  # Multiscale Numerical Methods for Inverse Problems Governed By Partial Differential Equations

## Periodic Report Summary 1 - MULTISCALE INVERSION (Multiscale Numerical Methods for Inverse Problems Governed By Partial Differential Equations)

The project “multiscale inversion” deals with algorithms and software development for parameter estimation of large-scale systems of partial differential equations, known as inverse problems. Such problems arise in diverse areas such as aerodynamics, mathematical finance, medicine, and geoscience. The parameter estimation is obtained by fitting data measurements to the physical model, which requires numerical solutions of high resolution large scale PDE-constrained optimization problems. These problems are usually ill-posed and non-convex. In particular, we investigate new approaches for the solution of the 3D inverse wave equation, also known as Full Waveform Inversion (FWI). This problem is one of the most important and challenging problems in the field. Its goal is to determine coefficients of the material parameter field of a heterogeneous medium, given a source and waveform observations at receiver locations on its boundary. Inverse problems of this type arise in seismic exploration of oil and gas reservoirs, earth sub-surface mapping, earthquake modeling, ultrasound imaging and more. During the last two year we have made several contributions to this field.

One of our contribution includes adopting the eikonal equation as a reduced model of the wave equation. The eikonal equation models the first arrival time of the wave, and is much easier to solve than the Helmholtz equation. In addition, solving the eikonal’s corresponding inverse problem, called “Travel time tomography”, can reveal a lot of valuable information regarding the underlying medium, and is much more stable and computationally easier to solve. We developed a Fast Marching Solver for the factored eikonal equation, which is yields a much more accurate approximation of the travel time when used for a point source. In addition, we showed how to solve the tomography problem using the factored equation as the simulation model (also in 3D). This work is published in the paper

Eran Treister and Eldad Haber, A Fast Marching algorithm for the factored eikonal equation, Journal of computation Physics, 324, 210-225, 2016.

In addition, we developed a solution of a joint travel-time tomography and FWI problems to handle missing low frequency data in FWI. This missing data is causing a lot of difficulties in the FWI problem. The following paper (under review) summarizes the results:

Eran Treister and Eldad Haber, Full waveform inversion guided by travel time tomography, Under review, 2016.

Another one of our efforts is to deliver a flexible and user friendly open source software package for solving inverse problems in parallel. Our software, called “jInv” includes a general framework for solving various types of inverse problems, including the ones mentioned above. Although our software is written in the simple and user friendly language “Julia”, it is efficiently programmed, parallelized in a few levels, and includes some of the state-of-the-art algorithms for the considered problems. More information can be found in the following paper:

Lars Ruthotto, Eran Treister and Eldad Haber, jInv--a flexible Julia package for PDE parameter estimation, Under review, 2016.

On a different stream, we worked on methods for learning inverse covariance matrices for adapting noise parameters and regularization techniques in the inverse problem. So far we focused on learning general sparse inverse covariance matrices at large scales, which is especially challenging because of memory limitations. This problem at such large scales also appears in fMRI and gene expression analysis applications. To solve the inverse covariance estimation problem at this scale, we developed a new block-coordinate-descent method, which uses low memory footprint, and adapted a multilevel acceleration for general sparse optimization. All these methods outperform the existing state of the art methods, especially for large-scale problems. For more information see

1) Eran Treister and Javier Turek, A Block-Coordinate Descent Approach for Large-scale Sparse Inverse Covariance Estimation, Neural Information Processing Systems (NIPS), Dec. 2014.
2) Eran Treister, Javier Turek and Irad Yavneh, A multilevel framework for sparse optimization with application to inverse covariance estimation. SIAM Sci. Comput. 38 (5), S566-S592, 2016.

In light of the contributions above, the PI Dr. Eran Treister accepted a position at the Computer Science Department at Ben Gurion University, which will host the third and last year of the project.