January  2018, 12(1): 175-204. doi: 10.3934/ipi.2018007

On the parameter estimation problem of magnetic resonance advection imaging

1. 

Doctoral Program Computational Mathematics, Johannes Kepler University Linz, Altenberger Strasse 69, A-4040 Linz, Austria

2. 

Industrial Mathematics Institute, Johannes Kepler University Linz, Altenberger Strasse 69, A-4040 Linz, Austria

3. 

Johann Radon Institute, Altenberger Strasse 69, A-4040 Linz, Austria

4. 

Department of Radiology, Weill Cornell Medical College, 516 E 72nd Street New York, NY 10021, USA

* Corresponding author: Simon Hubmer

Received  October 2016 Revised  September 2017 Published  December 2017

Fund Project: The first author is funded by the Austrian Science Fund (FWF): W1214-N15, project DK8. The fourth author acknowledges support by the Nancy M. and Samuel C. Fleming Research Scholar Award in Intercampus Collaborations, Cornell University

We present a reconstruction method for estimating the pulse-wave velocity in the brain from dynamic MRI data. The method is based on solving an inverse problem involving an advection equation. A space-time discretization is used and the resulting largescale inverse problem is solved using an accelerated Landweber type gradient method incorporating sparsity constraints and utilizing a wavelet embedding. Numerical example problems and a real-world data test show a significant improvement over the results obtained by the previously used method.

Citation: Simon Hubmer, Andreas Neubauer, Ronny Ramlau, Henning U. Voss. On the parameter estimation problem of magnetic resonance advection imaging. Inverse Problems & Imaging, 2018, 12 (1) : 175-204. doi: 10.3934/ipi.2018007
References:
[1]

R. Aaslid, Transcranial Doppler Sonography Springer-Verlag, Wien; New York, 1986. doi: 10. 1007/978-3-7091-8864-4. Google Scholar

[2]

G. A. Bateman, Pulse-wave encephalopathy: A comparative study of the hydrodynamics of leukoaraiosis and normal-pressure hydrocephalus, Neuroradiology, 44 (2002), 740-748. doi: 10.1007/s00234-002-0812-0. Google Scholar

[3]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202. doi: 10.1137/080716542. Google Scholar

[4]

M. S. DagliJ. E. Ingeholm and J. V. Haxby, Localization of cardiac-induced signal change in fMRI, NeuroImage, 9 (1999), 407-415. doi: 10.1006/nimg.1998.0424. Google Scholar

[5]

I. Daubechies, Ten Lectures on Wavelets Society for Industrial and Applied Mathematics, 1992. doi: 10. 1137/1. 9781611970104. Google Scholar

[6]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835. Google Scholar

[7]

W. A. EdelsteinG. H. GloverC. J. Hardy and R. W. Redington, The intrinsic signal-to-noise ratio in NMR imaging, Magnetic Resonance in Medicine, 3 (1986), 604-618. doi: 10.1002/mrm.1910030413. Google Scholar

[8]

P. Elter, Methoden und Systeme zur Nichtinvasiven, Kontinuierlichen und Belastungsfreien Blutdruckmessung Thesis, 2001.Google Scholar

[9]

S. Engblom and D. Lukarski, FSPARSE, http://user.it.uu.se/~stefane/freeware.html, 2014.Google Scholar

[10]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems Dordrecht: Kluwer Academic Publishers, 1996. Google Scholar

[11]

A. FrydrychowiczC. J. Francois and P. A. Turski, Four-dimensional phase contrast magnetic resonance angiography: Potential clinical applications, European Journal of Radiology, 80 (2011), 24-35. doi: 10.1016/j.ejrad.2011.01.094. Google Scholar

[12]

M. HankeF. J. BaumgartnerP. IbeF. R. KauleS. PollmannO. SpeckW. Zinke and J. Stadler, A high-resolution 7-Tesla fMRI dataset from complex natural stimulation with an audio movie, Scientific Data, 1 (2014), 1-18. doi: 10.1038/sdata.2014.3. Google Scholar

[13]

