June 2019, 13(3): 513-543. doi: 10.3934/ipi.2019025

On periodic parameter identification in stochastic differential equations

Shanghai Key Laboratory for Contemporary Applied Mathematics, Key Laboratory of Mathematics for Nonlinear Sciences and School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Corresponding author: Shuai Lu

Received  April 2018 Revised  October 2018 Published  March 2019

Fund Project: S. Lu is supported by National Key Research and Development Program of China (No. 2017YFC1404103), NSFC (No.91730304, 11522108), Shanghai Municipal Education Commission (No.16SG01) and Special Funds for Major State Basic Research Projects of China (2015CB856003). J. Cheng is supported by NSFC (key projects no.11331004, no.11421110002) and the Programme of Introducing Talents of Discipline to Universities (number B08018)

Periodic parameters are common and important in stochastic differential equations (SDEs) arising in many contemporary scientific and engineering fields involving dynamical processes. These parameters include the damping coefficient, the volatility or diffusion coefficient and possibly an external force. Identification of these periodic parameters allows a better understanding of the dynamical processes and their hidden intermittent instability. Conventional approaches usually assume that one of the parameters is known and focus on the recovery of rest parameters. By introducing the decorrelation time and calculating the standard Gaussian statistics (mean, variance) explicitly for the scalar Langevin equations with periodic parameters, we propose a parameter identification approach to simultaneously recovering all these parameters by observing a single trajectory of SDEs. Such an approach is summarized in form of regularization schemes with noisy operators and noisy right-hand sides and is further extended to parameter identification of SDEs which are indirectly observed by other random processes. Numerical examples show that our approach performs well in stable and weakly unstable regimes but may fail in strongly unstable regime which is induced by the strong intermittent instability itself.

Citation: Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025
References:
[1]

G. BaoC. Chen and P. Li, Inverse random source scattering problems in several dimensions, SIAM/ASA Journal on Uncertainty Quantification, 4 (2016), 1263-1287. doi: 10.1137/16M1067470.

[2]

G. BaoS. N. Chow and H. Zhou, An inverse random source problem for the Helmholtz equation, Mathematics of Computation, 83 (2014), 215-233. doi: 10.1090/S0025-5718-2013-02730-5.

[3]

G. Bao and X. Xu, An inverse random source problem in quantifying the elastic modulus of nanomaterials, Inverse Problems, 29 (2013), 015006, 16 pp. doi: 10.1088/0266-5611/29/1/015006.

[4]

G. Bao and X. Xu, Identification of the material properties in nonuniform nanostructures, Inverse Problems, 31 (2015), 125003, 11 pp. doi: 10.1088/0266-5611/31/12/125003.

[5]

M. Branicki and A. J. Majda, Quantifying Bayesian filter performance for turbulent dynamical systems through information theory, Commun. Math. Sci., 12 (2014), 901-978. doi: 10.4310/CMS.2014.v12.n5.a6.

[6]

D. G. Cacuci, I. M. Navon and M. Ionescu-Bujor, Computational Methods for Data Evaluation and Assimilation, CRC Press, Boca Raton, FL, 2014.

[7]

N. ChenD. GiannakisR. Herbei and A. J. Majda, An MCMC algorithm for parameter estimation in signals with hidden intermittent instability, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), 647-669. doi: 10.1137/130944977.

[8]

M. Cristofol and L. Roques, Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations, Inverse Problems, 33 (2017), 095006, 12 pp. doi: 10.1088/1361-6420/aa7a1c.

[9]

F. Dunker and T. Hohage, On parameter identification in stochastic differential equations by penalized maximum likelihood, Inverse Problems, 30 (2014), 095001, 20 pp. doi: 10.1088/0266-5611/30/9/095001.

[10]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. viii+321 pp.

[11]

H. GaoK. WangF. Wei and X. Ding, Massera-type theorem and asymptotically periodic logistic equations, Nonlinear Analysis: Real World Applications, 7 (2006), 1268-1283. doi: 10.1016/j.nonrwa.2005.11.008.

[12]

B. GershgorinJ. Harlim and A. J. Majda, Test models for improving filtering with model errors through stochastic parameter estimation, Journal of Computational Physics, 229 (2010), 1-31. doi: 10.1016/j.jcp.2009.08.019.

[13]

A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximation, Biometrics, 61 (2005), 781-788. doi: 10.1111/j.1541-0420.2005.00345.x.

