Eun Jae Park

Optimal harvesting for periodic age-dependent population dynamics

Here we investigate the optimal harvesting problem for some periodic age-dependent population dynamics; namely, we consider the linear Lotka--McKendrick model with periodic vital rates and a periodic forcing term that sustains oscillations. Existence and uniqueness of a positive periodic solution are demonstrated and the existence and uniqueness of the optimal control are established. We also state necessary optimality conditions. A numerical algorithm is developed to approximate the optimal control and the optimal harvest. Some numerical results are presented.

Static conductivity imaging using variational gradient Bz algorithm in magnetic resonance electrical impedance tomography

A new image reconstruction algorithm is proposed to visualize static conductivity images of a subject in magnetic resonance electrical impedance tomography (MREIT). Injecting electrical current into the subject through surface electrodes, we can measure the induced internal magnetic flux density B = (Bx, By, Bz) using an MRI scanner. In this paper, we assume that only the z-component Bz is measurable due to a practical limitation of the measurement technique in MREIT. Under this circumstance, a constructive MREIT imaging technique called the harmonic Bz algorithm was recently developed to produce high-resolution conductivity images. The algorithm is based on the relation between inverted delta2Bz and the conductivity requiring the computation of inverted delta2Bz. Since twice differentiations of noisy Bz data tend to amplify the noise, the performance of the harmonic Bz algorithm is deteriorated when the signal-to-noise ratio in measured Bz data is not high enough. Therefore, it is highly desirable to develop a new algorithm reducing the number of differentiations. In this work, we propose the variational gradient Bz algorithm where Bz is differentiated only once. Numerical simulations with added random noise confirmed its ability to reconstruct static conductivity images in MREIT. We also found that it outperforms the harmonic Bz algorithm in terms of noise tolerance. From a careful analysis of the performance of the variational gradient Bz algorithm, we suggest several methods to further improve the image quality including a better choice of basis functions, regularization technique and multilevel approach. The proposed variational framework utilizing only Bz will lead to different versions of improved algorithms.

Mixed finite element methods for nonlinear second-order elliptic problems

Mixed finite element methods are developed to approximate the solution of the Dirichlet problem for the most general quasi-linear second-order elliptic operator in divergence form. Existence and uniqueness of the approximation are proved, and optimal error estimates in

A mixed finite element method for a strongly nonlinear second-order elliptic problem

L^2

An upwind scheme for a nonlinear model in age-structured population dynamics

are demonstrated for both the scalar and vector functions approximated by the method. Error estimates are also derived in

Splitting methods for the numerical approximation of some models of age-structured population dynamics and epidemiology

L^q

A Priori and A Posteriori Pseudostress-velocity Mixed Finite Element Error Analysis for the Stokes Problem

,

A Hybrid Discontinuous Galerkin Method for Elliptic Problems

2 \leq q \leq + \infty

Fully discrete mixed finite element approximations for non-Darcy flows in porous media

. Newton’s method is presented and analyzed to solve the nonlinear algebraic equations.

A multiscale mortar mixed finite element method for slightly compressible flows in porous media

The approximation of the solution of the first boundary value problem for a strongly nonlinear second-order elliptic problem in divergence form by the mixed finite element method is considered. Existence and uniqueness of the approximation are proved and optimal error estimates in L 2 are established for both the scalar and vector functions approximated by the method. Error estimates are also derived in L q , 2 ≤ q ≤ +∞.