M. C. Henry FeugeasG. De MarcoI. I. PerettiS. Godon-HardyD. Fredy and E. S. Claeys, Age-related cerebral white matter changes and pulse-wave encephalopathy: Observations with three-dimensional MRI, Magnetic Resonance Imaging, 23 (2005), 929-937. doi: 10.1016/j.mri.2005.09.002. Google Scholar

[14]

L. H. G. HenskensA. A. KroonR. J. van OostenbruggeE. H. B. M. GronenschildM. M. J. J. Fuss-LejeuneP. A. M. HofmanJ. Lodder and P. W. de Leeuw, Increased aortic pulse wave velocity is associated with silent cerebral small-vessel disease in hypertensive patients, Hypertension, 52 (2008), 1120-1126. doi: 10.1161/HYPERTENSIONAHA.108.119024. Google Scholar

[15]

E. M. C. Hillman, Coupling mechanism and significance of the BOLD Signal: A status report, Annual Review of Neuroscience, 37 (2014), 161-181. doi: 10.1146/annurev-neuro-071013-014111. Google Scholar

[16]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear ill-posed Problems Berlin: de Gruyter, 2008. Google Scholar

[17]

D. J. Korteweg, Über die Fortpflanzungsgeschwindigkeit des Schalles in Elastischen Röhren, Annalen der Physik, 241 (1878), 525-542. Google Scholar

[18]

S. LaurentJ. CockcroftL. Van BortelP. BoutouyrieC. GiannattasioD. HayozB. PannierC. VlachopoulosI. WilkinsonH. Struijker-Boudier and E. N. Non-invasive, Expert consensus document on arterial stiffness: Methodological issues and clinical applications, European Heart Journal, 27 (2006), 2588-2605. doi: 10.1093/eurheartj/ehl254. Google Scholar

[19]

J. K. J. Li, Dynamics of the Vascular System Series on Bioengineering and Biomedical Engineering, World Scientific, River Edge, N. J., 2004.Google Scholar

[20]

D. A. LorenzP. Maass and P. Q. Muoi, Gradient descent for Tikhonov functionals with sparsity constraints: Theory and numerical comparison of step size rules., Electron. Trans. Numer. Anal., 39 (2012), 437-463. Google Scholar

[21]

D. W. McRobbie, E. A. Moore and M. J. Graves, MRI from Picture to Proton 3rd edition, University Printing House, Cambridge University Press, Cambridge; New York, 2016.Google Scholar

[22]

S. MoellerE. YacoubC. A. OlmanE. AuerbachJ. StruppN. Harel and K. Ugurbil, Multiband multislice GE-EPI at 7 Tesla, with 16-fold acceleration using partial parallel imaging with application to high spatial and temporal whole-brain fMRI, Magnetic Resonance in Medicine, 63 (2010), 1144-1153. doi: 10.1002/mrm.22361. Google Scholar

[23]

A. I. Moens, Over de Voortplantingssnelheid van den Pols [On the Speed of Propagation of the Pulse] Thesis, 1877.Google Scholar

[24]

Y. Nesterov, A method of solving a convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547. Google Scholar

[25]

S. OgawaT. M. LeeA. R. Kay and D. W. Tank, Brain magnetic-resonance-imaging with contrast dependent on blood oxygenation, Proceedings of the National Academy of Sciences of the United States of America, 87 (1990), 9868-9872. doi: 10.1073/pnas.87.24.9868. Google Scholar

[26]

S. W. Rabkin, Arterial stiffness: Detection and consequences in cognitive impairment and dementia of the elderly, Journal of Alzheimers Disease, 32 (2012), 541-549. Google Scholar

[27]

R. Ramlau, Regularization properties of Tikhonov regularization with sparsity constraints., Electron. Trans. Numer. Anal., 30 (2008), 54-74. Google Scholar

[28]

R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints, Numer. Math., 104 (2006), 177-203. doi: 10.1007/s00211-006-0016-3. Google Scholar

[29]

K. SagawaR. K. Lie and J. Schaefer, Translation of Otto Frank's paper "Die Grundform des Arteriellen Pulses" Zeitschrift für Biologie 37: 483-526 (1899), Jounral of Molecular and Cellular Cardiology, 22 (1990), 253-254. doi: 10.1016/0022-2828(90)91459-K. Google Scholar