[14]

B. Kaltenbacher and B. Pedretscher, Parameter estimation in SDEs via the Fokker-Planck equation: Likelihood function and adjoint based gradient computation, Journal of Mathematical Analysis and Applications, 465 (2018), 872-884. doi: 10.1016/j.jmaa.2018.05.048.

[15]

W. J. Lee and A. Stuart, Derivation and analysis of simplified filters, Communications in Mathematical Sciences, 15 (2017), 413-450. doi: 10.4310/CMS.2017.v15.n2.a6.

[16]

P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27 (2011), 035004, 22 pp. doi: 10.1088/0266-5611/27/3/035004.

[17]

S. Lu and S. V. Pereverzev, Regularization Theory for Ill-Posed Problems: Selected Topics, Inverse and Ill-posed Problems Series, 58 De Gruyter, Berlin, 2013. doi: 10.1515/9783110286496.

[18]

S. LuS. V. PereverzevY. Shao and U. Tautenhahn, On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales, Journal of Integral Equations and Application, 22 (2010), 483-517. doi: 10.1216/JIE-2010-22-3-483.

[19]

A. J. Majda and J. Harlim, Filtering Complex Turbulent Systems, Cambridge University Press, Cambridge, 2012. x+357 pp. doi: 10.1017/CBO9781139061308.

[20] A. J. Majda and X. Wang, Non-linear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511616778.
[21]

B. Øksendal, Stochastic Differential Equations, 6$^{th}$ ed., Springer, Heidelberg, 2003. doi: 10.1007/978-3-642-14394-6.

[22]

O. PapaspiliopoulosY PokernG. O. Roberts and A. M. Stuart, Nonparametric estimation of diffusions: A differential equations approach, Biometrika, 99 (2012), 511-531. doi: 10.1093/biomet/ass034.

[23]

G. O. Roberts and O. Stramer, On inference for partially observed nonlinear diffusion model using Metropolis–Hastings algorithms, Biometrika, 88 (2001), 603-621. doi: 10.1093/biomet/88.3.603.

[24]

H. Sorensen, Parametric inference for diffusion processes observed at discrete points in time: A survey, International Statistical Review, 72 (2004), 337-354. doi: 10.1111/j.1751-5823.2004.tb00241.x.

[25]

O. Stramer and M. Bognar, Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach, Bayesian Analysis, 6 (2011), 231-258. doi: 10.1214/11-BA608.

[26]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559. doi: 10.1017/S0962492910000061.

[27]

U. Tautenhahn, Regularization of linear ill-posed problems with noisy right hand side and noisy operator, J. Inv. Ill-Posed Problems, 16 (2008), 507-523. doi: 10.1515/JIIP.2008.027.

[28]

C. R. Vogel, Computational Methods for Inverse Problems, With a foreword by H. T. Banks. Frontiers in Applied Mathematics, 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717570.

show all references

References:
[1]

G. BaoC. Chen and P. Li, Inverse random source scattering problems in several dimensions, SIAM/ASA Journal on Uncertainty Quantification, 4 (2016), 1263-1287. doi: 10.1137/16M1067470.

[2]

G. BaoS. N. Chow and H. Zhou, An inverse random source problem for the Helmholtz equation, Mathematics of Computation, 83 (2014), 215-233. doi: 10.1090/S0025-5718-2013-02730-5.

[3]

G. Bao and X. Xu, An inverse random source problem in quantifying the elastic modulus of nanomaterials, Inverse Problems, 29 (2013), 015006, 16 pp. doi: 10.1088/0266-5611/29/1/015006.

[4]

G. Bao and X. Xu, Identification of the material properties in nonuniform nanostructures, Inverse Problems, 31 (2015), 125003, 11 pp. doi: 10.1088/0266-5611/31/12/125003.

[5]

M. Branicki and A. J. Majda, Quantifying Bayesian filter performance for turbulent dynamical systems through information theory, Commun. Math. Sci., 12 (2014), 901-978. doi: 10.4310/CMS.2014.v12.n5.a6.

[6]

D. G. Cacuci, I. M. Navon and M. Ionescu-Bujor, Computational Methods for Data Evaluation and Assimilation, CRC Press, Boca Raton, FL, 2014.

[7]

N. ChenD. GiannakisR. Herbei and A. J. Majda, An MCMC algorithm for parameter estimation in signals with hidden intermittent instability, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), 647-669. doi: 10.1137/130944977.

