American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Robust reconstruction of fluorescence molecular tomography with an optimized illumination pattern
  • IPI Home
  • This Issue
  • Next Article
    Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency
June  2020, 14(3): 511-533. doi: 10.3934/ipi.2020024

Statistical characterization of scattering delay in synthetic aperture radar imaging

Mikhail Gilman , and  Semyon Tsynkov

Department of Mathematics, North Carolina State University, Campus Box 8205, Raleigh, NC 27695, USA

* Corresponding author: Mikhail Gilman

Received  September 2019 Revised  December 2019 Published  March 2020

Distinguishing between the instantaneous and delayed scatterers in synthetic aperture radar (SAR) images is important for target identification and characterization. To perform this task, one can use the autocorrelation analysis of coordinate-delay images. However, due to the range-delay ambiguity the difference in the correlation properties between the instantaneous and delayed targets may be small. Moreover, the reliability of discrimination is affected by speckle, which is ubiquitous in SAR images, and requires statistical treatment.

Previously, we have developed a maximum likelihood based approach for discriminating between the instantaneous and delayed targets in SAR images. To test it, we employed simple statistical models. They allowed us to simulate ensembles of images that depend on various parameters, including aperture width and target contrast.

In the current paper, we enhance our previously developed methodology by establishing confidence levels for the discrimination between the instantaneous and delayed scatterers. Our procedure takes into account the difference in thresholds for different target contrasts without making any assumptions about the statistics of those contrasts.

Keywords: SAR imaging, scattering delay, speckle, confidence level, Monte-Carlo.
Mathematics Subject Classification: Primary: 65R32, 94A08; Secondary: 65C05.
Citation: Mikhail Gilman, Semyon Tsynkov. Statistical characterization of scattering delay in synthetic aperture radar imaging. Inverse Problems & Imaging, 2020, 14 (3) : 511-533. doi: 10.3934/ipi.2020024
References:
[1]

J. C. Allen and S. L. Hobbs, Spectral estimation of non-stationary white noise, J. Franklin Inst. B, 334 (1997), 99-116.  doi: 10.1016/S0016-0032(96)00060-9.  Google Scholar

[2]

M. Basu, Gaussian-based edge-detection methods–a survey, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 32 (2002), 252-260.  doi: 10.1109/TSMCC.2002.804448.  Google Scholar

[3]

C. M. Bishop, Pattern Recognition and Machine Learning, Information Science and Statistics, Springer, New York, 2006.  Google Scholar

[4]

B. Borden, Dispersive scattering for radar-based target classification and duct-induced image artifact mitigation, NATO Symposium on Non-Cooperative Air Target Identfication Using Radar, Mannheim, Germany, 1998, 14.1–14.7. Google Scholar

[5]

J. Canny, A computational approach to edge detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-8 (1986), 679-698.  doi: 10.1109/TPAMI.1986.4767851.  Google Scholar

[6]

S. Chen and H. Wang, SAR target recognition based on deep learning, 2014 International Conference on Data Science and Advanced Analytics (DSAA), IEEE, Shanghai, China, 2014, 541–547. doi: 10.1109/DSAA.2014.7058124.  Google Scholar

[7]

V. C. Chen and H. Ling, Time-Frequency Transforms for Radar Imaging and Signal Analysis, Artech House Radar Library, Artech House, Norwood, MA, 2002.  Google Scholar

[8]

M. Cheney and B. Borden, Fundamentals of Radar Imaging, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 79, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898719291.  Google Scholar

[9]

F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, (eds.). NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.24 of 2019-09-15. Google Scholar

[10]

M. Ferrara, A. Homan and M. Cheney, Hyperspectral SAR, IEEE Trans. Geoscience and Remote Sensing, 55 (2017), 1682–1695. doi: 10.1109/TGRS.2016.2629265.  Google Scholar

[11]

J. Garnier and K. Sølna, A multiscale approach to synthetic aperture radar in dispersive random media, Inverse Problems, 29 (2013), 054006, (18pp). doi: 10.1088/0266-5611/29/5/054006.  Google Scholar

