated for spring-mass dynamics, which include stable, or invariant, vertical ground forces. Subjects were equally able to stabilize vertical forces at all hopping frequencies (2.2, 2.8, 3.2 Hz) by selecting force-equivalent joint torque combinations. Using a hybrid-uncontrolled manifold permutation analysis, however, we discovered that force stabilization relies less on interjoint coordination at greater hopping frequencies and more on selection of appropriate ankle joint torques. We conclude that control strategies for selecting the joint torques that stabilize forces generated on the ground are adjusted to the rate of movement. Moreover, this indicates that legged locomotion may involve the differential regulation of several redundant motor control strategies that are accessed as needed to match changing environmental conditions.     

Invariant manifolds describing the dynamics of a hyperbolic-parabolic equation from nonlinear viscoelasticity

The governing equations for a collection of dynamical problems for heavy rigid attachments carried by light, deformable, nonlinearly viscoelastic bodies are studied. These equations are a discretization of a nonlinear hyperbolic-parabolic partial differential equation coupled to a dynamical boundary condition. A small parameter measuring the ratio of the mass of the deformable body to the mass of the rigid attachment is introduced, and geometric singular perturbation theory is applied to reduce the dynamics to the dynamics of the slow system. Fenichel theory is then applied to the regular perturbation of the slow system to prove the existence of a low-dimensional invariant manifold within the dynamics of the high-dimensional discretization.     

Probabilistic nonconvex constrained optimization with fixed number of function evaluations

A methodology is proposed for the efficient solution of probabilistic nonconvex constrained optimization problems with uncertain. Statistical properties of the underlying stochastic generator are characterized from an initial statistical sample of function evaluations. A diffusion manifold over the initial set of data points is first identified and an associated basis computed. The joint probability density function of this initial set is estimated using a kernel density model and an Itô stochastic differential equation (ISDE) constructed with this model as its invariant measure. This ISDE is adapted to fluctuate around the manifold yielding additional joint realizations of the uncertain parameters, design variables, and function values, which are obtained as solutions of the ISDE. The expectations in the objective function and constraints are then accurately evaluated without performing additional function evaluations. The methodology brings together novel ideas from manifold learning and stochastic Hamiltonian dynamics to tackle an outstanding challenge in stochastic optimization. Three examples are presented to highlight different aspects of the proposed methodology.     

Regularizing active set method for retrieval of the atmospheric aerosol particle size distribution function

The determination of the aerosol particle size distribution function by using the particle spectrum extinction equation is an ill-posed integral equation of the first kind [S. Twomey, J. Comput. Phys.18, 188 (1975);Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Education Press, 2007)], since we are often faced with limited or insufficient observations in remote sensing and the observations are contaminated. To overcome the ill-posed nature of the problem, regularization techniques were developed. However, most of the literature focuses on the application of Phillips-Twomey regularization and its variants, which are unstable in several cases. As is known, the particle size distribution is always nonnegative, and we are often faced with incomplete data. Therefore, we study the active set method and propose a regularizing active set algorithm for ill-posed particle size distribution function retrieval and for enforcing nonnegativity in computation. Our numerical tests are based on synthetic data for theoretical simulations and the field data obtained with a CE 318 Sun photometer for the Po Yang lake region of Jiang Xi Province, China, and are performed to show the efficiency and feasibility of the proposed algorithms.     

Numerical solution of two-dimensional inverse force function in the wave equation with nonlocal boundary conditions

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse time-dependent force function in the wave equation on regular and irregular domains. The SMRPI is developed for identifying the force function which satisfies in the wave equation subject to the integral overspecification over a portion of the spatial domain or to the overspecification at a point in the spatial domain. This method is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system. Three numerical examples are tested to show that numerical results are accurate for exact data and stable with noisy data.     

Dynamic estimation of the modeling error statistics in Diffuse Optical Tomography

Diffuse Optical Tomography (DOT) is a non-invasive imaging technique that suffers from a typical large-scale and ill-posed inverse problem with low spatial resolution. In DOT, the inverse problem is computationally intensive and decreasing the computation complexity and making it well-posed is the one of the most challenging research areas. More precisely, one of the well-known complexity reduction techniques is defined as applying modelling error originated from discretization of forward problem. Applying the discretization error in Bayesian inference has already been discussed; the method in which the likelihood is modified by an off-line prior density estimation. This paper implements a new method to enhance the modelling error approach using an iterative scheme to update statistical parameters of modelling discrepancy in DOT. The algorithm is very similar to Ensemble Kalman Filter. Moreover, the reconstruction process in the applied method is conducted by a small sample size rather than off-line method. Hence, the computation complexity is decreased and the algorithm converges in few iterations. The efficiency of the proposed method is illustrated by simulations.     

A regularization method for the approximate particular solution of nonhomogeneous Cauchy problems of elliptic partial differential equations with variable coefficients

Radial basis functions (RBFs) have proved to be very flexible in representing functions. Based on the idea of the analog equation method and radial basis functions, in this paper, ill-posed Cauchy problems of elliptic partial differential equations (PDEs) with variable coefficients are considered for the first time using the method of approximate particular solutions (MAPS). We show that, using the Tikhonov regularization, the MAPS results an effective and accurate numerical algorithm for elliptic PDEs and irregular solution domains. Comparing the proposed MAPS with Kansa's method, numerical results show that the proposed MAPS is effective, accurate and stable to solve the ill-posed Cauchy problems.     

Slow dynamics simulation of power systems

This paper investigates the simulation of slow dynamics in two-time-scale power systems. A new approach is proposed to obtain the slow dynamics by projecting the trajectory of the post-fault system onto its slow manifold. This is achieved by a nonlinear projection of the full order system initial condition onto the slow manifold, such that the fast intraarea dynamics are not excited. A projection scheme is developed and applied to two-test power systems.     

Conjugate Gradient Method for Estimation of Robin Coefficients

We consider a Robin inverse problem associated with the Laplace equation, which is a severely ill-posed and nonlinear. We formulate the problem as a boundary integral equation, and introduce a functional of the Robin coefficient as a regularisation term. A conjugate gradient method is proposed for solving the consequent regularised nonlinear least squares problem. Numerical examples are presented to illustrate the effectiveness of the proposed method.