[8]

M. Cristofol and L. Roques, Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations, Inverse Problems, 33 (2017), 095006, 12 pp. doi: 10.1088/1361-6420/aa7a1c.

[9]

F. Dunker and T. Hohage, On parameter identification in stochastic differential equations by penalized maximum likelihood, Inverse Problems, 30 (2014), 095001, 20 pp. doi: 10.1088/0266-5611/30/9/095001.

[10]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. viii+321 pp.

[11]

H. GaoK. WangF. Wei and X. Ding, Massera-type theorem and asymptotically periodic logistic equations, Nonlinear Analysis: Real World Applications, 7 (2006), 1268-1283. doi: 10.1016/j.nonrwa.2005.11.008.

[12]

B. GershgorinJ. Harlim and A. J. Majda, Test models for improving filtering with model errors through stochastic parameter estimation, Journal of Computational Physics, 229 (2010), 1-31. doi: 10.1016/j.jcp.2009.08.019.

[13]

A. Golightly and D. J. Wilkinson, Bayesian inference for stochastic kinetic models using a diffusion approximation, Biometrics, 61 (2005), 781-788. doi: 10.1111/j.1541-0420.2005.00345.x.

[14]

B. Kaltenbacher and B. Pedretscher, Parameter estimation in SDEs via the Fokker-Planck equation: Likelihood function and adjoint based gradient computation, Journal of Mathematical Analysis and Applications, 465 (2018), 872-884. doi: 10.1016/j.jmaa.2018.05.048.

[15]

W. J. Lee and A. Stuart, Derivation and analysis of simplified filters, Communications in Mathematical Sciences, 15 (2017), 413-450. doi: 10.4310/CMS.2017.v15.n2.a6.

[16]

P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27 (2011), 035004, 22 pp. doi: 10.1088/0266-5611/27/3/035004.

[17]

S. Lu and S. V. Pereverzev, Regularization Theory for Ill-Posed Problems: Selected Topics, Inverse and Ill-posed Problems Series, 58 De Gruyter, Berlin, 2013. doi: 10.1515/9783110286496.

[18]

S. LuS. V. PereverzevY. Shao and U. Tautenhahn, On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales, Journal of Integral Equations and Application, 22 (2010), 483-517. doi: 10.1216/JIE-2010-22-3-483.

[19]

A. J. Majda and J. Harlim, Filtering Complex Turbulent Systems, Cambridge University Press, Cambridge, 2012. x+357 pp. doi: 10.1017/CBO9781139061308.

[20] A. J. Majda and X. Wang, Non-linear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511616778.
[21]

B. Øksendal, Stochastic Differential Equations, 6$^{th}$ ed., Springer, Heidelberg, 2003. doi: 10.1007/978-3-642-14394-6.

[22]

O. PapaspiliopoulosY PokernG. O. Roberts and A. M. Stuart, Nonparametric estimation of diffusions: A differential equations approach, Biometrika, 99 (2012), 511-531. doi: 10.1093/biomet/ass034.

[23]

G. O. Roberts and O. Stramer, On inference for partially observed nonlinear diffusion model using Metropolis–Hastings algorithms, Biometrika, 88 (2001), 603-621. doi: 10.1093/biomet/88.3.603.

[24]

H. Sorensen, Parametric inference for diffusion processes observed at discrete points in time: A survey, International Statistical Review, 72 (2004), 337-354. doi: 10.1111/j.1751-5823.2004.tb00241.x.

[25]

O. Stramer and M. Bognar, Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach, Bayesian Analysis, 6 (2011), 231-258. doi: 10.1214/11-BA608.

[26]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559. doi: 10.1017/S0962492910000061.

[27]

U. Tautenhahn, Regularization of linear ill-posed problems with noisy right hand side and noisy operator, J. Inv. Ill-Posed Problems, 16 (2008), 507-523. doi: 10.1515/JIIP.2008.027.

[28]

C. R. Vogel, Computational Methods for Inverse Problems, With a foreword by H. T. Banks. Frontiers in Applied Mathematics, 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717570.