[30]

R. SladkyK. J. FristonJ. TroestlR. CunningtonE. Moser and C. Windischberger, Slice-time effects and their correction in functional MRI, NeuroImage, 58 (2011), 588-594. Google Scholar

[31]

M. K. StehlingR. Turner and P. Mansfield, Echo-planar imaging -magnetic-resonance-imaging in a fraction of a second, Science, 254 (1991), 43-50. doi: 10.1126/science.1925560. Google Scholar

[32]

Y. J. TongL. M. Hocke and B. D. Frederick, Short repetition time multiband echo-planar imaging with simultaneous pulse recording allows dynamic imaging of the cardiac pulsation signal, Magnetic Resonance in Medicine, 72 (2014), 1268-1276. doi: 10.1002/mrm.25041. Google Scholar

[33]

H. U. Voss, J. P. Dyke, K. Tabelow, N. D. Schiff and D. J. Ballon, Mapping cerebrovascular dynamics with magnetic resonance advection imaging (MRAI): Modeling challenges and estimation bias, Meeting of the Society for Neuroscience.Google Scholar

[34]

H. U. VossJ. P. DykeK. TabelowN. D. Schiff and D. J. Ballon, Magnetic resonance advection imaging (MRAI) of cerebrovascular pulse dynamics, Journal of Cerebral Blood Flow and Metabolism, 37 (2017), 1223-1235. Google Scholar

[35]

H. U. Voss and N. D. Schiff, Searching for conservation laws in brain dynamics-BOLD flux and source imaging, Entropy, 16 (2014), 3689-3709. doi: 10.3390/e16073689. Google Scholar

[36]

E. A. H. WarnertK. MurphyJ. E. Hall and R. G. Wise, Noninvasive assessment of arterial compliance of human cerebral arteries with short inversion time arterial spin labeling, Journal of Cerebral Blood Flow and Metabolism, 35 (2015), 461-468. doi: 10.1038/jcbfm.2014.219. Google Scholar

[37]

E. C. WongR. B. Buxton and L. R. Frank, Quantitative perfusion imaging using arterial spin labeling, Neuroimaging Clinics of North America, 9 (1999), 333-342. Google Scholar

[38]

L. YanC. Y. LiuR. X. SmithM. JogM. LanghamK. KrasilevaY. ChenJ. M. Ringman and D. J. J. Wang, Assessing intracranial vascular compliance using dynamic arterial spin labeling, NeuroImage, 124 (2016), 433-441. doi: 10.1016/j.neuroimage.2015.09.008. Google Scholar

[39]

X. YuY. HeM. WangH. MerkleS. J. DoddA. C. Silva and A. P. Koretsky, Sensory and optogenetically driven single-vessel fMRI, Nature Methods, 13 (2016), 337-340. doi: 10.1038/nmeth.3765. Google Scholar

[40]

M. Zamir, The Physics of Pulsatile Flow, Biological physics series, AIP Press; Springer, New York, 2000, URL http://www.loc.gov/catdir/enhancements/fy0816/99042457-d.html. Google Scholar

show all references

References:
[1]

R. Aaslid, Transcranial Doppler Sonography Springer-Verlag, Wien; New York, 1986. doi: 10. 1007/978-3-7091-8864-4. Google Scholar

[2]

G. A. Bateman, Pulse-wave encephalopathy: A comparative study of the hydrodynamics of leukoaraiosis and normal-pressure hydrocephalus, Neuroradiology, 44 (2002), 740-748. doi: 10.1007/s00234-002-0812-0. Google Scholar

[3]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202. doi: 10.1137/080716542. Google Scholar

[4]

M. S. DagliJ. E. Ingeholm and J. V. Haxby, Localization of cardiac-induced signal change in fMRI, NeuroImage, 9 (1999), 407-415. doi: 10.1006/nimg.1998.0424. Google Scholar

[5]

I. Daubechies, Ten Lectures on Wavelets Society for Industrial and Applied Mathematics, 1992. doi: 10. 1137/1. 9781611970104. Google Scholar

