Modifications of Steepest Descent Method and Conjugate Gradient Method Against Noise for Ill-posed Linear Systems

It is well known that the numerical algorithms of the steepest descent method (SDM), and the conjugate gradient method (CGM) are effective for solving well-posed linear systems. However, they are vulnerable to noisy disturbance for solving ill-posed linear systems. We propose the modifications of SDM and CGM, namely the modified steepest descent method (MSDM), and the modified conjugate gradient method (MCGM). The starting point is an invariant manifold defined in terms of a minimum functional and a fictitious time-like variable; however, in the final stage we can derive a purely iterative algorithm including an acceleration parameter. Through the Hopf bifurcation, this parameter indeed plays a major role to switch the situation of slow convergence to a new situation that the functional is stepwisely decreased very fast. Several numerical examples are examined and compared with exact solutions, revealing that the new algorithms of MSDM and MCGM have good computational efficiency and accuracy, even for the highly ill-conditioned linear equations system with a large noise being imposed on the given data

A Fictitious Time Integration Method for a Quasilinear Elliptic Boundary Value Problem, Defined in an Arbitrary Plane Domain

Motivated by the evolutionary and dissipative properties of parabolic type partial differential equation (PDE), Liu (2008a) has proposed a natural and mathematically equivalent approach by transforming the quasilinear elliptic PDE into a parabolic one. However, the above paper only considered a rectangular domain in the plane, and did not treat the difficulty arisen from the quasilinear PDE defined in an arbitrary plane domain. In this paper we propose a new technique of internal and boundary residuals in a fictitious rectangular domain, which are driving forces for the ordinary differential equations based on the Fictitious Time Integration Method (FTIM). Several numerical examples validate the performance of the FTIM, which can easily handle the nonlinear boundary value problem, defined in an arbitrary plane domain

A Sliding Mode Control Algorithm for Solving an Ill-posed Positive Linear System

For the numerical solution of an ill-posed positive linear system we combine the methods from invariant manifold theory and sliding mode control theory, developing an affine nonlinear dynamical system with a positive control force and with the residual vector as being a gain vector. This system is proven asymptotically stable to the zero residual vector by using an argument from the Lyapunov stability theory. We find that the system fast tends to the sliding surface and then moves with a sliding mode, such that the resultant sliding mode control algorithm (SMCA) is robust against large noise and stable to find the numerical solution of an ill-posed linear system. It is interesting that even under a random noise with an intensity 10-5 we can obtain a quite accurate solution of the linear Hilbert problem with dimension n = 500. For this highly ill-conditioned problem the number of iterations is still smaller than 100. Numerical tests, including the inverse problems of backward heat conduction problem and Cauchy problems, confirm that the present SMCA has superior computational efficiency and accuracy even for a highly ill-conditioned linear equations system under a large noise

The Solution of SO

In many applications we need to solve an orthogonal transformation tensor Q∈SO(3) from a tensorial equation Q˙ = WQ under a given spin history W. In this paper, we address some interesting issues about this equation. A general solution of Q is obtained by transforming the governing equation into a new one in the space of ℝP3. Then, we develop a novel method to solve Q in terms of a single parameter, whose governing equation is a single nonlinear ordinary differential equation (ODE)

A Lie-Group Adaptive Method for Imaging a Space-Dependent Rigidity Coefficient in an Inverse Scattering Problem of Wave Propagation

We are concerned with the reconstruction of an unknown space-dependent rigidity coefficient in a wave equation. This problem is known as one of the inverse scattering problems. Based on a two-point Lie-group equation we develop a Lie-group adaptive method (LGAM) to solve this inverse scattering problem through iterations, which possesses a special character that by using onlytwo boundary conditions and two initial conditions, as those used in the direct problem, we can effectively reconstruct the unknown rigidity function by aself-adaption between the local in time differential governing equation and the global in time algebraic Lie-group equation. The accuracy and efficiency of the present LGAM are assessed by comparing the imaged results with some postulated exact solutions. By means of LGAM, it is quite versatile to handle the wave inverse scattering problem for the image of the rigidity coefficient without needing any extra information from the wave motion

