Estimation Of Differential Equations Research Articles

Domain decomposition modified with characteristic mixed finite element of compressible oil-water seepage displacement and its numerical analysis

Abstract A parallel algorithm, structured by domain decomposition and characteristic mixed finite element, is presented for solving three-dimensional displacement problem of compressible seepage flow. Decomposing the computational domain into several subdomains, we define a characteristic function to approximate the value on interior boundary at previous time level and obtain numerical solutions implicitly in subdomains in parallel. The flow equation is treated by the method of mixed finite element and the saturation equation is approximated by the method of characteristic finite element. For a model problem we apply variation form, domain decomposition, the method of characteristics, the principle of energy, negative norm estimates, induction hypothesis, the theory and technique of priori estimates of differential equations to derive optimal error estimate in l2 norm. Numerical data are consistent with theoretical analysis and show that this method is effective in solving actual applications. Then it can solve the well-known problem.

A Modified Characteristics-mixed Finite Element for Semiconductor Device of Heat Conduction and Numerical Analysis

Numerical simulation of a three-dimensional semiconductor device of heat conduction is a fundamental problem in modern information science. The mathematical model is formulated by a nonlinear system of initial-boundary problem, which is interpreted by four partial differential equations: an elliptic equation for electrostatic potential, two convection-diffusion equations for electron concentration and hole concentration, a heat conduction equation for temperature. The electrostatic potential appears within the latter three equations, and the electric field strength controls the concentrations and the temperature. The electric field potential is solved by a mixed finite element method, and the electric field strength is obtained simultaneously. The first order of the accuracy is improved for the latter. The concentrations and temperature are computed by the characteristics-finite element method, where the characteristic approximation is adopted for the hyperbolic term and finite element method is use to treat the diffusion. The composite computational scheme can solve the convection-dominated diffusion equations well because it can cancel numerical dispersion and nonphysical oscillation. The temperature is computed by finite element method, and an interesting simulation tool is proposed for solving semiconductor device problem numerically. By using the technique of a priori estimates of differential equations, an optimal order error estimates is obtained. A theoretical work is shown for numerical simulation of information science, and the actual problem is solved well.

Convergence Analysis for Mixed Finite Element Method of Positive Semi-definite Problems

A mixed element-characteristic finite element method is put forward to approximate three-dimensional incompressible miscible positive semi-definite displacement problems in porous media. The mathematical model is formulated by a nonlinear partial differential system. The flow equation is approximated by a mixed element scheme, and the pressure and Darcy velocity are computed at the same time. The concentration equation is treated by the method of characteristic finite element, where the convection term is discretized along the characteristics and the diffusion term is computed by the scheme of finite element. The method of characteristics can confirm strong computation stability at the sharp fronts and avoid numerical dispersion and nonphysical oscillation. Furthermore, a large step is adopted while small time truncation error and high order accuracy are obtained. It is an important feature in numerical simulation of seepage mechanics that the mixed volume element can compute the pressure and Darcy velocity simultaneously and the accuracy of Darcy velocity is improved one order. Using the form of variation, energy method, $L^2$ projection and the technique of priori estimates of differential equations, we show convergence analysis for positive semi-definite problems. Then a powerfu tool is given to solve international famous problems.

Domain Decomposition Modified with Characteristic Mixed Finite Element and Numerical Analysis for Three-Dimensional Slightly Compressible Oil-Water Seepage Displacement

A parallel algorithm is presented to solve three-dimensional slightly compressible seepage displacement where domain decomposition and characteristics-mixed finite element are combined. Decomposing the computational domain into several subdomains, we define a special function to approximate the derivative at interior boundary explicitly and obtain numerical solutions of the saturation implicitly on subdomains in parallel. The method of characteristics can confirm strong stability at the fronts, and can avoid numerical dispersion and nonphysical oscillation. It can adopt large-time step but can obtain small time truncation error. So a characteristic domain decomposition finite element scheme is put forward to compute the saturation. The flow equation is computed by the method of mixed finite element and numerical accuracy of Darcy velocity is improved one order. For a model problem we apply some techniques such as variation form, domain decomposition, the method of characteristics, the principle of energy, negative norm estimates, induction hypothesis, and the theory of priori estimates of differential equations to derive optimal error estimate in $l^2$ norm. Numerical example is given to testify theoretical analysis and numerical data show that this method is effective in solving actual applications. Then it can solve the well-known problem.