[12]

M. Gilman and S. Tsynkov, Mathematical analysis of SAR imaging through a turbulent ionosphere, Application of Mathematics in Technical and Natural Sciences: 9th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences– AMiTaNS'17, Vol. 1895, AIP Conference Proceedings, Albena, Bulgaria, 2017, 020003 (23pp). doi: 10.1063/1.5007357.  Google Scholar

[13]

M. Gilman, E. Smith and S. Tsynkov, Transionospheric Synthetic Aperture Imaging, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, Switzerland, 2017. doi: 10.1007/978-3-319-52127-5.  Google Scholar

[14]

M. Gilman and S. Tsynkov, Detection of delayed target response in SAR, Inverse Problems, 35 (2019), 085005, (38pp). doi: 10.1088/1361-6420/ab1c80.  Google Scholar

[15]

J. W. Goodman, Statistical properties of laser speckle patterns, Topics in Applied Physics: Laser Speckle and Related Phenomena (ed. J. C. Dainty, ), Vol. 9, Springer-Verlag, Berlin, 1984, 9–75. Google Scholar

[16]

T. Hastie, R. Tibshirani and J. Friedman, The Elements Of Statistical Learning. Data Mining, Inference, and Prediction, 2$^nd$ edition, Springer Series in Statistics, Springer, New York, 2009. doi: 10.1007/978-0-387-84858-7.  Google Scholar

[17]

A. Krizhevsky, I. Sutskever and G. E. Hinton, ImageNet classification with deep convolutional neural networks, Advances in Neural Information Processing Systems 25, Curran Associates, Inc., 2012, 1097–1105. Google Scholar

[18]

D. Marr and E. Hildreth, Theory of edge detection, Proc. R. Soc. Lond. B, 207 (1980), 187-217.  doi: 10.1098/rspb.1980.0020.  Google Scholar

[19] W. Mendenhall and R. L. Scheaffer, Mathematical Statistics with Applications, Duxbury Press, North Scituate, Mass., 1973.   Google Scholar
[20]

F. J. Meyer, K. Chotoo, S. D. Chotoo, B. D. Huxtable and C. S. Carrano, The influence of equatorial scintillation on L-band SAR image quality and phase, IEEE Trans. Geoscience and Remote Sensing, 54 (2016), 869-880.  doi: 10.1109/TGRS.2015.2468573.  Google Scholar

[21]

D. Mishkin, N. Sergievskiy and J. Matas, Systematic evaluation of convolution neural network advances on the Imagenet, Computer Vision and Image Understanding, 161 (2017), 11-19.  doi: 10.1016/j.cviu.2017.05.007.  Google Scholar

[22]

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, Dover Publications, Inc., Mineola, NY, 2007.  Google Scholar

[23]

C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images, Artech House Remote Sensing Library, Artech House, Boston, 1988. Google Scholar

[24]

S. M. Rytov, Y. A. Kravtsov and V. I. Tatarskii, Principles of Statistical Radiophysics, Volume 4. Wave Propagation Through Random Media, Springer-Verlag, Berlin, 1989.  Google Scholar

[25]

T. Scarnati and B. Lewis, A deep learning approach to the synthetic and measured paired and labeled experiment (SAMPLE) challenge problem, Algorithms for Synthetic Aperture Radar Imagery XXVI, 10987 (2019), 109870G. doi: 10.1117/12.2523458.  Google Scholar

[26]

F. T. Ulaby and M. C. Dobson, Handbook of Radar Scattering Statistics for Terrain, Artech House Remote Sensing Library, Artech House, Norwood, MA, USA, 1989. Google Scholar

[27]

X. X. Zhu, D. Tuia, L. Mou, G.-S. Xia, L. Zhang, F. Xu and F. Fraundorfer, Deep learning in remote sensing: A comprehensive review and list of resources, IEEE Geoscience and Remote Sensing Magazine, 5 (2017), 8-36.  doi: 10.1109/MGRS.2017.2762307.  Google Scholar

