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SPARSE GRID STOCHASTIC COLLOCATION METHOD FOR STOCHASTIC BURGERS EQUATION
이형천,남윤 대한수학회 2017 대한수학회지 Vol.54 No.1
We investigate an efficient approximation of solution to stochastic Burgers equation driven by an additive space-time noise. We discuss existence and uniqueness of a solution through the Orstein-Uhlenbeck (OU) process. To approximate the OU process, we introduce the \linebreak Karhunen-Lo{\`{e}}ve expansion, and sparse grid stochastic collocation method. About spatial discretization of Burgers equation, two separate finite element approximations are presented: the conventional Galerkin method and Galerkin-conservation method. Numerical experiments are provided to demonstrate the efficacy of schemes mentioned above.
A cell boundary element method for a flux control problem
전영목,이형천 대한수학회 2013 대한수학회지 Vol.50 No.1
We consider a distributed optimal flux control problem: finding the potential of which gradient approximates the target vector field under an elliptic constraint. Introducing the Lagrange multiplier and a change of variables the Euler-Lagrange equation turns into a coupled equation of an elliptic equation and a reaction diffusion equation. The change of variables reduces iteration steps dramatically when the Gauss-Seidel iteration is considered as a solution method. For the elliptic equation solver we consider the Cell Boundary Element(CBE) method, which is the finite element type flux preserving methods.
최영미,김상동,이형천 대한수학회 2009 대한수학회지 Vol.46 No.5
First-order least-squares method of a distributed optimal control problem for the incompressible Stokes equations is considered. An optimality system for the optimal solution are reformulated to the equivalent first-order system by introducing the vorticity and then the least-squares functional corresponding to the system is defined in terms of the sum of the squared H^{-1} and L^2 norms of the residual equations of the system. Finite element approximations are studied and optimal error estimates are obtained. Resulting linear system of the optimality system is symmetric and positive definite. The V-cycle multigrid method is applied to the system to test computational efficiency. First-order least-squares method of a distributed optimal control problem for the incompressible Stokes equations is considered. An optimality system for the optimal solution are reformulated to the equivalent first-order system by introducing the vorticity and then the least-squares functional corresponding to the system is defined in terms of the sum of the squared H^{-1} and L^2 norms of the residual equations of the system. Finite element approximations are studied and optimal error estimates are obtained. Resulting linear system of the optimality system is symmetric and positive definite. The V-cycle multigrid method is applied to the system to test computational efficiency.
A MULTIGRID METHOD FOR AN OPTIMAL CONTROL PROBLEM OF A DIFFUSION-CONVECTION EQUATION
백헌기,김상동,이형천 대한수학회 2010 대한수학회지 Vol.47 No.1
In this article, an optimal control problem associated with convection-diffusion equation is considered. Using Lagrange multiplier,the optimality system is obtained. The derived optimal system becomes coupled, non-symmetric partial differential equations. For discretizations and implementations, the finite element multigrid V -cycle is employed. The convergence analysis of finite element multigrid methods for the derived optimal system is shown. Some numerical simulations are performed.
Finite element approximations of the optimal control problems for stochastic Stokes equations
최영미,김수현,이형천 대한수학회 2014 대한수학회보 Vol.51 No.3
Finite element approximation solutions of the optimal control problems for stochastic Stokes equations with the forcing term perturbed by white noise are considered. Error estimates are established for the fully coupled optimality system using Brezzi-Rappaz-Raviart theory. Numerical examples are also presented to examine our theoretical results.
Convergence of the Newton's method for an optimal control problems for Navier-Stokes equations
최영미,김상동,이형천 대한수학회 2011 대한수학회보 Vol.48 No.5
We consider the Newton's method for an direct solver of the optimal control problems of the Navier-Stokes equations. We show that the finite element solutions of the optimal control problem for Stokes equations may be chosen as the initial guess for the quadratic convergence of Newton's algorithm applied to the optimal control problem for the Navier-Stokes equations provided there are sufficiently small mesh size h and the moderate Reynold's number.
연구논문 : 악취 배출구의 합리적인 배출규제를 위한 사례연구
박정호 ( Jeong Ho Park ),이형천 ( Hyung Chun Lee ) 한국환경과학회 2016 한국환경과학회지 Vol.25 No.1
In this study, field experiment, odor simulator, and dispersion modeling were used to evaluate the odor impact from J sewage sludge treatment facility. The height and flow rate of exhaust stack at this facility were 22.3 m and 100 Nm^{3}/min. The mean odor concentrations of the wet scrubber inlet and exhaust stack were 267±160 and 93±44 OU/m^{3}, respectively. The odor removal efficiency of wet scrubber showed 65%. The odor simulator is used for the regulated standard calculation of the exhaust pipe(stack). Resulting odor emission rate(OER) by odor simulator was 2.4×106(24,000 OU/m^{3}). The forecasting result by Screen3 modeling showed that odor exhaust concentration up to 30,000 OU/m^{3} was’t exceeded maximum allowable emission level on site boundary(15 OU/m^{3}).