[6]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835. Google Scholar

[7]

W. A. EdelsteinG. H. GloverC. J. Hardy and R. W. Redington, The intrinsic signal-to-noise ratio in NMR imaging, Magnetic Resonance in Medicine, 3 (1986), 604-618. doi: 10.1002/mrm.1910030413. Google Scholar

[8]

P. Elter, Methoden und Systeme zur Nichtinvasiven, Kontinuierlichen und Belastungsfreien Blutdruckmessung Thesis, 2001.Google Scholar

[9]

S. Engblom and D. Lukarski, FSPARSE, http://user.it.uu.se/~stefane/freeware.html, 2014.Google Scholar

[10]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems Dordrecht: Kluwer Academic Publishers, 1996. Google Scholar

[11]

A. FrydrychowiczC. J. Francois and P. A. Turski, Four-dimensional phase contrast magnetic resonance angiography: Potential clinical applications, European Journal of Radiology, 80 (2011), 24-35. doi: 10.1016/j.ejrad.2011.01.094. Google Scholar

[12]

M. HankeF. J. BaumgartnerP. IbeF. R. KauleS. PollmannO. SpeckW. Zinke and J. Stadler, A high-resolution 7-Tesla fMRI dataset from complex natural stimulation with an audio movie, Scientific Data, 1 (2014), 1-18. doi: 10.1038/sdata.2014.3. Google Scholar

[13]

M. C. Henry FeugeasG. De MarcoI. I. PerettiS. Godon-HardyD. Fredy and E. S. Claeys, Age-related cerebral white matter changes and pulse-wave encephalopathy: Observations with three-dimensional MRI, Magnetic Resonance Imaging, 23 (2005), 929-937. doi: 10.1016/j.mri.2005.09.002. Google Scholar

[14]

L. H. G. HenskensA. A. KroonR. J. van OostenbruggeE. H. B. M. GronenschildM. M. J. J. Fuss-LejeuneP. A. M. HofmanJ. Lodder and P. W. de Leeuw, Increased aortic pulse wave velocity is associated with silent cerebral small-vessel disease in hypertensive patients, Hypertension, 52 (2008), 1120-1126. doi: 10.1161/HYPERTENSIONAHA.108.119024. Google Scholar

[15]

E. M. C. Hillman, Coupling mechanism and significance of the BOLD Signal: A status report, Annual Review of Neuroscience, 37 (2014), 161-181. doi: 10.1146/annurev-neuro-071013-014111. Google Scholar

[16]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear ill-posed Problems Berlin: de Gruyter, 2008. Google Scholar

[17]

D. J. Korteweg, Über die Fortpflanzungsgeschwindigkeit des Schalles in Elastischen Röhren, Annalen der Physik, 241 (1878), 525-542. Google Scholar

[18]

S. LaurentJ. CockcroftL. Van BortelP. BoutouyrieC. GiannattasioD. HayozB. PannierC. VlachopoulosI. WilkinsonH. Struijker-Boudier and E. N. Non-invasive, Expert consensus document on arterial stiffness: Methodological issues and clinical applications, European Heart Journal, 27 (2006), 2588-2605. doi: 10.1093/eurheartj/ehl254. Google Scholar

[19]

J. K. J. Li, Dynamics of the Vascular System Series on Bioengineering and Biomedical Engineering, World Scientific, River Edge, N. J., 2004.Google Scholar

[20]

D. A. LorenzP. Maass and P. Q. Muoi, Gradient descent for Tikhonov functionals with sparsity constraints: Theory and numerical comparison of step size rules., Electron. Trans. Numer. Anal., 39 (2012), 437-463. Google Scholar

[21]

D. W. McRobbie, E. A. Moore and M. J. Graves, MRI from Picture to Proton 3rd edition, University Printing House, Cambridge University Press, Cambridge; New York, 2016.Google Scholar

[22]

S. MoellerE. YacoubC. A. OlmanE. AuerbachJ. StruppN. Harel and K. Ugurbil, Multiband multislice GE-EPI at 7 Tesla, with 16-fold acceleration using partial parallel imaging with application to high spatial and temporal whole-brain fMRI, Magnetic Resonance in Medicine, 63 (2010), 1144-1153. doi: 10.1002/mrm.22361. Google Scholar