[28]

D. Ziou and S. Tabbone, et al., Edge detection techniques-an overview, Pattern Recognition and Image Analysis C/C of Raspoznavaniye Obrazov I Analiz Izobrazhenii, 8 (1998), 537-559. Google Scholar

show all references

References:
[1]

J. C. Allen and S. L. Hobbs, Spectral estimation of non-stationary white noise, J. Franklin Inst. B, 334 (1997), 99-116.  doi: 10.1016/S0016-0032(96)00060-9.  Google Scholar

[2]

M. Basu, Gaussian-based edge-detection methods–a survey, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 32 (2002), 252-260.  doi: 10.1109/TSMCC.2002.804448.  Google Scholar

[3]

C. M. Bishop, Pattern Recognition and Machine Learning, Information Science and Statistics, Springer, New York, 2006.  Google Scholar

[4]

B. Borden, Dispersive scattering for radar-based target classification and duct-induced image artifact mitigation, NATO Symposium on Non-Cooperative Air Target Identfication Using Radar, Mannheim, Germany, 1998, 14.1–14.7. Google Scholar

[5]

J. Canny, A computational approach to edge detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-8 (1986), 679-698.  doi: 10.1109/TPAMI.1986.4767851.  Google Scholar

[6]

S. Chen and H. Wang, SAR target recognition based on deep learning, 2014 International Conference on Data Science and Advanced Analytics (DSAA), IEEE, Shanghai, China, 2014, 541–547. doi: 10.1109/DSAA.2014.7058124.  Google Scholar

[7]

V. C. Chen and H. Ling, Time-Frequency Transforms for Radar Imaging and Signal Analysis, Artech House Radar Library, Artech House, Norwood, MA, 2002.  Google Scholar

[8]

M. Cheney and B. Borden, Fundamentals of Radar Imaging, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 79, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898719291.  Google Scholar

[9]

F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, (eds.). NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.24 of 2019-09-15. Google Scholar

[10]

M. Ferrara, A. Homan and M. Cheney, Hyperspectral SAR, IEEE Trans. Geoscience and Remote Sensing, 55 (2017), 1682–1695. doi: 10.1109/TGRS.2016.2629265.  Google Scholar

[11]

J. Garnier and K. Sølna, A multiscale approach to synthetic aperture radar in dispersive random media, Inverse Problems, 29 (2013), 054006, (18pp). doi: 10.1088/0266-5611/29/5/054006.  Google Scholar

[12]

M. Gilman and S. Tsynkov, Mathematical analysis of SAR imaging through a turbulent ionosphere, Application of Mathematics in Technical and Natural Sciences: 9th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences– AMiTaNS'17, Vol. 1895, AIP Conference Proceedings, Albena, Bulgaria, 2017, 020003 (23pp). doi: 10.1063/1.5007357.  Google Scholar

[13]

M. Gilman, E. Smith and S. Tsynkov, Transionospheric Synthetic Aperture Imaging, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, Switzerland, 2017. doi: 10.1007/978-3-319-52127-5.  Google Scholar

[14]

M. Gilman and S. Tsynkov, Detection of delayed target response in SAR, Inverse Problems, 35 (2019), 085005, (38pp). doi: 10.1088/1361-6420/ab1c80.  Google Scholar

[15]

J. W. Goodman, Statistical properties of laser speckle patterns, Topics in Applied Physics: Laser Speckle and Related Phenomena (ed. J. C. Dainty, ), Vol. 9, Springer-Verlag, Berlin, 1984, 9–75. Google Scholar

[16]

T. Hastie, R. Tibshirani and J. Friedman, The Elements Of Statistical Learning. Data Mining, Inference, and Prediction, 2$^nd$ edition, Springer Series in Statistics, Springer, New York, 2009. doi: 10.1007/978-0-387-84858-7.  Google Scholar

[17]

