**Room**: Old Main Academic Center 3050

**Chair**: ** Kobra Rabiei**

**Sylvia Amihere,** The University of Alabama

**Time:** 4:00 pm - 4:20 pm (CST)

**Title:** Benchmarking electrostatic free energy of the nonlinear Poisson-Boltzmann model for the Kirkwood sphere

**Abstract:**

Various numerical packages have been developed to solve the Poisson-Boltzmann equation (PBE) for the electrostatic analysis of solvated biomolecules. A common benchmark test for the PBE solvers is the Kirkwood sphere, for which analytical potential and free energy are available for the linearized PBE. However, the Kirkwood sphere does not admit analytical solution for the nonlinear PBE involving a hyperbolic sine term. In this talk, we will introduce a simple numerical approach, so that the energy of the Kirkwood sphere for the nonlinear PBE can be calculated at a very high precision. This provides a new benchmark test for the future developments of nonlinear PBE solvers.

**Yiming Ren, **The University of Alabama

**Time:** 4:20 pm - 4:40 pm (CST)

**Title**: A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains

**Abstract***:*

For elliptic boundary value problems (BVPs) involving irregular domains and Robin boundary condition, no numerical method is known to deliver a fourth order convergence and O(N log N) efficiency, where N stands for the total degree-of-freedom of the system. Based on the matched interface and boundary (MIB) and fast Fourier transform (FFT) schemes, a new finite difference method is introduced for such problems, which involves two main components. First, a ray-casting MIB scheme is proposed to handle different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mix combinations. By enclosing the concerned irregular domain by a large enough cubic domain, the ray-casting MIB scheme generates necessary fictitious values outside the irregular domain by imposing boundary conditions along the normal direction of the boundary, so that a high order central difference discretization of the Laplacian can be formed. Second, an augmented MIB formulation is built, in which Cartesian derivative jumps are reconstructed on the boundary as auxiliary variables. By treating such variables as unknowns, the discrete Laplacian can be efficiently inverted by the FFT algorithm, in the Schur complement solution of the augmented system. The accuracy and efficiency of the proposed augmented MIB method are numerically examined by considering various elliptic BVPs in two and three dimensions. Numerical results indicate that the new algorithm not only achieves a fourth order of accuracy in treating irregular domains and complex boundary conditions, but also maintains the FFT efficiency

**Da Li,** University of Calgary

**Time:** 4:40 pm - 5:00 pm (CST)

**Title**: Incorporating Multiple A Priori Information for Inverse Problem by Inexact Gradient Projection

**Abstract***:*

Many inverse problems can be formulated as a constrained optimization problem, and the feasible set can be a description of its a priori information. In this work, we investigate the constrained optimization problem, where the feasible set is the intersection of several convex sets. Gradient projection methods are standard ways to solve the constrained optimization problem, and the closed-form projection is necessary since the projection has to be evaluated exactly. The numerical methods of projection onto the intersection of convex sets usually have an iterative structure which leads to an inexact projection result in practice. A feasible set changing strategy is designed to overcome this issue. Based on this strategy, an inexact gradient projection method and an inexact scaled gradient projection method are proposed. The convergence analysis is provided with proper assumptions. Numerical results for seismic inversion are provided.

*(joint work with Michael P. Lamoureux, and Wenyuan Liao).*