Research Interests:

My research interests cover numerical algorithms and their applications to problems in a number of additional circumstances. I have made dedicated effort to apply fast algorithms to interesting problems. I am concerned with optimal algorithms, optimal meshes, and a number of interesting application areas as described below.

Optimal Algorithms

Hierarchy of meshes for an "interesting" domain.

The applicability of geometric multigrid methods is limited by the degree to which one can guarantee good behavior. I have combined technologies from computational geometry, automated finite element methods, and multilevel methods to attempt to create coarse spaces that work for a variety of problems. I focused on problems with non-quasi-uniform meshes, and am extending it to anisotropy and nonlinear problems.

Optimal Meshes

A simple, extremely graded mesh in 2D

One may reap huge benefits by using meshes optimized for a particular problem. The benefits only increase as the problems considered increase in dimensions. An area of inquiry I have undertaken is the optimal meshing strategy for high-dimensional problems involving solutions that necessarily die off exponentially. This includes small quantum systems. I have developed generalized finite element software for treating the high dimensional problems, and the rammifications of high-dimensionality combined with H-P adaptivity are an interesting line of continued research.

Integral Operators and Bioelectrostatics

Mesh of a Trypsin molecule generated from a PDB file.

Integral operators for bioelectrostatics allow for the computation of nonlocal solvent and ion effects required to accurately capture solvation dynamics in biological systems such as proteins. In particular this work is aimed at providing reasonable computational and algorithmic frameworks to calculate solvation effects given interesting models of water as a dielectric. In order to do this, I have developed a finite element implementation of these models and am in the process of getting it to the point where I can calculate physical quantities, like the solvation free energy, small and medium sized molecules. This includes automating the mesh generation of interesting geometries, as shown above.

Efficient Classical DFT

Cartoon of an Ion Channel Configuration.

The other part of this effort is speeding up the calculation of the non-local effects in classical DFT calculations around ion-channels. The reference-density approach to this problem requires the construction of a "reference density", which is basically an ion concentration modified to remove screening effects. The interaction of the ions is expanded around this. The calculation of this is a difficult problem as it is essentially a pseudoconvolution; a convolution where the filter is dependent upon the position in the domain. I have explored two interesting options for doing this pseudoconvolution: acceleration using CUDA, and fast approximation.

Automation of Scientific Computing

Isoparametric Boundaries for Flow Problems.

I have also put a fair amount of effort into automating other interesting problems in scientific computing, including working on the automated use of parametric geometry information with standard FEM software.

Education:

Grad:

Ph.D. Student; Department of Computer Science, University of Chicago
GRADUATION DATE: August, 2011

M.S., Computer Science (Winter 2008), University of Chicago

Undergrad:

B.S. Computer Science (Spring 2005), Minors in Physics and Math, Pennsylvania State University

Publications:

  1. Peter Brune, Enabling Unstructured Multigrid Under The Sieve Framework,
    M.S. Thesis, Feb. 2008 (PDF)
  2. Peter Brune, Matthew Knepley, L. Ridgway Scott, Unstructured Geometric Multigrid in Two and Three Dimensions on Complex and Graded Meshes,
    Submitted, Mar. 2011 (arXiv)
  3. Peter Brune, L. Ridgway Scott, Matt Knepley, Exponential Grids in High Dimensional Space,
    In Preparation
  4. Peter Brune, Matt Knepley, Fast Reparametrized Approach to Approximating the Reference Density in Classical Density Functional Theory,
    In Preparation

Presentations

  1. P. Brune, M. Knepley, L. R. Scott
    A Topologically Inspired Approach to Geometric Unstructured Multigrid
    USNCCM9, San Francisco, CA, 2007 (PDF)
  2. P. Brune, M. Knepley, Geometric Multigrid on Interesting Meshes
    AMS Spring Section, Bloomington, Indiana, 2008 (PDF)
  3. P. Brune, L. R. Scott, Dimension-Independent FEM
    FEniCS '09, Oslo, Norway, 2009 (PDF)
  4. P. Brune, J. Hoffman, J. Jansson, Automating Parametric Geometry using FEniCS Tools
    FEniCS '10, Stockholm, Sweden, 2010 (PDF)
  5. P. Brune, Computation of Rho^ref using GPUs
    Rush University Medical Center, Chicago, IL, 2010 (PDF)
  6. P. Brune, Fast Solvers and Electrochemical Problems
    Argonne National Lab, Argonne, IL, 2011 (PDF)

Teaching:

  1. TA: CSMC 2/32200, Computer Architecture, Spring 2008
  2. TA: CSMC 2/32200, Computer Architecture, Fall 2009

Collaborations:

Job Materials:

Will solve your PDEs for food: CV | Research Statement