June  2019, 13(3): 635-652. doi: 10.3934/ipi.2019029

Inverse random source problem for biharmonic equation in two dimensions

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author: xxu@zju.edu.cn

Received  May 2018 Revised  December 2018 Published  March 2019

Fund Project: The authors are partly supported by NSFC grant 11471284, 11421110002, 11621101, 91630309 and the Fundamental Research Funds for the Central Universities.

The establishment of relevant model and solving an inverse random source problem are one of the main tools for analyzing mechanical properties of elastic materials. In this paper, we study an inverse random source problem for biharmonic equation in two dimension. Under some regularity assumptions on the structure of random source, the well-posedness of the forward problem is established. Moreover, based on the explicit solution of the forward problem, we can solve the corresponding inverse random source problem via two transformed integral equations. Numerical examples are presented to illustrate the validity and effectiveness of the proposed inversion method.

Citation: Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029
References:
[1]

J. E. AllenD. E. PereaE. R. Hemesath and L. J. Lauhon, Nonuniform nanowire doping profiles revealed by quantitative scanning photocurrent microscopy, Advanced Materials, 21 (2009), 3067-3072.  doi: 10.1002/adma.200803865.  Google Scholar

[2]

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

[3]

G. BaoC. Chen and P. Li, Inverse random source scattering for elastic waves, SIAM J. Numer. Anal., 55 (2017), 2616-2643.  doi: 10.1137/16M1088922.  Google Scholar

[4]

G. BaoS. N. ChowP. Li and H. Zhou, An inverse random source problem for the helmholtz equation, Math. Comp., 83 (2013), 215-233.  doi: 10.1090/S0025-5718-2013-02730-5.  Google Scholar

[5]

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

[6]

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

[7]

X. DengV. R. JosephW. MaiZ. L. WangF. C. Jeff Wu and P. J. Bickel, Statistical approach to quantifying the elastic deformation of nanomaterials, Proc. Natl. Acad. Sci. USA, 106 (2009), 11845-11850.  doi: 10.1073/pnas.0808758106.  Google Scholar

[8]

K. L EkinciX. M. H Huang and M. L Roukes, Ultrasensitive nanoelectromechanical mass detection, Appl. Phys. Lett., 84 (2004), 4469-4471.  doi: 10.1063/1.1755417.  Google Scholar

[9]

M. Hairer, An introduction to stochastic pdes, Moss, F.; Gielen, S. (ed.), Handbook of Biological Physics, 2009,517–552. Google Scholar

[10]

J. HuX. W. Liu and B. C. Pan, A study of the size-dependent elastic properties of zno nanowires and nanotubes, Nanotechnology, 19 (2008), 285710.  doi: 10.1088/0957-4484/19/28/285710.  Google Scholar

[11]

M. Li, C. Chen and P. Li, Inverse random source scattering for the Helmholtz equation in inhomogeneous media, Inverse Problems, 34 (2018), 015003, 19pp. doi: 10.1088/1361-6420/aa99d2.  Google Scholar

[12]

P. Li and G. Yuan, Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation, J. Math. Anal. Appl., 450 (2017), 872-887.  doi: 10.1016/j.jmaa.2017.01.074.  Google Scholar

[13]

P. Li and G. Yuan, Increasing stability for the inverse source scattering problem with multi-frequencies, Inverse Problems and Imaging, 11 (2017), 745-759.  doi: 10.3934/ipi.2017035.  Google Scholar

[14]

W. Mai and X. Deng, The applications of statistical quantification techniques in nanomechanics and nanoelectronics, Nanotechnology, 21 (2010), 405704.  doi: 10.1088/0957-4484/21/40/405704.  Google Scholar

[15]

V. Mlinar, Utilization of inverse approach in the design of materials over nano to macro scale, Annalen Der Physik, 527 (2015), 187-204.  doi: 10.1002/andp.201400190.  Google Scholar

[16]

S. Pereverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal., 43 (2005), 2060-2076.  doi: 10.1137/S0036142903433819.  Google Scholar

[17]

P. PoncharalZ. WangD. Ugarte and W. Heer, Electrostatic deflections and electromechanical resonances of carbon nanotubes, Science, 283 (1999), 1513-1516.  doi: 10.1126/science.283.5407.1513.  Google Scholar

[18]

M. TreacyT. Ebbesen and J. Gibson, Exceptionally high Young's modulus observed for individual carbon nanotubes, Nature, 381 (1996), 678-680.  doi: 10.1038/381678a0.  Google Scholar

[19]

E. W. WongP. E. Sheehan and C. M. Lieber, Nanobeam mechanics: Elasticity, strength, and toughness of nanorods and nanotubes, Science, 277 (1997), 1971-1975.  doi: 10.1126/science.277.5334.1971.  Google Scholar

show all references

References:
[1]

