ECCOMAS International Conference IPM 2013 on Inverse Problems in Mechanics of Structure and Materials 24-27 April 2013, Rzeszów–Baranów Sandomierski, Poland
SOLVING INVERSE PROBLEMS USING COMPUTING AGENTS: AN ATTEMPT TO A DEDICATED HIERARCHIC MEMETIC STRATEGY Robert Schaefer1 , Maciej Smołka1∗ , Ewa Gajda-Zagórska1 , Maciej Paszyński1 , David Pardo2 1
2
1
AGH University of Science and Technology, Kraków, Poland e-mail: {schaefer,smolka,gajda,paszynsk}@agh.edu.pl
University of the Basque Country UPV/EHU and Ikerbasque, Bilbao, Spain e-mail:
[email protected]
INTRODUCTION
There are at least two main difficulties in inverse problem solving. The first of them is the high computational cost of solving a series of direct problems. In addition, this cost increases significantly with direct problem solution accuracy improvement. The second difficulty is caused by the nonconvexity (i.e. potential multimodality) of the cost functional. This makes fast local optimization methods (such as gradient ones) less usable as in general they may discover only some local minimizers, probably far from real solutions. One method to avoid the latter obstacle is to use stochastic algorithms, but they tend to be either inaccurate or expensive. We propose a hybrid method for solving inverse problems, which tries to combine advantages of global stochastic search (i.e. the guarantee of finding all local minimizers) with the effectiveness of local methods. 2 SAMPLE INVERSE PROBLEM AND A PROPOSED SOLUTION STRATEGY We shall present our inverse problem solution strategy on a case study. Let us consider the following problem: with D = {σ ∈ L∞ (Ω)| ess inf σ > 0} find σ ˆ ∈ D such that C(ˆ σ) =
k X i=1
|Qi (ui (ˆ σ )) − mi |2 ≤ C(σ) =
k X
|Qi (ui (σ)) − mi |2
for all σ ∈ D
(1)
i=1
where Qi ∈ H −1 (Ω) are quantities of interest, mi are their measured values, and ui (σ) are solutions of the variational problems of finding ui (σ) ∈ H01 (Ω) such that σ∇ui (σ), ∇v L2 (Ω) = F i (v) for all v ∈ H01 (Ω) (2) with F i ∈ H −1 (Ω). Such problems are known to have singular solutions when the geometry of Ω is sufficiently ’irregular’ (e.g. L-shaped). The solution of (2) is effectively approximated by means of hp-FEM, where the mesh adaptation is based on minimizing the H 1 -norm of
the relative error of the solution of (2) multiplied by the solution of the dual problem. The solution of the dual problem is pi (σ) ∈ H01 (Ω) such that (3) ∇pi (σ), σ∇w L2 (Ω) = Qi (w) for all w ∈ H01 (Ω). Note that (2) and (3) differ only in the right-hand side, hence the additional cost of solving (3) is not significant when using direct solvers. Moreover, the cost functional C is Fr´echet differentiable and its derivative can be easily computed (also without a noticeable cost) using the formula DC(σ)(h) = −2
k X (Qi (ui (σ)) − mi ) h∇ui (σ), ∇pi (σ) L2 (Ω) .
(4)
i=1
A proposed hybrid strategy called Hierarchic Memetic Strategy (HMS) is based on the experience with developing HGS and hp-HGS [1], and, on the other hand, evolutionary multi-agent systems (EMAS) [2]. Namely, it makes use of the multi-agent paradigm as an architectural principle. The agents carry (encoded) solution space points as their genotypes. The accuracy of an agent’s solution depends on the value of the cost functional (which serves as the agent’s fitness): agents with the cost closer to 0 (which is in our case the global minimum) are allowed to compute their solutions with a higher accuracy. In order to obtain a proper degree of stochastic diversity, we force agents to perform at random moments some operations inspired by those used in genetic strategies, i.e. the mutation, the crossover and a kind of distributed selection. On the other hand, in order to cause the agents’ population to concentrate around the local minimizers, we allow the best agents to perform a sort of distributed clustering. Every discovered cluster is equipped with a local approximation of the cost functional, which is computed using the already known values of the cost functional at the cluster’s elements. This approximation is then used to execute a rough stage of a gradient method within a cluster, which is supposed to be very cheap while not very accurate. Nevertheless, it can result in new promising solutions. Finally, the best agents in a cluster are used as starting points for an exact stage of a gradient method, which uses the real cost functional and its gradient computed using (4) instead of the approximation. During the computations, clusters are rearranged (i.e. merged, split or even deleted) and at the end they are supposed to approximate local minimizers’ basins of attraction. Those algorithmic results are accompanied by some benchmark tests of a prototypic HMS implementation. ACKNOWLEDGEMENTS The work has been partially supported by Polish National Science Center grants Nos. NN 519 447 739 and DEC-2011/03/B/ST6/01393. REFERENCES [1] B. Barabasz, S. Migórski, R. Schaefer, and M. Paszyński. Multi-deme, twin adaptive strategy hp-HGS. Inverse Problems in Science and Engineering, 19:3–16, 2011. [2] R. Schaefer, A. Byrski, J. Kołodziej, and M. Smołka. An agent-based model of hierarchic genetic search. Computers & Mathematics with Applications (CAMWA), 64(12):3763–3776, 2012.
