Parallel application of block-iterative methods in medical imaging and radiation therapy

Some row-action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems. Recently developed block-iterative versions of such special-purpose methods enable parallel computation when the underlying problem is appropriately decomposed. This opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning, all in their fully discretized modelling approach. Since there is more than one way of deriving block-iterative versions of any row-action method, the choice has to be made with reference to the underlying real-world problem.This research was done with partial support of National Institutes of Health, Grant HL-28438 while the author was with the Medical Image Processing Group (MIPG) at the Department of Radiology, Hospital of the University of Pennsylvania, Philadelphia, PA, USA.     

Strong-norm error estimates for the projective-difference method for approximately solving abstract parabolic equations

Solutions continuously differentiable with respect to time of parabolic equations in Hilbert space are obtained by the projective-difference method approximately. The discretization of the problem is carried out in the spatial variables using Galerkin's method, and in the time variable using Euler's implicit method. Strong-norm error estimates for approximate solutions are obtained. These estimates not only allow one to establish the convergence of the approximate solutions to the exact ones but also yield numerical characteristics of the rates of convergence. In particular, order-sharp error estimates for finite element subspaces are obtained. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 898–909, March, 1998. Translated by N. K. Kulman     

SMOOTHNESS OF INERTIAL MANIFOLD UNDER TIME DISCRETIZATION

0IntroductionInertialmanifolds(IM)arenewobjectsthathavebeenintroducedinrelationwiththestudyoflargetimebehaviorofnonlineardissipativePDE,see[1].In[2],theyareextendedtodiscretizeddynamicsstemmingfromdissipativePDE.Frommathematicalpointofview,thesearesmootllfiniteLipschitzmanifoldsthatareinviantbytheflowofassociatedsolutions,andattractallthetrajectorieswithexponentialspeed,see[1].Fromphysicalpointofview,IMmodeltheinteractionlawsbetweensmallandlargestructureofaturbulentflow,andrepresentsitspe…     

具有易损坏储备部件复杂可修系统解的半离散化

讨论了易损坏部件对系统的影响,且故障系统的修复时间是任意分布的.并对修复率μi(x)用初等阶梯函数进行逼近,给出了系统的半离散化模型,为进一步的数值计算打下基础.     

带导数记忆项抛物型积分微分方程分数次欧拉时间离散

我们研究一类带导数记忆项抛物型偏积分微分方程欧拉时间离散，记忆项通过Ｌｕｂｉｃｈ建议的分数次卷积求积逼近。使用谱表示技术导出最优阶误差估计。     

Stabilized approximations of strongly continuous semigroups

This paper introduces stabilization techniques for intrinsically unstable, high accuracy rational approximation methods for strongly continuous semigroup. The methods not only stabilize the approximations, but improve their speed of convergence by a magnitude of up to 1/2.     

The boundary integral equation method for the solution of wave-obstacle interaction

The boundary integral equation method constitutes the basis of a number of computer programs used for the solution of wave-obstacle interaction problems. For the case of obstacles in a constant depth fluid, the method assumes that the velocity potential at any point in the fluid may be represented by a distribution of Green's function sources over the immersed surface of the obstacle. Application of the obstacle kinematic boundary condition gives rise to an integral equation which may be solved, using numerical discretization, for the unknown source strength distribution function. Subsequent evaluation of the discretized velocity potential permits evaluation of the hydrodynamic interaction parameters. A series of numerical solutions have been carried out for a range of substantially rectangular obstacles, in a two-dimensional domain, using varying levels of immersed profile discretization. The results, presented in the form of fixed and floating mode wave reflection and transmission, together with the motion response of the floating obstacle, demonstrate the significant sensitivity of the evaluated parameters to variations in the level of discretization.     

HIGH RESOLUTION SCHEMES FOR CONSERVATION LAWS AND CONVECTION-DIFFUSION EQUATIONS WITH VARYING TIME AND SPACE GRIDS

This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection-diffusion equations, by projecting the solution increments of the underlying partial differential equations （PDE） at each local time step. The main advantages are that they are of good consistency, and it is convenient to implement them. The schemes are L^∞ stable, satisfy a cell entropy inequality, and may be extended to the initial boundary value problem of general unsteady PDEs with higher-order spatial derivatives. The high resolution schemes are given by combining the reconstruction technique with a second order TVD Runge-Kutta scheme or a Lax-Wendroff type method, respectively. The schemes are used to solve a linear convection-diffusion equation, the nonlinear inviscid Burgers＇ equation, the one- and two-dimensional compressible Euler equations, and the two-dimensional incompressible Navier-Stokes equations. The numerical results show that the schemes are of higher-order accuracy, and efficient in saving computational cost, especially, for the case of combining the present schemes with the adaptive mesh method [15]. The correct locations of the slow moving or stronger discontinuities are also obtained, although the schemes are slightly nonconservative.     

在有限差分法下一类抛物型方程的离散化

本文对于一类带有狄拉克函数δ０初值的抛物型方程，在有限差分法下进行离散化，证明了其数值解的存在性、唯一性，尤其是它的稳定性．     

Approximate Gradient Projection Method with Runge-Kutta Schemes for Optimal Control Problems

We consider an optimal control problem for systems governed by ordinary differential equations with control constraints. The state equation is discretized by the explicit fourth order Runge-Kutta scheme and the controls are approximated by discontinuous piecewise affine ones. We then propose an approximate gradient projection method that generates sequences of discrete controls and progressively refines the discretization during the iterations. Instead of using the exact discrete directional derivative, which is difficult to calculate, we use an approximate derivative of the cost functional defined by discretizing the continuous adjoint equation by the same Runge-Kutta scheme and the integral involved by Simpson's integration rule, both involving intermediate approximations. The main result is that accumulation points, if they exist, of sequences constructed by this method satisfy the weak necessary conditions for optimality for the continuous problem. Finally, numerical examples are given.