Figure 1.  Pathwise solutions of the stochastic differential equation (4) with different parameters in Table 1. Upper (stable regime); middle (weakly unstable regime) and bottom (strongly unstable regime). Each panel presents a segment of $ v(t) $ for $ t\in [100,102] $ whereas the long path of the solution $ v(t) $ for $ t\in [0,20000] $ is presented in the small picture in each panel
Figure 2.  Empirical values of $ 5 $ trajectories of the random process $ v(t) $ in (4) with different parameters in Table 1. Upper row (stable regime); middle row (weakly unstable regime) and bottom row (strongly unstable regime). The red solid line in each panel is the exact value. The blue dashed lines are empirical values of $ 5 $ different trajectories
Figure 3.  Pathwise solutions of the stochastic differential equations (23)-(24) with different parameters in Table 1 and (30). Upper row: $ v(t) $ and $ u(t) $ (stable regime); middle row: $ v(t) $ and $ u(t) $ (weakly unstable regime) and bottom row: $ v(t) $ and $ u(t) $ (strongly unstable regime). Each panel presents a segment of $ v(t) $ or $ u(t) $ for $ t\in [100,102] $ whereas the long path of the solutions for $ t\in [0,40000] $ is presented in the small picture in each panel
Figure 4.  Empirical values of $ 5 $ trajectories of the random process $ u(t) $ in (23)-(24) with different parameters in Table 1 and (30). Upper row (stable regime); middle row (weakly unstable regime) and bottom row (strongly unstable regime). The red solid line in each panel is the exact value. The blue dashed lines are empirical values of $ 5 $ different trajectories
Figure 5.  Parameter identification approach for direct observation $ v(t) $ in (4). Upper row: exact and reconstructed $ \gamma_v $, $ f_{v} $ and $ \sigma_v^2 $ (stable regime); middle row: exact and reconstructed $ \gamma_v $, $ f_{v} $ and $ \sigma_v^2 $ (weakly unstable regime) and bottom row: exact and reconstructed $ \gamma_v $, $ f_{v} $ and $ \sigma_v^2 $ (strongly unstable regime). Small figures in each panel are the exact (red solid line) and empirical (blue dashed line) decorrelation time, mean and variance of $ v(t) $ which are used in the parameter identification approach (20)
Figure 6.  Parameter identification approach for indirect observation $ u(t) $ in (23)-(24). Upper row: exact and reconstructed $ \gamma_v $, $ f_{v} $ and $ \sigma_v^2 $ (stable regime); middle row: exact and reconstructed $ \gamma_v $, $ f_{v} $ and $ \sigma_v^2 $ (weakly unstable regime) and bottom row: exact and reconstructed $ \gamma_v $, $ f_{v} $ and $ \sigma_v^2 $ (strongly unstable regime). Small figures in each panel are the exact (red solid line) and empirical (blue dashed line) decorrelation time, mean and variance of $ u(t) $ which are used in the parameter identification approach (28)
Figure 7.  Empirical values of $ 5 $ trajectories of the random process $ v(t) $ (upper row) and $ u(t) $ (bottom row) with reconstructed parameters of the strongly unstable regime. The red solid line in each panel is the exact Gaussian statistics. The blue dashed lines are empirical Gaussian statistics of $ 5 $ different trajectories by the reconstructed parameters
Table 1.  Parameters of the stochastic differential equations (4) and (23)
Parameters Value
Damping parameter (stable regime): $ \gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+2.05 $
(weakly unstable regime): $ \gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+1.9 $
(strongly unstable regime): $ \gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+0.05 $
Force parameter $ f_v(\zeta)=0.1\sin(4\pi\zeta)+0.2 $
Volatility parameter $ \sigma_v^2(\zeta)=\left(0.1\sin(2\pi\zeta)+0.3\right)^2 $.
Parameters Value
Damping parameter (stable regime): $ \gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+2.05 $
(weakly unstable regime): $ \gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+1.9 $
(strongly unstable regime): $ \gamma_v(\zeta)=2\sin(2\pi\zeta+\pi)+0.05 $
Force parameter $ f_v(\zeta)=0.1\sin(4\pi\zeta)+0.2 $
Volatility parameter $ \sigma_v^2(\zeta)=\left(0.1\sin(2\pi\zeta)+0.3\right)^2 $.
Table 2.  Direct observation. Columns 2-4: $ L^2- $relative errors of exact and empirical Gaussian statistics of $ v(t) $. Final three columns: $ L^2- $relative errors of the exact and reconstructed parameters. The observation time is $ [15000,20000] $
Decorrelation time Mean Variance $ \gamma_v $ $ f_v $ $ \sigma_v^2 $
stable 0.0287 0.0170 0.0173 0.0347 0.0706 0.0271
weakly unstable 0.0347 0.0295 0.0173 0.0601 0.0949 0.0662
strongly unstable 0.0049 0.0274 0.0310 0.0178 0.5762 0.3591
Decorrelation time Mean Variance $ \gamma_v $ $ f_v $ $ \sigma_v^2 $
stable 0.0287 0.0170 0.0173 0.0347 0.0706 0.0271
weakly unstable 0.0347 0.0295 0.0173 0.0601 0.0949 0.0662
strongly unstable 0.0049 0.0274 0.0310 0.0178 0.5762 0.3591
Table 3.  Indirect observation. Columns 2-4: $ L^2- $relative errors of exact and empirical Gaussian statistics of $ u(t) $. Final three columns: $ L^2- $relative errors of the exact and reconstructed parameters. The observation time is $ [30000,40000] $
Decorrelation time Mean Variance $ \gamma_v $ $ f_v $ $ \sigma_v^2 $
stable 0.0108 0.0043 0.0098 0.0496 0.0916 0.0613
weakly unstable 0.0066 0.0099 0.0151 0.0208 0.1724 0.0773
strongly unstable 0.0090 0.0142 0.0575 0.0187 0.3866 0.1462
Decorrelation time Mean Variance $ \gamma_v $ $ f_v $ $ \sigma_v^2 $
stable 0.0108 0.0043 0.0098 0.0496 0.0916 0.0613
weakly unstable 0.0066 0.0099 0.0151 0.0208 0.1724 0.0773
strongly unstable 0.0090 0.0142 0.0575 0.0187 0.3866 0.1462
[1]