The Lie-Group Shooting Method for Thermal Stress Evaluation Through an Internal Temperature Measurement

In the present work we study numerical computations of inverse thermal stress problems. The unknown boundary conditions of an elastically deformable heat conducting rod are not given a priori and are not allowed to measure directly, because the boundary may be not accessible to measure. However, an internal measurement of temperature is available. We treat this inverse problem by using a semi-discretization technique, of which the time domain is divided into many sub-intervals and the physical quantities are discretized at these node points of discrete times. Then the resulting ordinary differential equations in the discretized space are numerically integrated towards the spatial direction by the Lie-group shooting method to find unknown boundary conditions. The key point is based on one-step Lie group elements: G(r) = G(y0,yl). We are able to search missing boundary conditions through a minimum discrepancy from the targets in terms of a weighting factor r ∈ (0,1). Several numerical examples were worked out to persuade that this novel approach has good efficiency and accuracy. Although the measured temperature is disturbed by large noise, the Lie group shooting method is stable to recover the boundary conditions very well

A Fictitious Time Integration Method for the Burgers Equation

When the given input data are corrupted by an intensive noise, most numerical methods may fail to produce acceptable numerical solutions. Here, we propose a new numerical scheme for solving the Burgers equation forward in time and backward in time. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ )u(x,t) =: v(x,t,τ), such that the original Burgers equation is written as a new parabolic type partial differential equation in the space of (x,t,τ). A fictitious damping coefficient can be used to strengthen the stability in the numerical integration of a semi-discretized ordinary differential equations set on the spatial-temporal grid points. Even for a very large final time and under a large noise, the present Fictitious Time Integration Method (FTIM) can be used to retrieve the initial data very well. When the FTIM is used to solve the direct problems of Burgers equation, with a large Reynolds number and the input data being noised seriously, we can still reconstruct the solution rather accurately. This result however cannot be achieved by other conventional numerical methods. It is interesting that both the forward and backward problems of Burgers equation can be unifiedly treated by the FTIM

An LGDAE Method to Solve Nonlinear Cauchy Problem Without Initial Temperature

We recover an unknown initial temperature for a nonlinear heat conduction equation ut(x,t) = uxx(x,t) + H(x,t,u,ux), under the Cauchy boundary conditions specified on the left-boundary. The method in the present paper transforms the Cauchy problem into an inverse heat source problem to find F(x) in Tt(x,t) = Txx(x,t) + H + F(x). By using the GL(N,R) Lie-group differential algebraic equations (LGDAE) algorithm to integrate the numerical method of lines discretized equations from sideways heat equation, we can fast recover the initial temperature and two boundary conditions on the right-boundary. The accuracy and efficiency are confirmed by comparing the exact solutions with the recovered results, where a large noisy disturbance is imposed on the Cauchy data

A Fictitious Time Integration Method for Solving Delay Ordinary Differential Equations

A new numerical method is proposed for solving the delay ordinary differential equations (DODEs) under multiple time-varying delays or state-dependent delays. The finite difference scheme is used to approximate the ODEs, which together with the initial conditions constitute a system of nonlinear algebraic equations (NAEs). Then, a Fictitious Time Integration Method (FTIM) is used to solve these NAEs. Numerical examples confirm that the present approach is highly accurate and efficient with a fast convergence

An LGEM to Identify Time-Dependent Heat Conductivity Function by an Extra Measurement of Temperature Gradient

We consider an inverse problem for estimating an unknown heat conductivity parameter α(t) in a heat conduction equation Tt(x,t) = α(t)Txx(x,t) with the aid of an extra measurement of temperature gradient on boundary. Basing on an establishment of the one-step Lie-group elements G(r) and G(l) for the semi-discretization of heat conduction equation in time domain, we can derive algebraic equations from G(r) = G(l). The new method, namely the Lie-group estimation method (LGEM), is examined through numerical examples to convince that it is highly accurate and efficient; the maximum estimation error is smaller than 10-5 for smooth parameter and for discontinuous and oscillatory parameter the accuracy is still in the order of 10-2. Although the estimation is carried out under a large measurement noise, the LGEM is also stable