Research accomplishments and objectives

This is a statement of research accomplishments and objectives, current as of the summer of 2007. Over time, I hope to extend some of the sections in more detail, but for now this must suffice. Note also that there are some images created in the various areas of my research on my Picture Gallery page.

This document has different sections for each of the fields in which I work (see the table of contents for this page at the right). For a while, I had a bibliography of my publications in each of these sections, but it turned out to be too hard to maintain alongside the other lists of publications one has in various places. An up-to-date version of my entire bibliography can be found here.

Research Accomplishments and Objectives

My research centers around the development of mathematical and computational methods for the solution of partial differential equations, and their application to a wide range of problems in engineering and sciences. In particular, this involves (i) the development of theory and efficient implementation of adaptive finite element methods, in order to allow resolving scales that are too small for ordinary, uniform-mesh simulations, but are nevertheless driving the dynamics of systems; and (ii) the application of these methods to a number of projects that I am currently involved in, as listed below. Most of this work is based on the Open Source deal.II finite element library of which I am the project leader and principal author.

My personal career objective is to form a group focused on the development and application of modern numerical algorithms and computational science tools to relevant problems in the applied sciences, and in particular to the development of methods for modern nonlinear imaging methods in the biomedical and other sciences. Typically, this includes close interdisciplinary interaction with researchers from various fields as well as a strong foundation in mathematical modeling and numerical solver techniques.

Inverse problems and biomedical imaging

Mathematical and practical background. In many applications, one needs to know about the interior composition of a body without invasive procedures. A typical example is medical imaging, in particular cancer detection, which uses X-rays, MRI, or a number of other techniques to "see" into a body without opening it (this is called tomography). Another important application is imaging of technical objects for cracks or of cargo and suitcases to determine the presence of prohibited objects such as nuclear material or bombs.

Mathematically, imaging is usually posed in the form of inverse problems. The inverse problems I am particularly interested in are stated as nonlinear partial differential equations; consequently, these cases require solution using numerical methods, and the complexity of the problem often results in very large and expensive computations. Therefore, there is a clear need for advanced numerical methods.

I have been working in collaboration with scientists at Baylor College of Medicine in Houston, Texas, since 2003 on the development of optical tomography methods for the detection of breast cancer. Traditionally, breast cancer is detected using mammograms. However, using X-rays, this technique might induce cancer itself. Equally importantly, what the X-rays "see" are only changes in the density of material, typically secondary calcification of vessels in the vicinity of a tumor, but not the tumor structure itself. Thus, one would like to a) use non-ionizing radiation, and b) use methods that are specific to tumor tissue types, not just secondary effects of tumors. Optical tomography using light in the visible and near-infrared range, for which human tissue has a high scattering but low absorption coefficient can satify the first condition. The idea for the second requirement is to use fluorescent agents that bind selectively to cancer tissue, excite the agents with light of one wave length and detect the fluorescent light at a different wave length.

The mathematical description of this leads to an inverse problem where one wants to infer the spatial distribution of the fluorescent dye concentration, which is a coefficient in a diffusion-type equation. The accurate resolution of this inherently three-dimensional problem requires the use of adaptive meshes to avoid excessive computing times. In this on-going collaboration, we have shown that the results of adaptive forward simulations closely match actual measurements. More importantly, we were the first to demonstrate that adaptive codes work for optical tomography, and our results show excellent resolution of tumor location and size, much better than the results of competing groups who do not use adaptivity.

Some results. The following pictures show how adaptivity works for detecting a target in a phantom geometry of cubical shape: we illuminate the right face of the geometry with laser light, which propagates into the tissue, and excites fluorescence in the target molecules; the fluorescent light diffuses back to the surface and is detected there. From this detected light, we reconstruct the target molecule concentrations throughout the body. The following pictures show a sequence of meshes used to simulate the light diffusion problem (top row) and to discretize the sought parameter concentrations (bottom row); clicking on a picture yields an enlarged version:

The locally refined nature of the meshes is clearly seen. In the bottom row, the red cells show the location of the reconstructed tumor. The following pictures show the accuracy of the reconstruction of location and size of two small tumors located at distances of 10, 6.6, 3.1, and 1.6mm from each other (again, get larger version by clicking on a picture):

The red cubes denote the identified targets, while black wireframes show exact location and radius of the targets. Obviously, our method is able to determine these parameters to excellent accuracy except for the last cases where the two identified targets merge. As a sidenote, for practitioners the holy grail is to get to a resolution of about 1mm, so we are pretty close!

