Error Analysis of MLFMA With Closed-Form Expressions

Kalfa, Mert; Ertürk, Vakur B.; Ergül, Özgür

doi:10.1109/tap.2021.3070086

Cited by 4 publications

(2 citation statements)

References 11 publications

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“…For the translation function calculation, the truncation number is calculated via excess bandwidth formula [6] until the level with λ/4 box length. After that level, the truncation number is set as 7 to decrease the error in the Green's function calculation for the double precision arithmetic by considering the results in [7].…”

Section: Methods and Numerical Resultsmentioning

confidence: 99%

Trimmed Multilevel Fast Multipole Algorithm for D-Type Volume Integral Equations

Topozlu

Karaosmanoğlu²,

Ertürk

2022

2022 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (AP-S/URSI)

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In this work, we present a trimming scheme for the multilevel tree structure of multilevel fast multipole algorithm (MLFMA), which is applied on D-type volume integral equations. With this approach, the number of iterations and the durations of matrix-vector multiplications are significantly reduced for the solution of multi-scale volumetric problems. The trimming operation is performed on rows and columns of the impedance matrix. In order to eliminate the matrix columns, the current coefficients are estimated via machine learning techniques. The implementation particularly provides significant acceleration for the iterative solutions of electrically large volumetric problems.

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Section: Methods and Numerical Resultsmentioning

confidence: 99%

Trimmed Multilevel Fast Multipole Algorithm for D-Type Volume Integral Equations

Topozlu

Karaosmanoğlu²,

Ertürk

2022

2022 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (AP-S/URSI)

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“…For quadruple precision, we use the Advanpix multiprecision computing toolbox for MATLAB 20 . To simulate half precision floating point arithmetic, we use the chop library and associated functions from, 21 available at https://github.com/higham/chop and https://github.com/SrikaraPranesh/LowPrecision_Simulation.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Multistage mixed precision iterative refinement

Oktay

Carson

2022

Numerical Linear Algebra App

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Low precision arithmetic, in particular half precision (16-bit) floating point arithmetic, is now available in commercial hardware. Using lower precision can offer significant savings in computation and communication costs with proportional savings in energy. Motivated by this, there has been a renewed interest in mixed precision iterative refinement schemes for solving linear systems Ax = b, and new variants of GMRES-based iterative refinement have been developed. Each particular variant with a given combination of precisions leads to different condition number-based constraints for convergence of the backward and forward errors, and each has different performance costs. The constraints for convergence given in the literature are, as an artifact of the analyses, often overly strict in practice, and thus could lead a user to select a more expensive variant when a less expensive one would have sufficed. In this work, we develop a multistage mixed precision iterative refinement solver which aims to combine existing mixed precision approaches to balance performance and accuracy and improve usability. For a user-specified initial combination of precisions, the algorithm begins with the least expensive approach and convergence is monitored via inexpensive computations with quantities produced during the iteration. If slow convergence or divergence is detected using particular stopping criteria, the algorithm switches to use a more expensive, but more reliable variant. A novel aspect of our approach is that, unlike existing implementations, our algorithm first attempts to use "stronger" GMRES-based solvers for the solution update before resorting to increasing the precision(s). In some scenarios, this can avoid the need to refactorize the matrix in higher precision. We perform extensive numerical experiments on a variety of random dense problems and problems from real applications which confirm the benefits of the multistage approach.

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