J. E. AllenD. E. PereaE. R. Hemesath and L. J. Lauhon, Nonuniform nanowire doping profiles revealed by quantitative scanning photocurrent microscopy, Advanced Materials, 21 (2009), 3067-3072.  doi: 10.1002/adma.200803865.  Google Scholar

[2]

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

[3]

G. BaoC. Chen and P. Li, Inverse random source scattering for elastic waves, SIAM J. Numer. Anal., 55 (2017), 2616-2643.  doi: 10.1137/16M1088922.  Google Scholar

[4]

G. BaoS. N. ChowP. Li and H. Zhou, An inverse random source problem for the helmholtz equation, Math. Comp., 83 (2013), 215-233.  doi: 10.1090/S0025-5718-2013-02730-5.  Google Scholar

[5]

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

[6]

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

[7]

X. DengV. R. JosephW. MaiZ. L. WangF. C. Jeff Wu and P. J. Bickel, Statistical approach to quantifying the elastic deformation of nanomaterials, Proc. Natl. Acad. Sci. USA, 106 (2009), 11845-11850.  doi: 10.1073/pnas.0808758106.  Google Scholar

[8]

K. L EkinciX. M. H Huang and M. L Roukes, Ultrasensitive nanoelectromechanical mass detection, Appl. Phys. Lett., 84 (2004), 4469-4471.  doi: 10.1063/1.1755417.  Google Scholar

[9]

M. Hairer, An introduction to stochastic pdes, Moss, F.; Gielen, S. (ed.), Handbook of Biological Physics, 2009,517–552. Google Scholar

[10]

J. HuX. W. Liu and B. C. Pan, A study of the size-dependent elastic properties of zno nanowires and nanotubes, Nanotechnology, 19 (2008), 285710.  doi: 10.1088/0957-4484/19/28/285710.  Google Scholar

[11]

M. Li, C. Chen and P. Li, Inverse random source scattering for the Helmholtz equation in inhomogeneous media, Inverse Problems, 34 (2018), 015003, 19pp. doi: 10.1088/1361-6420/aa99d2.  Google Scholar

[12]

P. Li and G. Yuan, Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation, J. Math. Anal. Appl., 450 (2017), 872-887.  doi: 10.1016/j.jmaa.2017.01.074.  Google Scholar

[13]

P. Li and G. Yuan, Increasing stability for the inverse source scattering problem with multi-frequencies, Inverse Problems and Imaging, 11 (2017), 745-759.  doi: 10.3934/ipi.2017035.  Google Scholar

[14]

W. Mai and X. Deng, The applications of statistical quantification techniques in nanomechanics and nanoelectronics, Nanotechnology, 21 (2010), 405704.  doi: 10.1088/0957-4484/21/40/405704.  Google Scholar

[15]

V. Mlinar, Utilization of inverse approach in the design of materials over nano to macro scale, Annalen Der Physik, 527 (2015), 187-204.  doi: 10.1002/andp.201400190.  Google Scholar

[16]

S. Pereverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal., 43 (2005), 2060-2076.  doi: 10.1137/S0036142903433819.  Google Scholar

[17]

P. PoncharalZ. WangD. Ugarte and W. Heer, Electrostatic deflections and electromechanical resonances of carbon nanotubes, Science, 283 (1999), 1513-1516.  doi: 10.1126/science.283.5407.1513.  Google Scholar

[18]

M. TreacyT. Ebbesen and J. Gibson, Exceptionally high Young's modulus observed for individual carbon nanotubes, Nature, 381 (1996), 678-680.  doi: 10.1038/381678a0.  Google Scholar

[19]

E. W. WongP. E. Sheehan and C. M. Lieber, Nanobeam mechanics: Elasticity, strength, and toughness of nanorods and nanotubes, Science, 277 (1997), 1971-1975.  doi: 10.1126/science.277.5334.1971.  Google Scholar

Figure 1.  The model for the two-dimensional biharmonic equation
Figure 2.  The mesh generation under the polar coordination
Figure 3.  The solution to direct problem with random source
Figure 4.  Inverse stiffness D(The exact solution is 0.05)
Figure 5.  The left subfigure is the L-curve for $ g $ and the right subfigure is the inverse mean(The dotted plots are accurate values)
Figure 6.  The left subfigure is the L-curve for $ h^2 $ and the right subfigure is the inverse variance(The dotted plots are accurate values)
Figure 7.  The left subfigure is the L-curve for $ g $ and the right subfigure is the inverse mean(The dotted plots are accurate values)
Figure 8.  The left subfigure is the L-curve for $ h^2 $ and the right subfigure is the inverse variance(The dotted plots are accurate values)
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