Hedia Fgaier, Hermann J. Eberl. Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation. Conference Publications, 2009, 2009 (Special) : 230-239. doi: 10.3934/proc.2009.2009.230

[2]

Angelo Favini. A general approach to identification problems and applications to partial differential equations. Conference Publications, 2015, 2015 (special) : 428-435. doi: 10.3934/proc.2015.0428

[3]

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

[4]

Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005

[5]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025

[6]

Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182

[7]

Davide Guidetti. Some inverse problems of identification for integrodifferential parabolic systems with a boundary memory term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 749-756. doi: 10.3934/dcdss.2015.8.749

[8]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19

[9]

Tayel Dabbous. Identification for systems governed by nonlinear interval differential equations. Journal of Industrial & Management Optimization, 2012, 8 (3) : 765-780. doi: 10.3934/jimo.2012.8.765

[10]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135

[11]

Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272

[12]

Yanqing Wang, Donghui Yang, Jiongmin Yong, Zhiyong Yu. Exact controllability of linear stochastic differential equations and related problems. Mathematical Control & Related Fields, 2017, 7 (2) : 305-345. doi: 10.3934/mcrf.2017011

[13]

Farid Tari. Two-parameter families of implicit differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 139-162. doi: 10.3934/dcds.2005.13.139

[14]

Farid Tari. Two parameter families of binary differential equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 759-789. doi: 10.3934/dcds.2008.22.759

[15]

Sergiy Zhuk. Inverse problems for linear ill-posed differential-algebraic equations with uncertain parameters. Conference Publications, 2011, 2011 (Special) : 1467-1476. doi: 10.3934/proc.2011.2011.1467

[16]

Jaan Janno, Kairi Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems & Imaging, 2009, 3 (1) : 17-41. doi: 10.3934/ipi.2009.3.17

[17]

Mohammed Al Horani, Angelo Favini. Inverse problems for singular differential-operator equations with higher order polar singularities. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2159-2168. doi: 10.3934/dcdsb.2014.19.2159

[18]

Kenrick Bingham, Yaroslav Kurylev, Matti Lassas, Samuli Siltanen. Iterative time-reversal control for inverse problems. Inverse Problems & Imaging, 2008, 2 (1) : 63-81. doi: 10.3934/ipi.2008.2.63

[19]

Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1

[20]

Yuepeng Wang, Yue Cheng, I. Michael Navon, Yuanhong Guan. Parameter identification techniques applied to an environmental pollution model. Journal of Industrial & Management Optimization, 2018, 14 (2) : 817-831. doi: 10.3934/jimo.2017077

2017 Impact Factor: 1.465

Article outline

Figures and Tables

[Back to Top]