Quasi-Newton for Multi-Physics

Quasi-Newton Methods for Coupled Multi-Physics Problems

In scientific computing, an increasing need for ever more detailed insights and optimization leads to improved models often including several effects / components described by different types of equations. In most applications, we have a strong coupling between all involved effects, such that neglecting the coupling would lead to inaccurate or wrong results. My primary focus of research are strongly coupled multi-component problems that are inherently hard to solve as different equations might require different kinds of approaches while the dependencies amongst the equations (coupling conditions) need to be fulfilled at all times. Those systems have a natural need for high performance computing, since the smaller modeling error we get when switching from a single-component to a multi-component model is useless if we cannot sustain a high resolution. The complexity of the corresponding solver algorithms and implementations typically can be tackled with so-called partitioned simulations reusing existing and established software codes for different components. This approach profits from decades of experience and development in terms of models, numerical methods and parallel scalability of the single components. Ensuring the fulfillment of coupling conditions and overall stability requires sophisticated coupling numerics. Quasi-Newton approaches are a particularly promising numerical approach for the outer coupling iterations as they require only input and output values of each component in order to estimate the underlying Jacobian or their inverses and execute an approximate Newton iteration for the coupled system.

Two exemplary applications are addressed in this project: surface coupled multi-physics problems such as fluid-structure interactions and a volume-coupling between brain tumor growth and image registration with the final task to identify growth parameters based on an adjoint approach. The SIBIA (Scalable Integrated Biophysics-based Image Analysis) framework is developed to provide methods and algorithms for the joint image registration and biophysical inversion. Its application is in analysing MR images of glioblastomas (primary brain tumors). Given the segmentation of a normal brain MRI and the segmentation of a cancer patient MRI, we wish to determine tumor growth parameters and a registration map so that if we grow a tumor (using our tumor model) in the normal segmented image and then register it to the segmented patient image, then the registration mismatch is as small as possible. We call this the coupled problembecause it two-way couples the biophysical inversion and registration problems. In the image registration step we solve a large-deformation diffeomorphic registration problem parameterized by an Eulerian velocity  field. In the biophysical inversion step we estimate parameters in a reaction-diffusion tumor growth model that is formulated as a partial differential equation (PDE). In SIBIA, we couple these two steps in an iterative manner.


  • Miriram Mehl
  • Klaudius Scheufele
  • George Biros, ICES, UT Austin

Selected Publications

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