Iterative Solution of Multiple Linear Systems: Theory, Practice, Parallelism and Applications

AbstractAdvancements in ultra-high field (7 T and higher) magnetic resonance imaging (MRI) scanners have made it possible to investigate both the structure and function of the human brain at a sub-millimeter scale. As neuronal feedforward and feedback information arrives in different layers, sub-millimeter functional MRI has the potential to uncover information processing between cortical micro-circuits across cortical depth, i.e. laminar fMRI. For nearly all conventional fMRI analyses, the main assumption is that the relationship between local neuronal activity and the blood oxygenation level dependent (BOLD) signal adheres to the principles of linear systems theory. For laminar fMRI, however, directional blood pooling across cortical depth stemming from the anatomy of the cortical vasculature, potentially violates these linear system assumptions, thereby complicating analysis and interpretation. Here we assess whether the temporal additivity requirement of linear systems theory holds for laminar fMRI. We measured responses elicited by viewing stimuli presented for different durations and evaluated how well the responses to shorter durations predicted those elicited by longer durations. We find that BOLD response predictions are consistently good predictors for observed responses, across all cortical depths, and in all measured visual field maps (V1, V2, and V3). Our results suggest that the temporal additivity assumption for linear systems theory holds for laminar fMRI. We thus show that the temporal additivity assumption holds across cortical depth for sub-millimeter gradient-echo BOLD fMRI in early visual cortex.

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Teaching linear systems theory using Cramer's rule

IEEE Transactions on Education ◽

10.1109/13.57070 ◽

1990 ◽

Vol 33 (3) ◽

pp. 258-267 ◽

Cited By ~ 5

Author(s):

R.E. Klein

Keyword(s):

Systems Theory ◽

Linear Systems ◽

Cramer’S Rule

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Iterative Solution of Indefinite Symmetric Linear Systems by Methods Using Orthogonal Polynomials over Two Disjoint Intervals

SIAM Journal on Numerical Analysis ◽

10.1137/0720052 ◽

1983 ◽

Vol 20 (4) ◽

pp. 784-811 ◽

Cited By ~ 29

Author(s):

Youcef Saad

Keyword(s):

Orthogonal Polynomials ◽

Linear Systems ◽

Iterative Solution

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Study of an Iterative Solution for Boltzmann Transport Equation and Calculation of Thermal Conductivity

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.777.421 ◽

2018 ◽

Vol 777 ◽

pp. 421-425 ◽

Cited By ~ 1

Author(s):

Chhengrot Sion ◽

Chung Hao Hsu

Keyword(s):

Heat Transfer ◽

Thermal Conductivity ◽

Numerical Calculation ◽

Iterative Method ◽

Transport Equation ◽

Linear Systems ◽

Heat Transport ◽

Boltzmann Transport Equation ◽

Iterative Solution ◽

Boltzmann Transport

Many methods have been developed to predict the thermal conductivity of the material. Heat transport is complex and it contains many unknown variables, which makes the thermal conductivity hard to define. The iterative solution of Boltzmann transport equation (BTE) can make the numerical calculation and the nanoscale study of heat transfer possible. Here, we review how to apply the iterative method to solve BTE and many linear systems. This method can compute a sequence of progressively accurate iteration to approximate the solution of BTE.

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Generation of Synthetic Seismograms Using Linear Systems Theory

Canadian Journal of Earth Sciences ◽

10.1139/e71-130 ◽

1971 ◽

Vol 8 (11) ◽

pp. 1409-1422 ◽

Cited By ~ 4

Author(s):

O. G. Jensen ◽

R. M. Ellis

Keyword(s):

Wave Propagation ◽

Elastic Wave ◽

Systems Theory ◽

Linear Systems ◽

Time Domain ◽

Synthetic Seismograms ◽

Elastic Wave Propagation ◽

Arbitrary Angle ◽

Teleseismic Events

The linear systems theory for elastic wave propagation in a multilayered crust has been extended to time domain solutions. Attenuation is specifically included. This direct time domain approach allows the computation of synthetic seismograms for P or SV waveforms incident at an arbitrary angle at the base of the crustal section. To demonstrate the utility of the technique, seismograms are computed for various conditions and comparisons made with teleseismic events recorded in central Alberta.

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