A probing method for computing the diagonal of a matrix inverse

Tang, Jinyu; Saad, Yousef

doi:10.1002/nla.779

Cited by 120 publications

(97 citation statements)

References 45 publications

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“…The probing method explored here has been developed by Tang and Saad [14], and approximates the diagonal of the inverse matrix S denoted by…”

Section: Probingmentioning

confidence: 99%

“…. , v s } ∈ R N ×s , s ∈ N, the diagonal of the inverse of the N × N dimensional Dirac operator can be approximated as [14] …”

Section: Probingmentioning

confidence: 99%

“…In this work we apply a recently proposed "probing" algorithm [14] to the problem of approximating the diagonal entries of a matrix inverse to LQCD computations, and compare it to the use of SVS. In the probing method the linear system of eq.…”

Section: Introductionmentioning

confidence: 99%

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Variance reduction with practical all-to-all lattice propagators

Endress¹,

Peña²,

Sivalingam³

2015

Computer Physics Communications

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Abstract:We discuss all-to-all quark propagator techniques in two (related) contexts within Lattice QCD: the computation of closed quark propagators, and applications to the so-called "eye diagrams" appearing in the computation of non-leptonic kaon decay amplitudes. Combinations of low-mode averaging and diluted stochastic volume sources that yield optimal signal-to-noise ratios for the latter problem are developed. We also apply a recently proposed probing algorithm to compute directly the diagonal of the inverse Dirac operator, and compare its performance with that of stochastic methods. At fixed computational cost the two procedures yield comparable signal-to-noise ratios, but probing has practical advantages which make it a promising tool for a wide range of applications in Lattice

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“…The probing method explored here has been developed by Tang and Saad [14], and approximates the diagonal of the inverse matrix S denoted by…”

Section: Probingmentioning

confidence: 99%

“…. , v s } ∈ R N ×s , s ∈ N, the diagonal of the inverse of the N × N dimensional Dirac operator can be approximated as [14] …”

Section: Probingmentioning

confidence: 99%

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Variance reduction with practical all-to-all lattice propagators

Endress¹,

Peña²,

Sivalingam³

2015

Computer Physics Communications

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“…on a factorization). An iterative method has been proposed in Tang & Saad (2009) for matrices with a decay property. Some methods have also been developed for matrices arising from specific applications; a more detailed survey is given in Amestoy et al (2010).…”

Section: Computing Error Bars: Partial Computation Of the Inversementioning

confidence: 99%

Simultaneous analysis of large INTEGRAL/SPI11Based on observations with INTEGRAL, an ESA project with instruments and science data center funded by ESA member states (especially the PI countries: Denmark, France, Germany, Italy, Spain, and Switzerland), Czech Republic and Poland with participation of Russia and the USA. datasets: Optimizing the computation of the solution and its variance using sparse matrix algorithms

Bouchet¹,

Amestoy²,

Buttari³

et al. 2013

Astronomy and Computing

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Nowadays, analyzing and reducing the ever larger astronomical datasets is becoming a crucial challenge, especially for long cumulated observation times. The INTEGRAL/SPI X/γ-ray spectrometer is an instrument for which it is essential to process many exposures at the same time in order to increase the low signal-to-noise ratio of the weakest sources. In this context, the conventional methods for data reduction are inefficient and sometimes not feasible at all. Processing several years of data simultaneously requires computing not only the solution of a large system of equations, but also the associated uncertainties. We aim at reducing the computation time and the memory usage. Since the SPI transfer function is sparse, we have used some popular methods for the solution of large sparse linear systems; we briefly review these methods. We use the Multifrontal Massively Parallel Solver (MUMPS) to compute the solution of the system of equations. We also need to compute the variance of the solution, which amounts to computing selected entries of the inverse of the sparse matrix corresponding to our linear system. This can be achieved through one of the latest features of the MUMPS software that has been partly motivated by this work. In this paper we provide a brief presentation of this feature and evaluate its effectiveness on astrophysical problems requiring the processing of large datasets simultaneously, such as the study of the entire emission of the Galaxy. We used these algorithms to solve the large sparse systems arising from SPI data processing and to obtain both their solutions and the associated variances. In conclusion, thanks to these newly developed tools, processing large datasets arising from SPI is now feasible with both a reasonable execution time and a low memory usage.

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“…IfT is diagonally dominant and sparse, i.e., the number of nonzeros of each row ofT is bounded by a small constant, F p is then sparse and the entries shall decrease in value as p increases, as stated in [24] and elaborated in [7,14]. Hence, (T −1 ) (p) is also compressible.…”

Section: I-ltm Has a Sparse Representationmentioning

confidence: 99%

Compressive Inverse Light Transport

Chu¹,

Ng²,

Pahwa³

et al. 2011

Procedings of the British Machine Vision Conference 2011

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This paper explores the possibility of acquiring inverse light transport directly. The current strategy of obtaining an inverse light transport matrix involves two steps: First, acquire the forward light transport matrix (f-LTM) and then calculate the inverse of the f-LTM. Both steps of the strategy requires considerable computational power. In addition to computational cost, the measurement error incurred at the first step inevitably propagates to or potentially gets amplified in the matrix inversion step.In this paper, we propose a sensing strategy that acquires the inverse light transport matrix (i-LTM) directly, without reconstructing the f-LTM. Our direct strategy reduces both computational error and cost of acquiring i-LTM. For that, we propose a compressive inverse theory. Following the compressible property of i-LTM, a reconstruction condition for i-LTM is introduced. This new framework implies a trade-off between two factors: condition numbers of submatrices of f-LTM and the isometry constant of the illumination pattern. Our direct i-LTM reconstruction method is then demonstrated with a 2nd-bounce separation experiment on an M-shaped panel scene. Finally by quantitatively comparing our method with the existing two-stage approach, our method shows higher accuracy with lower complexity. The proofs of main theorem/lemma are contained in the supplementary material. The compressive inverse theory is general and potentially useful for wider application.

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A probing method for computing the diagonal of a matrix inverse

Cited by 120 publications

References 45 publications

Variance reduction with practical all-to-all lattice propagators

Variance reduction with practical all-to-all lattice propagators

Compressive Inverse Light Transport

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