[23]

A. I. Moens, Over de Voortplantingssnelheid van den Pols [On the Speed of Propagation of the Pulse] Thesis, 1877.Google Scholar

[24]

Y. Nesterov, A method of solving a convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547. Google Scholar

[25]

S. OgawaT. M. LeeA. R. Kay and D. W. Tank, Brain magnetic-resonance-imaging with contrast dependent on blood oxygenation, Proceedings of the National Academy of Sciences of the United States of America, 87 (1990), 9868-9872. doi: 10.1073/pnas.87.24.9868. Google Scholar

[26]

S. W. Rabkin, Arterial stiffness: Detection and consequences in cognitive impairment and dementia of the elderly, Journal of Alzheimers Disease, 32 (2012), 541-549. Google Scholar

[27]

R. Ramlau, Regularization properties of Tikhonov regularization with sparsity constraints., Electron. Trans. Numer. Anal., 30 (2008), 54-74. Google Scholar

[28]

R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints, Numer. Math., 104 (2006), 177-203. doi: 10.1007/s00211-006-0016-3. Google Scholar

[29]

K. SagawaR. K. Lie and J. Schaefer, Translation of Otto Frank's paper "Die Grundform des Arteriellen Pulses" Zeitschrift für Biologie 37: 483-526 (1899), Jounral of Molecular and Cellular Cardiology, 22 (1990), 253-254. doi: 10.1016/0022-2828(90)91459-K. Google Scholar

[30]

R. SladkyK. J. FristonJ. TroestlR. CunningtonE. Moser and C. Windischberger, Slice-time effects and their correction in functional MRI, NeuroImage, 58 (2011), 588-594. Google Scholar

[31]

M. K. StehlingR. Turner and P. Mansfield, Echo-planar imaging -magnetic-resonance-imaging in a fraction of a second, Science, 254 (1991), 43-50. doi: 10.1126/science.1925560. Google Scholar

[32]

Y. J. TongL. M. Hocke and B. D. Frederick, Short repetition time multiband echo-planar imaging with simultaneous pulse recording allows dynamic imaging of the cardiac pulsation signal, Magnetic Resonance in Medicine, 72 (2014), 1268-1276. doi: 10.1002/mrm.25041. Google Scholar

[33]

H. U. Voss, J. P. Dyke, K. Tabelow, N. D. Schiff and D. J. Ballon, Mapping cerebrovascular dynamics with magnetic resonance advection imaging (MRAI): Modeling challenges and estimation bias, Meeting of the Society for Neuroscience.Google Scholar

[34]

H. U. VossJ. P. DykeK. TabelowN. D. Schiff and D. J. Ballon, Magnetic resonance advection imaging (MRAI) of cerebrovascular pulse dynamics, Journal of Cerebral Blood Flow and Metabolism, 37 (2017), 1223-1235. Google Scholar

[35]

H. U. Voss and N. D. Schiff, Searching for conservation laws in brain dynamics-BOLD flux and source imaging, Entropy, 16 (2014), 3689-3709. doi: 10.3390/e16073689. Google Scholar

[36]

E. A. H. WarnertK. MurphyJ. E. Hall and R. G. Wise, Noninvasive assessment of arterial compliance of human cerebral arteries with short inversion time arterial spin labeling, Journal of Cerebral Blood Flow and Metabolism, 35 (2015), 461-468. doi: 10.1038/jcbfm.2014.219. Google Scholar

[37]

E. C. WongR. B. Buxton and L. R. Frank, Quantitative perfusion imaging using arterial spin labeling, Neuroimaging Clinics of North America, 9 (1999), 333-342. Google Scholar

[38]

L. YanC. Y. LiuR. X. SmithM. JogM. LanghamK. KrasilevaY. ChenJ. M. Ringman and D. J. J. Wang, Assessing intracranial vascular compliance using dynamic arterial spin labeling, NeuroImage, 124 (2016), 433-441. doi: 10.1016/j.neuroimage.2015.09.008. Google Scholar

[39]

X. YuY. HeM. WangH. MerkleS. J. DoddA. C. Silva and A. P. Koretsky, Sensory and optogenetically driven single-vessel fMRI, Nature Methods, 13 (2016), 337-340. doi: 10.1038/nmeth.3765. Google Scholar

[40]

M. Zamir, The Physics of Pulsatile Flow, Biological physics series, AIP Press; Springer, New York, 2000, URL http://www.loc.gov/catdir/enhancements/fy0816/99042457-d.html. Google Scholar

Figure 1.  Example image of a clinical MRI scanner
Figure 2.  Simulation phantom: Magnitude of the norm of the velocity vector field (left figure) and colour direction MIP of the velocity (right figure)
Figure 12.  Result of the algorithm applied to the modified problem ($\delta =1%$), where all involved velocities were multiplied by a factor of $10^4$, using the weak divergence-free, the wavelet embedding and the sparsity option with $\alpha =10^{-3}$. Velocity norm MIP (left) and colour direction MIP (right)
Figure 3.  Magnitudes of the velocity vector field components. Left: First component. Middle: Second component. Right: Third component
Figure 4.  Result of the algorithm applied to the test problem ($\delta =1%$), using no additional options. Velocity norm MIP (left) and colour direction MIP (right)
Figure 5.  Result of the algorithm applied to the test problem ($\delta =1%$), using the weak divergence-free option. Velocity norm MIP (left) and colour direction MIP (right)
Figure 6.  Result of the algorithm applied to the test problem ($\delta =1%$), using the wavelet embedding option. Velocity norm MIP (left) and colour direction MIP (right)
Figure 7.  Result of the algorithm applied to the test problem ($\delta =1%$), using the weak divergence-free and the wavelet embedding option. Velocity norm MIP (left) and colour direction MIP (right)
Figure 8.  Result of the algorithm applied to the test problem ($\delta =1%$), using the sparsity option with $\alpha =10^{-3}$. Velocity norm MIP (left) and colour direction MIP (right)
Figure 9.  Result of the algorithm applied to the test problem ($\delta =1%$), using the wavelet embedding and the sparsity option with $\alpha =10^{-3}$. Velocity norm MIP (left) and colour direction MIP (right)
Figure 10.  Result of the algorithm applied to the test problem ($\delta =1%$), using the weak divergence-free and the sparsity option with $\alpha =10^{-3}$. Velocity norm MIP and colour direction MIP
Figure 11.  Result of the algorithm applied to the test problem ($\delta =1%$), using the weak divergence-free, the wavelet embedding and the sparsity option with $\alpha =10^{-3}$. Velocity norm MIP and colour direction MIP
Figure 13.  Result of the algorithm applied to the modified problem ($\delta =1%$), where all involved velocities were multiplied by a factor of ^4$ with initial signal (58), using the weak divergence-free and the wavelet embedding option. Velocity norm MIP (left) and colour direction MIP (right)
Figure 14.  Results of our proposed algorithm (upper two figures, 20 seconds of data) and the regression-based algorithm (lower two figures, 15 minutes of data), applied to subject 16 of the data set. Velocity norm MIPs (left) and colour direction MIPs (right)
Figure 15.  Results of our proposed algorithm (upper two figures, 20 seconds of data) and the regression-based algorithm (lower two figures, 15 minutes of data), applied to subject 2 of the data set. Velocity norm MIPs (left) and colour direction MIPs (right)
Table 6.1.  Comparison of the results of the reconstruction algorithm applied to the test problem ($\delta =1%$), achieved using combinations of the different computation options
div-free wavelets sparsity $k_*$ ${\left\| {\left( {\vec v_{{k_*}}^\delta ,\vec \rho _{0,{k_*}}^\delta } \right) - \left( {{{\vec v}^\dagger },\vec \rho _0^\dagger } \right)} \right\|_{{\ell _2}}}$
Figure 4 no no no 90 16.9658
Figure 5 yes no no 126 9.2904
Figure 6noyesno12116.4324
Figure 7yesyesno1628.8878
Figure 8nonoyes7117.179
Figure 9noyesyes10016.7834
Figure 10yesnoyes995.6577
Figure 11yesyesyes1385.5245
div-free wavelets sparsity $k_*$ ${\left\| {\left( {\vec v_{{k_*}}^\delta ,\vec \rho _{0,{k_*}}^\delta } \right) - \left( {{{\vec v}^\dagger },\vec \rho _0^\dagger } \right)} \right\|_{{\ell _2}}}$
Figure 4 no no no 90 16.9658
Figure 5 yes no no 126 9.2904
Figure 6noyesno12116.4324
Figure 7yesyesno1628.8878
Figure 8nonoyes7117.179
Figure 9noyesyes10016.7834
Figure 10yesnoyes995.6577
Figure 11yesyesyes1385.5245
[1]

