© 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.This contribution identifies an often ignored source of uncertainty in the accuracy of the adaptive cross approximation algorithm, and proposes a combination of adaptations that reduce this uncertainty with negligible additional computational cost.Peer ReviewedPostprint (author's final draft

Heldring, Alexander

Rius Casals, Juan Manuel

Sekulic, Ivan

Úbeda Farré, Eduard

UPCommons (Universitat Politècnica de Catalunya)

On the Accuracy of the Adaptive CrossApproximation AlgorithmAlex Heldring, Eduard Ubeda, Ivan Sekulic, and Juan Manuel RiusUniversitat Politecnica de Catalunya, Barcelona, 08034, Spainheldring@tsc.upc.eduAbstract—This contribution identifies an often ignored sourceof uncertainty in the accuracy of the adaptive cross approxima-tion algorithm, and proposes a combination of adaptations thatreduce this uncertainty with negligible additional computationalcost.I. INTRODUCTIONThe adaptive cross approximation (ACA) [1] is an algorithmthat computes a low rank approximation to matrices or matrixsub-blocks. Due to its efficiency and its black-box nature (noprior knowledge of the matrix content or origin is necessary),it is gaining ground as a tool in accelerated electromagneticsimulations. As with all approximations, the ACA introducesan error that can only be known up to an order of magnitude,to be chosen by the user. With the growing popularity of theACA, so grows the importance of assessing the reliability ofthe algorithm under all circumstances. In this contribution, weintend to draw attention to one aspect of the ACA algorithmthat introduces a variation in the actual error around the user-chosen error threshold of about one order of magnitude. Thisvariation is triggered by an initial choice inside the algorithmwhich is not left to the user but instead hard-wired in thealgorithm formulation. There is presently no known recipe foroptimizing this choice. This means that the ACA algorithmis, to some extent, a randomized algorithm, with a probabilitydistribution for the true residual error. The reason that theseobservations have not been made before, even though theACA algorithm has been receiving ample attention for sometime now, is twofold: Firstly, since the initial setting is hard-wired, the user will always obtain identical results on identicalproblems and the algorithm will seem perfectly deterministicto her. Secondly, the ACA is typically used to compressoff-diagonal blocks of a larger linear system. How exactlythe relative error in the blocks propagates into the solutionof the linear system cannot be known but it will often beconsiderably smaller, as the matrix is often dominated bythe non-approximated on-diagonal blocks. So it is unlikely,though not impossible, that an error that is one order ofmagnitude larger than the user intended, will result in an errorof the same magnitude in the final solution. In Section II weillustrate the phenomenon with a simple numerical exampleand in Section III we propose an adaptation to the ACAalgorithm and the ACA convergence criterion which, althoughit does not solve the problem, seems to dampen its effect, atnegligible extra computational cost. Remains to be mentionedthat the possible unreliability of the estimated error in the ACAalgorithm has been noted before, although without linking itto the initial choice and consequently without addressing itsstatistical nature, notably in [2] and [3].II. NUMERICAL EXAMPLEAs an illustration, the ACA algorithm is invoked to com-press the mutual interaction matrix of the two square PECplates shown in Fig. 1, in the EFIE formulation and usingRWG basis functions. The plates are discretized into 1160basis functions each, and the working frequency is such thatthe plate edges span two wave lengths. Fig. 2 shows thedistribution of the true relative error after convergence ofthe ACA algorithm with a threshold of 10−4, when all 1160possible random choices of the initial column are tried. Notethe variation in the true error of more than one order ofmagnitude. The number of iterations needed for convergencesimilarly shows a wide distribution, in this case ranging fromk = 41 to 58 steps, with an average over all 1160 initialchoices of k̄ = 37.8 steps.III. PROPOSED ADAPTATION OF THE ACAThe first intervention that we attempted was based onthe observation that the conventional ACA estimation of theresidual error sometimes shows sharp negative peaks leading toa highly premature declaration of convergence, as was notedin [2], [3]. An example of this phenomenon is observed atstep 32 in Fig. 3. Considering that the global convergencerate is theoretically expected and observed in practice toFig. 1. Two coplanar 2λ× 2λ PEC plates at a mutual distance of 2λ.Fig. 2. Distribution of the true relative error after ACA convergence withthreshold τ = 10−4, for the two plates in Fig. 1.be exponential, a cumulative linear fit is computed of thelogarithm of the estimated residual error versus the step index,including all estimates up to the current step. The resultingcurve is also shown in Fig. 3. This fit is then used to estimatethe current error. This slightly narrows the distribution oftrue relative errors, but it also leads to an increase of theaverage number of steps before convergence, to k̄ = 40.4 inthe example. Considering that the computational cost of theACA scales with k2, this approach unacceptably increases thecomputational effort.Fig. 3. Convergence of the ACA for the two plates in Fig. 1, for arandomly chosen initial column. Shown are the true error at every iteration,the conventional ACA estimator and the cumulative linear fit proposed in thispaper.Subsequently we explored an entirely different idea; observ-ing that changing the initial column, the sequence of selectedcolumns and rows often changes drastically, regularly resultingin a zero overlap between the columns and rows used intwo equivalent ACA decompositions, we conjectured that thechoice of the next column or row to be included is not unique.The ACA algorithm prescribes to choose that column/rowwhose common element in the current row/column shows thelargest absolute error. We ran a series of experiments pickingthe elements corresponding to the nth largest error instead, andwe found the counter-intuitive outcome that up to n ≈ 50 thedistribution of the true error narrowed and at the same time theaverage number of steps decreased (For example, with n = 10,the extremes of the distribution decreased to {−44 dB, −36dB} and k̄ decreased to 36.5). The effect was too small toprovide a substantial improvement to the ACA efficiency, butit was persistent.The most surprising result was obtained when we combinedthe two approaches hereabove: Picking the 10th best elementto select the rows and columns and using the linear fit toestimate the residual error. The distribution of the true errorafter convergence is shown in Fig. 4. There is a considerablenarrowing and a left-shift of the distribution, which now spansapproximately {−45 dB, −38 dB}, but this has not been atthe expense of the convergence rate which, with k̄ = 38.3, isonly half a step above the conventional ACA.Fig. 4. Distribution of the true relative error as in Fig. 2, with and withoutthe adaptations proposed in this paper.IV. CONCLUSIONSThe aim of this contribution was to draw attention to theoften overlooked statistical uncertainty in the ACA algorithm.Although no definitive solution has been found to this problem,a combination of two adaptations to the ACA algorithm hasbeen proposed, that reduces its impact without compromisingthe efficiency of the ACA. Further investigation is necessaryto understand the mechanisms behind these adaptations andperhaps to reduce the remaining uncertainty.ACKNOWLEDGMENTThis work was supported by the SpanishInterministerial Commission on Science and Technology(CICYT) under projects TEC2013-47360-C3-1-P andTEC2016-78028-C3-1-P.REFERENCES[1] M. Bebendorf, “Approximation of boundary element matrices,” Numer.Math., vol. 86, no. 4, pp. 565–589, Oct. 2000.[2] J. Laviada, R. Mittra, M. R. Pino, and F. Las-Heras, “On the convergenceof the ACA,” Microwave Opt. Technol. Lett., vol. 51, no. 10, pp. 2458–2460, Oct. 2009.[3] A. Heldring, E. Ubeda, and J. M. Rius, “On the convergence of the ACAalgorithm for radiation and scattering problems,” IEEE Trans. AntennasPropag., vol. 62, no. 7, pp. 3806–3809, July 2014.

