Multi-Physics, Multi-Objective, Multi-Adaptive Strategies for Solving Inverse Problems

Principal Investigator: Robert Schaefer,

Catalogue number: UMO-2011/03/B/ST6/01393,

Scientific goals of the project:
- We will develop an inverse analysis strategy based on a multi-physics, multi-objective, multi-adaptive approach (MMM) that utilizes information extracted from multiple physical processes
appearing in  a studied  phenomenon. We will formulate MMM strategy as a solver for  multi-objective global optimization problems. We envision M MM to be applicable to solving as diverse
problems as oil deposit discovery, tumor diagnostics, defectoscopy, etc.
- We will develop multilevel adaptive methods coupling  hp-FEM algorithms (for  solving direct
problems) with stochastic adaptive algorithms HGS-EMOA (for  solving multi-objective global
optimization problems),  as well as with advanced techniques for speeding up genetic
computations (e.g., fitness approximation, fitness deterioration, gradient mutation).
- We will analyze algorithmic interconnections and error propagation in MMM.
- We will perform computational complexity analysis of MMM components necessary for its
configuration and tuning.
- We will design software architecture allowing for parallel MMM implementation (dedicated to
Linux clusters and hybrid environments including NVIDIA CUDA technology).
- We will test our approach by solving an important two-physics, three-objective case of a problem
of discovering oil deposits using sonic and electromagnetic measurements.
The significance of the project:
Inverse solvers are an important tool in science, engineering, and medicine. They have found
applications as diverse as discovering fuel resources, identification of tumor tissues, detecting
damages in mechanical parts and buildings, and many others.
Our proposed inverse problem-solving strategy, MMM, will allow  one  to solve an important class of
such problems.  To evaluate the performance of our approach, we plan to focus on oil deposits
identification using sonic and electromagnetic measurements.
Solving inverse problems is difficult mainly due to their ill-posedness, manifested by low regularity
and multi-modality of the objective function (for example, the Pareto set is not necessarily connected
in the multi-objective case). To a large extent, this immanent ill-posedness is a consequence of the fact
that inverse problem solvers do not have access to sufficient data regarding the problem at hand.
One of the core assumptions of the MMM strategy is to consider data from many  physical processes,
and their models, jointly. Such multi- objective formulation allows one to develop algorithms with
increased robustness (in particular, with improved guarantees regarding finding a solution) and with
decreased computational cost (due to improved conditioning)
Solving our multi-objective problems may require novel algorithmic ideas and approaches to deal with
large computational cost imposed by our setting. In particular, standard numer ical methods are not
applicable  due to their lack of  robustness,  their  high regularity requirements, and their unacceptable
computational complexity. Similarly, single adaptive schemes for solving direct or inverse problems
are unsatisfactory because of their computational complexity; the speedup obtained by  a simple
composition of adaptive algorithms can be at most the sum of the speedups obtained during particular
stages.
We intend to tackle the computational complexity barrier inherent to multi-objective inverse problems
by using a multi-adaptive strategy  as follows. We plan to use a multilevel, strongly coupled
composition of hp-FEM adaptive algorithms (controlled by graph-grammar productions) with the
HGS-EMOA inverse solver (with adaptive accuracy). Our goal is to show that it is possible to obtain a
super-linear speedup proportional to the product of speedups obtained by component algorithms. 
Our previous results, obtained as part of Polish Ministry of Science and Higher Education's project
NN-519-447739, provide strong evidence that our research goals are feasible.