Vinicius Albani, Adriano De Cezaro, Jorge P. Zubelli. On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy. Inverse Problems & Imaging, 2016, 10 (1) : 1-25. doi: 10.3934/ipi.2016.10.1

[2]

Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353

[3]

Barbara Kaltenbacher, Ivan Tomba. Enhanced choice of the parameters in an iteratively regularized Newton-Landweber iteration in Banach space. Conference Publications, 2015, 2015 (special) : 686-695. doi: 10.3934/proc.2015.0686

[4]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[5]

Alberto M. Gambaruto, João Janela, Alexandra Moura, Adélia Sequeira. Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology. Mathematical Biosciences & Engineering, 2011, 8 (2) : 409-423. doi: 10.3934/mbe.2011.8.409

[6]

Alberto Gambaruto, João Janela, Alexandra Moura, Adélia Sequeira. Shear-thinning effects of hemodynamics in patient-specific cerebral aneurysms. Mathematical Biosciences & Engineering, 2013, 10 (3) : 649-665. doi: 10.3934/mbe.2013.10.649

[7]

Scott R. Pope, Laura M. Ellwein, Cheryl L. Zapata, Vera Novak, C. T. Kelley, Mette S. Olufsen. Estimation and identification of parameters in a lumped cerebrovascular model. Mathematical Biosciences & Engineering, 2009, 6 (1) : 93-115. doi: 10.3934/mbe.2009.6.93

[8]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

[9]

Plamen Stefanov, Yang Yang. Multiwave tomography with reflectors: Landweber's iteration. Inverse Problems & Imaging, 2017, 11 (2) : 373-401. doi: 10.3934/ipi.2017018

[10]

Yuming Zhang. On continuity equations in space-time domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212

[11]

David Maxwell. Kozlov-Maz'ya iteration as a form of Landweber iteration. Inverse Problems & Imaging, 2014, 8 (2) : 537-560. doi: 10.3934/ipi.2014.8.537

[12]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511

[13]

Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial & Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471

[14]

Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems & Imaging, 2007, 1 (1) : 1-11. doi: 10.3934/ipi.2007.1.1

[15]

Susanne Pumplün, Thomas Unger. Space-time block codes from nonassociative division algebras. Advances in Mathematics of Communications, 2011, 5 (3) : 449-471. doi: 10.3934/amc.2011.5.449

[16]

Gerard A. Maugin, Martine Rousseau. Prolegomena to studies on dynamic materials and their space-time homogenization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1599-1608. doi: 10.3934/dcdss.2013.6.1599

[17]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713

[18]

Frédérique Oggier, B. A. Sethuraman. Quotients of orders in cyclic algebras and space-time codes. Advances in Mathematics of Communications, 2013, 7 (4) : 441-461. doi: 10.3934/amc.2013.7.441

[19]

Grégory Berhuy. Algebraic space-time codes based on division algebras with a unitary involution. Advances in Mathematics of Communications, 2014, 8 (2) : 167-189. doi: 10.3934/amc.2014.8.167

[20]

David Grant, Mahesh K. Varanasi. Duality theory for space-time codes over finite fields. Advances in Mathematics of Communications, 2008, 2 (1) : 35-54. doi: 10.3934/amc.2008.2.35

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (56)
  • HTML views (315)
  • Cited by (1)

[Back to Top]