A more realistic situation, modeling an experimental setup, is shown here (taken from a SIAM J. Scient. Comput. paper):

Here, we scan a laser widened to illuminate a rectangle over an experimentally obtained geometry. At each position, we take an image of the light intensity in the infrared wavelength range. If we use this information to determine the original fluorescent dye concentration, we get the following image:

One can see the tumor in the center, which we found to be at a location and depth that is compatible to the location found by a surgeon after we had done our experiments.

Future directions. Although the results shown above are already pretty good, there are a number of directions where more research is needed in the future. In particular, the following topics are certainly open questions at present and will be worked on in our group:

Solvers: Solving the inverse problems outlined above is expensive: our already quite optimized algorithms take 10-20 minutes to compute an image as the one above. That's too much for clinical practice, where the goal is 1-2 minutes. Consequently, we need to investigate new solver techniques, such as multigrid, that can more efficiently solve the problems at hand.
Regularization: Inverse problems are typically ill-posed, i.e. a little bit of noise in measurement data will produce significant changes in the reconstructed images. To stabilize this process, we add "regularization" to the problem, but it is a difficult and open problem how mach regularization to add, and where. We have some ideas in this regard, but will have to investigate their viability in practice.
Other imaging modalities: The optical tomography application above is only one imaging modality. Others in biomedical research use radiofrequency waves, ultrasound, optoacoustics, and many other ways, and beyond that there are related modalities in neutron and X-ray imaging of cargo containers and suitcases for border security. In collaboration with computer scientists, biomedical engineers, and nuclear engineers, we are working on extending our methods to some of these other modalities.
Questions beyond imaging: Beyond creating images there are a number of questions that one can ask. For example, how do we optimize our experimental setup so that the reconstructed images are the most accurate or reliable? These are questions in "optimal experimental design" that lead to problems of enormous mathematical and computational complexity. We are working on approaches to tackle these problems, but there is much left to be solved in these areas.

Numerical software

The use of advanced numerical techniques such as adaptive finite element meshes, multigrid, hp-adaptivity, or distributed computing requires complex software. The resulting programs are often large and utilize complicated data structures. Advances in these areas are therefore only possible if one re-uses and extends existing software, rather than write it anew for each project.

During my time at the Institute for Applied Mathematics in Heidelberg, Germany, the deal.II project was started by myself and two others to provide such a code basis. I am the principal author and project leader of this Open Source C++ library that offers h-, p-, and hp-adaptive meshes in one, two, and three space dimensions, a variety of finite elements, multigrid, support for parallel computing, and many other components. In order to allow other researchers to use the library, it comes with extensive documentation of all interfaces (more than 5000 pages) and a collection of example programs that demonstrate the use of the library. Since its inception, deal.II has grown to more than 370,000 lines of code, and has become one of the best known and most widely used finite element libraries in the field of computational mathematics research and applied sciences (see, for example, the list of publications obtained with the help of deal.II).

With several hundred installations world-wide, more than 100 downloads per month, and several thousand hits per month on its homepage http://www.dealii.org, it has received positive response from researchers and students in a wide range of scientific fields, including computational fluid dynamics, glacier flow, fuel cell design, cancer imaging, and computational biology. Here are a few pictures from various application areas (from left to right: inverse problems, wave equation, error estimators for the Laplace equation, seismics, Euler equation of hypersonic gas flow, elastostatics):

We have recently added hp-adaptivity to the library. My current work is focused on practical aspects of hp-adaptivity, integrating external linear solver libraries, and in particular further pushing the limits of parallel computing into the range of significantly more than 10 million unknowns on fully adapted 3D meshes. The goal is to make deal.II a standard tool for adaptive finite element simulations in applied mathematics and related fields.

The deal.II project has been successful beyond expectations. It is part of the SPEC CPU2006 industry standard testsuite for computer systems and compilers. Its authors have also been awarded the 2007 J. H. Wilkinson Prize for Numerical Software. Furthermore, it will likely be part of the software framework of the Center for Computational Infrastructure in Geodynamics (CIG) operated at the California Institute of Technology.

Finite element analysis

Closely coupled to the development of software, I am involved in theoretical research on adaptivity and error estimation, in particular, goal-oriented error estimates for finite element discretizations (see the book with Rolf Rannacher on my publications page). My work in this area focuses on developing and analyzing new adaptive approaches to complex problems and demonstrating their superiority over more traditional methods. Finding efficient discretizations is particularly important for time-dependent problems such as the wave equation which is the central component of fast inversion algorithms in geophysical imaging. Adaptive mesh refinement methods such as the ones used in my research may be one of the solutions to alleviate this bottleneck.

This page:

Introduction

Inverse problems and biomedical imaging

Numerical software

Finite element analysis

This site:

The deal.II library

Curriculum Vitae

Personal notes

Some photos

About these pages

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