A. Krizhevsky, I. Sutskever and G. E. Hinton, ImageNet classification with deep convolutional neural networks, Advances in Neural Information Processing Systems 25, Curran Associates, Inc., 2012, 1097–1105. Google Scholar

[18]

D. Marr and E. Hildreth, Theory of edge detection, Proc. R. Soc. Lond. B, 207 (1980), 187-217.  doi: 10.1098/rspb.1980.0020.  Google Scholar

[19] W. Mendenhall and R. L. Scheaffer, Mathematical Statistics with Applications, Duxbury Press, North Scituate, Mass., 1973.   Google Scholar
[20]

F. J. Meyer, K. Chotoo, S. D. Chotoo, B. D. Huxtable and C. S. Carrano, The influence of equatorial scintillation on L-band SAR image quality and phase, IEEE Trans. Geoscience and Remote Sensing, 54 (2016), 869-880.  doi: 10.1109/TGRS.2015.2468573.  Google Scholar

[21]

D. Mishkin, N. Sergievskiy and J. Matas, Systematic evaluation of convolution neural network advances on the Imagenet, Computer Vision and Image Understanding, 161 (2017), 11-19.  doi: 10.1016/j.cviu.2017.05.007.  Google Scholar

[22]

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, Dover Publications, Inc., Mineola, NY, 2007.  Google Scholar

[23]

C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images, Artech House Remote Sensing Library, Artech House, Boston, 1988. Google Scholar

[24]

S. M. Rytov, Y. A. Kravtsov and V. I. Tatarskii, Principles of Statistical Radiophysics, Volume 4. Wave Propagation Through Random Media, Springer-Verlag, Berlin, 1989.  Google Scholar

[25]

T. Scarnati and B. Lewis, A deep learning approach to the synthetic and measured paired and labeled experiment (SAMPLE) challenge problem, Algorithms for Synthetic Aperture Radar Imagery XXVI, 10987 (2019), 109870G. doi: 10.1117/12.2523458.  Google Scholar

[26]

F. T. Ulaby and M. C. Dobson, Handbook of Radar Scattering Statistics for Terrain, Artech House Remote Sensing Library, Artech House, Norwood, MA, USA, 1989. Google Scholar

[27]

X. X. Zhu, D. Tuia, L. Mou, G.-S. Xia, L. Zhang, F. Xu and F. Fraundorfer, Deep learning in remote sensing: A comprehensive review and list of resources, IEEE Geoscience and Remote Sensing Magazine, 5 (2017), 8-36.  doi: 10.1109/MGRS.2017.2762307.  Google Scholar

[28]

D. Ziou and S. Tabbone, et al., Edge detection techniques-an overview, Pattern Recognition and Image Analysis C/C of Raspoznavaniye Obrazov I Analiz Izobrazhenii, 8 (1998), 537-559. Google Scholar