English

On the accuracy of the adaptive cross aproximation algorithm

© 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.This contribution identifies an often ignored source of uncertainty in the accuracy of the adaptive cross approximation algorithm, and proposes a combination of adaptations that reduce this uncertainty with negligible additional computational cost.Peer Reviewe

UPCommons

Alex Heldring

Eduard Ubeda

Ivan Sekulic

Juan Manuel Rius

Crossref

On the accuracy of the adaptive cross approximation algorithm

UPCommons. Portal del coneixement obert de la UPC

On the Accuracy of the Adaptive CrossApproximation AlgorithmAlex Heldring, Eduard Ubeda, Ivan Sekulic, and Juan Manuel RiusUniversitat Politecnica de Catalunya, Barcelona, 08034, Spainheldring@tsc.upc.eduAbstract—This contribution identifies an often ignored sourceof uncertainty in the accuracy of the adaptive cross approxima-tion algorithm, and proposes a combination of adaptations thatreduce this uncertainty with negligible additional computationalcost.I. INTRODUCTIONThe adaptive cross approximation (ACA) [1] is an algorithmthat computes a low rank approximation to matrices or matrixsub-blocks. Due to its efficiency and its black-box nature (noprior knowledge of the matrix content or origin is necessary),it is gaining ground as a tool in accelerated electromagneticsimulations. As with all approximations, the ACA introducesan error that can only be known up to an order of magnitude,to be chosen by the user. With the growing popularity of theACA, so grows the importance of assessing the reliability ofthe algorithm under all circumstances. In this contribution, weintend to draw attention to one aspect of the ACA algorithmthat introduces a variation in the actual error around the user-chosen error threshold of about one order of magnitude. Thisvariation is triggered by an initial choice inside the algorithmwhich is not left to the user but instead hard-wired in thealgorithm formulation. There is presently no known recipe foroptimizing this choice. This means that the ACA algorithmis, to some extent, a randomized algorithm, with a probabilitydistribution for the true residual error. The reason that theseobservations have not been made before, even though theACA algorithm has been receiving ample attention for sometime now, is twofold: Firstly, since the initial setting is hard-wired, the user will always obtain identical results on identicalproblems and the algorithm will seem perfectly deterministicto her. Secondly, the ACA is typically used to compressoff-diagonal blocks of a larger linear system. How exactlythe relative error in the blocks propagates into the solutionof the linear system cannot be known but it will often beconsiderably smaller, as the matrix is often dominated bythe non-approximated on-diagonal blocks. So it is unlikely,though not impossible, that an error that is one order ofmagnitude larger than the user intended, will result in an errorof the same magnitude in the final solution. In Section II weillustrate the phenomenon with a simple numerical exampleand in Section III we propose an adaptation to the ACAalgorithm and the ACA convergence criterion which, althoughit does not solve the problem, seems to dampen its effect, atnegligible extra computational cost. Remains to be mentionedthat the possible unreliability of the estimated error in the ACAalgorithm has been noted before, although without linking itto the initial choice and consequently without addressing itsstatistical nature, notably in [2] and [3].II. NUMERICAL EXAMPLEAs an illustration, the ACA algorithm is invoked to com-press the mutual interaction matrix of the two square PECplates shown in Fig. 1, in the EFIE formulation and usingRWG basis functions. The plates are discretized into 1160basis functions each, and the working frequency is such thatthe plate edges span two wave lengths. Fig. 2 shows thedistribution of the true relative error after convergence ofthe ACA algorithm with a threshold of 10−4, when all 1160possible random choices of the initial column are tried. Notethe variation in the true error of more than one order ofmagnitude. The number of iterations needed for convergencesimilarly shows a wide distribution, in this case ranging fromk = 41 to 58 steps, with an average over all 1160 initialchoices of k¯ = 37.8 steps.III. PROPOSED ADAPTATION OF THE ACAThe first intervention that we attempted was based onthe observation that the conventional ACA estimation of theresidual error sometimes shows sharp negative peaks leading toa highly premature declaration of convergence, as was notedin [2], [3]. An example of this phenomenon is observed atstep 32 in Fig. 3. Considering that the global convergencerate is theoretically expected and observed in practice toFig. 1. Two coplanar 2λ× 2λ PEC plates at a mutual distance of 2λ.Fig. 2. Distribution of the true relative error after ACA convergence withthreshold τ = 10−4, for the two plates in Fig. 1.be exponential, a cumulative linear fit is computed of thelogarithm of the estimated residual error versus the step index,including all estimates up to the current step. The resultingcurve is also shown in Fig. 3. This fit is then used to estimatethe current error. This slightly narrows the distribution oftrue relative errors, but it also leads to an increase of theaverage number of steps before convergence, to k¯ = 40.4 inthe example. Considering that the computational cost of theACA scales with k2, this approach unacceptably increases thecomputational effort.Fig. 3. Convergence of the ACA for the two plates in Fig. 1, for arandomly chosen initial column. Shown are the true error at every iteration,the conventional ACA estimator and the cumulative linear fit proposed in thispaper.Subsequently we explored an entirely different idea; observ-ing that changing the initial column, the sequence of selectedcolumns and rows often changes drastically, regularly resultingin a zero overlap between the columns and rows used intwo equivalent ACA decompositions, we conjectured that thechoice of the next column or row to be included is not unique.The ACA algorithm prescribes to choose that column/rowwhose common element in the current row/column shows thelargest absolute error. We ran a series of experiments pickingthe elements corresponding to the nth largest error instead, andwe found the counter-intuitive outcome that up to n ≈ 50 thedistribution of the true error narrowed and at the same time theaverage number of steps decreased (For example, with n = 10,the extremes of the distribution decreased to {−44 dB, −36dB} and k¯ decreased to 36.5). The effect was too small toprovide a substantial improvement to the ACA efficiency, butit was persistent.The most surprising result was obtained when we combinedthe two approaches hereabove: Picking the 10th best elementto select the rows and columns and using the linear fit toestimate the residual error. The distribution of the true errorafter convergence is shown in Fig. 4. There is a considerablenarrowing and a left-shift of the distribution, which now spansapproximately {−45 dB, −38 dB}, but this has not been atthe expense of the convergence rate which, with k¯ = 38.3, isonly half a step above the conventional ACA.Fig. 4. Distribution of the true relative error as in Fig. 2, with and withoutthe adaptations proposed in this paper.IV. CONCLUSIONSThe aim of this contribution was to draw attention to theoften overlooked statistical uncertainty in the ACA algorithm.Although no definitive solution has been found to this problem,a combination of two adaptations to the ACA algorithm hasbeen proposed, that reduces its impact without compromisingthe efficiency of the ACA. Further investigation is necessaryto understand the mechanisms behind these adaptations andperhaps to reduce the remaining uncertainty.ACKNOWLEDGMENTThis work was supported by the SpanishInterministerial Commission on Science and Technology(CICYT) under projects TEC2013-47360-C3-1-P andTEC2016-78028-C3-1-P.REFERENCES[1] M. Bebendorf, “Approximation of boundary element matrices,” Numer.Math., vol. 86, no. 4, pp. 565–589, Oct. 2000.[2] J. Laviada, R. Mittra, M. R. Pino, and F. Las-Heras, “On the convergenceof the ACA,” Microwave Opt. Technol. Lett., vol. 51, no. 10, pp. 2458–2460, Oct. 2009.[3] A. Heldring, E. Ubeda, and J. M. Rius, “On the convergence of the ACAalgorithm for radiation and scattering problems,” IEEE Trans. AntennasPropag., vol. 62, no. 7, pp. 3806–3809, July 2014.

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On the accuracy of the adaptive cross aproximation algorithm

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