Fast Iterative Solvers for Integral Equation Based Techniques in Electromagnetics

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1 Fast Iterative Solvers for Integral Equation Based Techniques in Electromagnetics Mario Echeverri, PhD. Student (2 nd year, presently doing a research period abroad) ID:30360 Tutor: Prof. Francesca Vipiana, Politecnico di Torino Collaborators: Prof. Giuseppe Vecchi, the Antenna and EMC research unit at ISMB.

2 Outline Attended classes Research context Addressed research: Questions and Problems State of the art Expected contributions (Novelty) Methodology Submitted and published conference/journal papers Future work 2

3 Attended classes:

4 Research context & motivation Solution of electromagnetic problems, where complex/large structures are involved. Integral Equation (IE) techniques in the frequency domain for the formulation of the problem. IE + variational formulation (weighted residuals): Method of Moments (MoM). MoM is suitable for Antenna and Scattering problems solution. Fast and accurate solutions are mandatory for industry applications and realistic modelling. 4

5 Addressed research: Questions & Problems Multis-scale MoM systems suffer of two main drawbacks (But have also many advantages, accuracy among them): a) Dense matrix, therefore time and memory consuming (fast iterative solvers required). b) Convergence problems due to ill conditioning (preconditioner is a must!). Use of scalable techniques (e.g. O(NLogN)) is mandatory. 5

6 Addressed research: Some insight Fast solver: accelerate matrix-vector products Preconditioner: improve convergence e.g. Compressive (Low rank) approx.: e.g. Change of basis preconditioner: Compress MoM matrix Compressed MoM matrix Change-of-basis preconditioner (MR) 6

7 State of the art I Fast solvers Low frequency (LF): D < λ EMC problems Kernel independent: Adaptive Cross Approximation ACA (NLogN), IE-QR (N 3/2 ). Kernel dependent: LF- MLFMA (NLogN). Intermediate frequency (IF): D ~ λ Antenna problems Kernel independent: Adaptive Cross Approximation ACA (N 4/3 LogN), IE-QR (N 3/2 ). Kernel dependent: MLFMA (NLogN); AIM (N 3/2 ); GIFFT (N 3/2 LogN). High frequency D > λ Scattering and radiation problems Kernel dependent: MLFMA (NLogN). D Analyzed structure 7

8 State of the art II Preconditioning M.S.: Multi-scale L.T.: Loop-Tree Low frequency (LF): D < λ EMC problems LF breakdown; Charge (numerical) cancellation; Dense/Non uniform meshes. Helmholtz decomposition: Multiresolution MR (L.T. don t work for dense/m.s. meshes) Intermediate frequency (IF): D ~ λ Antenna problems Combination of LF & HF effects: parts of the structure have D > λ and others D < λ. Combination of solutions: MR+ILU. Domain decomposition methods. High frequency D > λ Scattering and radiation problems Eigenvalues of MoM matrix cluster around zero ( h ~ λ/10 ). M.S. issues (disparate scales) ILU (Not suitable for M.S. meshes); Calderón MP; Sparse inverse. Domain decomposition methods. 8

9 Expected contributions and methodology Study and implementation of preconditioning techniques able to cure the ill conditioning problems of the MoM matrix in wideband simulations. Novelty: Combination of different techniques to achieve fast and accurate simulations in an effective way, e.g. using DDM + MR + ACA. First year contributions: Stabilization of a kernel independent fast solver (ACA) at low and very low frequencies, solving the charge cancellation problem by applying the MR preconditioner in a novel manner; two journal papers and two conference papers (see publications section ahead). Second year contributions: development of a domain decomposition framework for the solution of multi-scale problems at intermediate and high frequencies. A novel set of transmission conditions were proposed for the improvement of the convergence properties; one journal paper and one conference paper have been submitted, while two conference papers have been presented. 9

10 First year contributions: Low frequency problem Frequency: 1 khz Iterations: 45 10

11 Second year contributions: Intermediate & High frequency Domain decomposition + Preconditioner + Fast method Single domain S DDA iter 2 Decompose the problem. Analyze each domain (separately, therefore in parallel); preconditioners and fast solvers can be used as needed (e.g. can use MR+MLFMA). Connect independent solutions iteratively (using transmission conditions). Stop at a given tolerance. Versatile, inherently parallel and scalable. Tackle the Multi-scale conditioning Reference (Single domain) problem by separating the disparate scales. Decomposed domain E DDA iter 1 DDA: Domain decomposition algorithm DDA iter 5 11

12 Domain decomposition: Results

13 Submitted and published conference/journal papers Published: F. Vipiana, M. Bercigli, M. A. Echeverri Bautista, M. Bandinelli, and G. Vecchi, Incremental Multilevel Filling and Sparsification for the MoM Solution of Multi-Scale Structures at Low Frequencies, IEEE Transactions on Electromagnetic Compatibility, accepted for publication in IEEE Transactions on Electromagnetic Compatibility, 2014 [DOI: /TEMC ]. M. A. Echeverri Bautista, M. A. Francavilla, F. Vipiana, G. Vecchi, "A Hierarchical Fast Solver for EFIE-MoM Analysis of Multiscale Structures at Very Low Frequencies", IEEE Transactions on Antennas and Propagation, Vol. 62, No. 3, March M. A. Echeverri Bautista, F. Vipiana, M. A. Francavilla, G. Vecchi, A Domain Decomposition Framework for the Solution of Multi-scale Problems Using Integral Equation Formulations, accepted for IEEE International Conference on Computational Electromagnetics, Hong Kong, China, Feb M. A. Echeverri Bautista, F. Vipiana, M. A. Francavilla, G. Vecchi, Non Overlapping Domain Decomposition Based on Discontinuous Galerkin and Enhanced Transmission Conditions for Multi-Scale Structures, accepted for Proc. IEEE International Symposium on Antennas and Propagation and USNC-URSI National Radio Science Meeting, Memphis (TN, US), July

14 Submitted and published conference/journal papers M. A. Echeverri Bautista, M. A. Francavilla, F. Vipiana, G. Vecchi, Investigation of Split Potentials Low Frequency Stabilization of Kernel-Independent Low-Rank Compressive Solvers, accepted for Proc. of European Conf. Antennas Propag., (The Hague, The Netherlands), April M. A. Echeverri Bautista, M. A. Francavilla, F. Vipiana, G. Vecchi, MR-preconditioned ACA for Sub- Wavelength Problems, Proc. IEEE International Symposium on Antennas and Propagation and USNC- URSI National Radio Science Meeting, Orlando (FL, US), July Submitted: M. A. Echeverri Bautista, F. Vipiana, M. A. Francavilla, J. A. Tobón Vasquez, G. Vecchi, A Nonconformal Domain Decomposition Scheme for the Analysis of Multi-scale Structures, under review in IEEE Transactions on Antennas and Propagation, M. A. Echeverri Bautista, F. Vipiana, M. A. Francavilla, G. Vecchi, Domain Decomposition Method for Integral Equations Using Non-conformal Meshing, under review for Proc. of European Conf. Antennas Propag., (Lisbon, Portugal), April 2015.

15 Future work To investigate the feasibility of fast direct solver approaches, aiming to overcome the convergence issues of iterative solvers. Combination of different techniques in the developed DDM framework (MLFMA + MR, etc.). 15

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