# Boundary-Integral methods in molecular science and engineering

#### by Prof. Jaydeep Bardhan

**Rush University Medical Center**

This course will present an introduction to the theory and practice of solving boundary-integral equations (BIE) using boundary-element methods (BEM)—a popular and computationally efficient alternative to finite-element methods (FEM) for the solution of partial-differential equations. The course will present these methods in the context of studying electrostatic interactions between biological molecules such as proteins.

Electrostatic effects play key roles in determining a biomolecule’s behavior, but are strongly influenced by the water molecules and dissolved ions. Atomistic theories such as molecular dynamics (MD) offer high resolution but are computationally expensive. Macroscopic continuum theory (*e.g.*, the Poisson equation) is much faster to compute, and works remarkably well for many investigations.

### Boundary-integral equations

Whereas the solution to a PDE is usually sought throughout a region of space, the solution of a BIE lies only on a *surface* in that space. This difference has substantial implications, and we will illustrate the advantages and disadvantages of these complementary but equivalent approaches. The course will provide a brief survey of application domains where BIE has been particularly successful, including not only biophysics but also electromagnetics, fluids, and elasticity. We will also describe the basic approaches for converting a suitable PDE problem to a BIE, emphasizing that the mathematical techniques are quite accessible to students with basic PDE knowledge.

### Boundary element methods (BEM)

Solving a BIE numerically is not like solving a PDE. We will present some of the key differences and describe how modern numerical techniques and computer architectures, as well as open-source software, make fast, large-scale calculations not only possible but actually quite straightforward.

For example, the system matrix for a finite-difference or finite-element calculation is sparse, reflecting the local nature of the differential operator. In contrast, a BIE leads to a dense matrix, whose computation grows *quadratically* with the number of unknowns. This course will describe how to use algorithms such as the fast multipole method (FMM) to solve BEM’s dense matrix problems using only *linear* time and memory.

The course’s conclusion will highlight recent research on BEM techniques for biomolecular electrostatics.

### Printable course description