Numerical questions in ODE boundary value problems

Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di-stability) in problem discretizations. However, computational categories are not precise, and qualitative descriptors such as `of moderate size' cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive Newton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate boundary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood. References U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical solution of boundary value problems for ordinary differential equations , SIAM, Philadelphia, 1995. J. C. Butcher, Numerical methods for ordinary differential equations , John Wiley and Sons, 2003. G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33--53. G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), 27--43. F. R. de Hoog and R. M. M. Mattheij, On dichotomy and well-conditioning in BVP, SIAM J. Numer. Anal. 24 (1987), 89--105. P. Deuflhard, Newton methods for nonlinear problems , Springer--Verlag, Berlin Heidelberg, 2004. R. England and R. M. M. Mattheij, Boundary value problems and dichotomic stability, SIAM J. Numer. Anal. 25 (1988), 1037--1054. Z. Li, M. R. Osborne, and T. Prvan, Parameter estimation of ordinary differential equations, IMA J. Numer. Anal. 25 (2005), 264--285. J. Nocedal and S. J. Wright, Numerical optimization , Springer--Verlag, 1999. M. R. Osborne, On shooting methods for boundary value problems, J. Math. Analysis and Applic. 27 (1969), 417--433. M. R. Osborne, Cyclic reduction, dichotomy, and the estimation of differential equations, J. Comp. and Appl. Math. 86 (1997), 271--286. B. J. Quinn and E. J. Hannan, The estimation and tracking of frequency , Cambridge University Press, Cambridge, United Kingdom, 2001. J. O. Ramsay, G. Hooker, C. Cao, and C. Campbell, Estimating differential equations , preprint, Department of Psychology, McGill University, Montreal, Canada, 2005, p. 40.

Numerical questions in ODE boundary value problems

Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di-stability) in problem discretizations. However, computational categories are not precise, and qualitative descriptors such as `of moderate size' cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive Newton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate boundary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood. References U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical solution of boundary value problems for ordinary differential equations, SIAM, Philadelphia, 1995. J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley and Sons, 2003. G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33--53. G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), 27--43. F. R. de Hoog and R. M. M. Mattheij, On dichotomy and well-conditioning in BVP, SIAM J. Numer. Anal. 24 (1987), 89--105. P. Deuflhard, Newton methods for nonlinear problems, Springer--Verlag, Berlin Heidelberg, 2004. R. England and R. M. M. Mattheij, Boundary value problems and dichotomic stability, SIAM J. Numer. Anal. 25 (1988), 1037--1054. Z. Li, M. R. Osborne, and T. Prvan, Parameter estimation of ordinary differential equations, IMA J. Numer. Anal. 25 (2005), 264--285. J. Nocedal and S. J. Wright, Numerical optimization, Springer--Verlag, 1999. M. R. Osborne, On shooting methods for boundary value problems, J. Math. Analysis and Applic. 27 (1969), 417--433. M. R. Osborne, Cyclic reduction, dichotomy, and the estimation of differential equations, J. Comp. and Appl. Math. 86 (1997), 271--286. B. J. Quinn and E. J. Hannan, The estimation and tracking of frequency, Cambridge University Press, Cambridge, United Kingdom, 2001. J. O. Ramsay, G. Hooker, C. Cao, and C. Campbell, Estimating differential equations, preprint, Department of Psychology, McGill University, Montreal, Canada, 2005, p. 40.

Report on the session "Estimation of Differential Equations and Stochastic Processes: Methods and Applications"

Report on the session "Estimation of Differential Equations and Stochastic Processes: Methods and Applications"

Cyclic reduction, dichotomy, and the estimation of differential equations

In the estimation of ordinary differential equations given observed data it is necessary to parametrize the solutions in a computationally tractable way in order to make comparisons with the given data. Typically, this is done by adjoining initial or boundary conditions, and this requires additional information on the solution structure in order to do this in a manner that leads to stable computation of the comparison solutions. Here, an alternative approach is described in which cyclic reduction is used to reduce the estimation problem to an optimization problem subject to a fixed number of equality constraints. If orthogonal transformations are used in the cyclic reduction process then it appears that stable computations are possible without the need for the structural information needed to devise the stable imbeddings. The aim of this paper is to provide evidence in support of this claim. In particular, it is shown that the cyclic reduction process is linked to a new family of representation of the solutions of the ODE system. Some properties of members of this family are described. These provide insight into the particular advantages of the orthogonal reduction form of cyclic reduction.