Figure 1.  Plots of $ \left\langle{|I_t|^2}\right\rangle $ and $ \left\langle{|I_s|^2}\right\rangle $ for different values of $ \zeta_{\max} $ and $ \kappa $, see (39) and (27). The dashed lines passing through the origin indicate the ambiguity direction, see (14), (18). For the middle row of plots, the condition $ \kappa \zeta_{\max} \gtrsim 20 $ (see (40)) is satisfied, and the difference in the orientation of the parallelogram-shaped level lines is more apparent than for the top and bottom rows
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Simulated coordinate-delay SAR images with different contrasts. We use $ \kappa = 2.5 $ and $ \zeta_{\max} = 5\pi $, which corresponds to the middle row in Fig. 1. To build each image, we sample $ \zeta_\text{d} $ and $ \psi_\text{d} $ with a step of $ \pi $ each, and for every $ \zeta_\text{d} $ (i.e., each ambiguity line) generate the multivariate circular Gaussian image components according to (36) with the weights $ \sigma^2_ \alpha K_ \alpha $ calculated via (43). The total images are then computed according to the appropriate expression in (42). Simulation of multivariate normal random variables is performed using the $ \text{MATLAB}^© $ function $\mathtt{mvnpdf}$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Cumulative distribution functions (cdf) for ensembles generated from the s-model and t-model. (a) Examples of cdfs for $ \log \breve p_s $ and $ \log \breve p_t $, see (54). The notations $ \text{cdf}_s[\ldots] $ and $ \text{cdf}_t[\ldots] $ are similar to those in (60). (b) Examples of cdfs for $ l = \log \breve p_t - \log \breve p_s $, see (55) and (60). According to (61) (see also Table 1(a)), we have $ r_t = \text{cdf}_t(0; q) $ and $ r_s = 1- \text{cdf}_s(0; q) $. (c) A zoom-in to the central part of panel (b). The entries $ r'_s $, $ r''_s $, $ r'_t $, and $ r''_t $ from Table 1(b) are determined according to (63) and (65) with $ p = 0.05 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Separation between the graphs of $ \text{cdf}_s(\cdot\, ;q) $ and $ \text{cdf}_t(\cdot\, ;q) $ for different values of $ \kappa $ and $ q $. The thick colored vertical bars indicate the percentage of uncertain classifications for the ensembles generated from the s-model (the left set of bars in each plot) and t-model (the right set of bars), see also (65) and Fig. 3(c)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Dependence of the discrimination quality on $ \zeta_{\max} $, see (39), for two different target contrasts. The left column of plots corresponds to algorithm (56) and Table 1(a), and the right column corresponds to algorithm (57) and Table 1(b). The dashed vertical lines are drawn at $ \kappa \zeta_{\max} = b_\Phi \approx 23 $, cf. (40). The lower and upper parts of the colored panels represent ensembles generated from the s-model and t-model, respectively. The percentage of correct classifications is shown in green (two different shades are used to distinguish between the ensemble types), incorrect in red, and uncertain in yellow (only the right column of plots)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Dependence of discrimination quality on $ \kappa $, see (27). The notations are the same as in Fig. 5
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Confusion matrices: (a) for classification using algorithm (56); (b) for classification using algorithm (57). The rows correspond to the models in (42), whereas the columns indicate the outcomes of a particular classification algorithm. The entries are relative frequencies of the events calculated for two ensembles with the same contrast, i.e., $ q_s = q_t = q $
(a) output: s output: t
input: s $ 1 - r_s $ $ r_s $
input: t $ r_t $ $ 1 - r_t $
(b) output: s output: tuncertain
input: s $ 1 - r'_s -r''_s $ $ r'_s $ $ r''_s $
input: t $ r'_t $ $ 1 - r'_t- r''_t $ $ r''_t $
Table Options

Download as excel

(a) output: s output: t
input: s $ 1 - r_s $ $ r_s $
input: t $ r_t $ $ 1 - r_t $
(b) output: s output: tuncertain
input: s $ 1 - r'_s -r''_s $ $ r'_s $ $ r''_s $
input: t $ r'_t $ $ 1 - r'_t- r''_t $ $ r''_t $
[1]

Masaru Hamano, Satoshi Masaki. A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1415-1447. doi: 10.3934/dcds.2020323
[2]

Alexander Dabrowski, Ahcene Ghandriche, Mourad Sini. Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021005
[3]

Weihong Guo, Yifei Lou, Jing Qin, Ming Yan. IPI special issue on "mathematical/statistical approaches in data science" in the Inverse Problem and Imaging. Inverse Problems & Imaging, 2021, 15 (1) : I-I. doi: 10.3934/ipi.2021007
[4]

Xinlin Cao, Huaian Diao, Jinhong Li. Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29 (1) : 1753-1782. doi: 10.3934/era.2020090
[5]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004
[6]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032
[7]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073
[8]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070
[9]

Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347
[10]

Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390
[11]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351
[12]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258
[13]

Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021002
[14]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328
[15]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018
[16]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[17]

Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317
[18]

Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134
[19]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048
[20]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (101)
  • HTML views (110)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences