A novel localized collocation solver based on Trefftz basis for potential-based inverse electromyography. (English) Zbl 1508.65154
Summary: This paper introduces a novel localized collocation Trefftz method (LCTM) for potential-based inverse electromyography (PIE). PIE is a noninvasive technique to calculate the internal electrical potentials from measured body surface electromyographic data, which can be considered as an inverse Cauchy problem with potential equation. In the proposed LCTM, the electrical potential at every node is expressed as a linear combination of 3D Trefftz basis in each stencil support, and the sparse linear system is yield by satisfying governing equation at interior nodes and boundary conditions at boundary nodes. The proposed LCTM inherits the properties of easy-to-use and meshless from the collocation Trefftz method (CTM), and mitigates the ill-conditioning resultant matrix encountered in the CTM. Numerical efficiency of the proposed method is investigated in comparison with the CTM and experimental data.
MSC:
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 92C55 | Biomedical imaging and signal processing |
Keywords:
localized collocation scheme; potential-based inverse electromyography; inverse Cauchy problem; collocation Trefftz method; meshlessReferences:
| [1] | Blencowe, H.; Cousens, S.; Oestergaard, M. Z.; Chou, D.; Moller, A.; Narwal, R.; Adler, A.; Garcia, C. V.; Rohde, S.; Say, L.; ELawn, J., National, regional, and worldwide estimates of preterm birth rates in the year 2010 with time trends since 1990 for selected countries: a systematic analysis and implications, Lancet, 379, 2162-2172 (2012) |
| [2] | Saigal, S.; Doyle, L. W., An overview of mortality and sequelae of preterm birth from infancy to adulthood, Lancet, 371, 261-269 (2008) |
| [3] | Wu, W. J.; Wang, H.; Zhao, P.; Talcott, M.; Lai, S. S.; McKinstry, R. C.; Woodard, P. K.; Macones, G. A.; Schwartz, A. L.; Cahill, A. G.; Cuculich, P. S.; Wang, Y., Noninvasive high-resolution electromyometrial imaging of uterine contractions in a translational sheep model, Sci. Transl. Med., 11, eaau1428 (2019) |
| [4] | Wang, H.; Wu, W. J.; Talcott, M.; McKinstry, R. C.; Woodard, P. K.; Macones, G. A.; Schwartz, A. L.; Cuculich, P.; Cahill, A. G.; Wang, Y., Accuracy of electromyometrial imaging of uterine contractions in clinical environment, Comput. Biol. Med., 116, 103543 (2020) |
| [5] | Cuculich, P. S.; Zhang, J.; Wang, Y.; Desouza, K. A.; Vijayakumar, R.; Woodard, P. K.; Rudy, Y., The electrophysiological cardiac ventricular substrate in patients after myocardial infarction: noninvasive characterization with electrocardiographic imaging, J. Am. Coll. Cardiol., 58, 1893-1902 (2011) |
| [6] | Cuculich, P. S.; Wang, Y.; Lindsay, B. D.; Vijayakumar, R.; Rudy, Y., Noninvasive real-time mapping of an incomplete pulmonary vein isolation using electrocardiographic imaging (ECGI), Heart Rhythm., 7, 1316-1317 (2010) |
| [7] | Plonsey, R., Action potential sources and their volume conductor fields, Proc. IEEE, 65, 601-611 (1977) |
| [8] | Marin, L.; Cipu, C., Non-iterative regularized MFS solution of inverse boundary value problems in linear elasticity: a numerical study, Appl. Math. Comput., 293, 265-286 (2017) · Zbl 1411.74063 |
| [9] | Marin, L.; Karageorghis, A.; Lesnic, D., Regularized MFS solution of inverse boundary value problems in three-dimensional steady-state linear thermoelasticity, Int. J. Solids Struct., 91, 127-142 (2016) |
| [10] | Hadamard, J.; Morse, P. M., Lectures on Cauchys problem in linear partial differential equations, Phys. Today, 6, 18 (1953) |
| [11] | Burman, E., A stabilized nonconforming finite element method for the elliptic Cauchy problem, Math. Comput., 86, 75-96 (2017) · Zbl 1351.65074 |
| [12] | Bergam, A.; Chakib, A.; Nachaoui, A.; Nachaoui, M., Adaptive mesh techniques based on a posteriori error estimates for an inverse Cauchy problem, Appl. Math. Comput., 346, 865-878 (2019) · Zbl 1429.65270 |
| [13] | Zhou, S. W.; Zhuang, X. Y.; Rabczuk, T., Phase-field modeling of fluid-driven dynamic cracking in porous media, Comput. Methods Appl. Mech. Eng., 350, 169-198 (2019) · Zbl 1441.74222 |
| [14] | Marin, L.; Elliott, L.; Heggs, P. J.; Ingham, D. B.; Lesnic, D.; Wen, X., Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations, Comput. Mech., 31, 367-377 (2003) · Zbl 1047.65097 |
| [15] | Fu, Z. J.; Reutskiy, S.; Sun, H. G.; Ma, J.; Khan, M. A., A robust kernel-based solver for variable-order time fractional PDEs under 2D/3D irregular domains, Appl. Math. Lett., 94, 105-111 (2019) · Zbl 1411.65137 |
| [16] | Qiu, L.; Hu, C.; Qin, Q. H., A novel homogenization function method for inverse source problem of nonlinear time-fractional wave equation, Appl. Math. Lett., 109, 106554 (2020) · Zbl 1450.35016 |
| [17] | Marin, L.; Delvare, F.; Cimetiere, A., Fading regularization MFS algorithm for inverse boundary value problems in two-dimensional linear elasticity, Int. J. Solids Struct., 78, 9-20 (2016) |
| [18] | Alves, C. J.S.; Antunes, P. R.S., The method of fundamental solutions applied to some inverse eigenproblems, SIAM J. Sci. Comput., 35, A1689-A1708 (2013) · Zbl 1275.49075 |
| [19] | Alves, C. J.S.; Valtchevac, S. S., On the application of the method of fundamental solutions to boundary value problems with jump discontinuities, Appl. Math. Comput., 320, 61-74 (2018) · Zbl 1426.65199 |
| [20] | Fu, Z. J.; Xi, Q.; Chen, W.; Cheng, A. H.D., A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations, Comput. Math. Appl., 76, 760-773 (2018) · Zbl 1430.65009 |
| [21] | Fu, Z. J.; Chen, W.; Wen, P. H.; Zhang, C. Z., Singular boundary method for wave propagation analysis in periodic structures, J. Sound Vib., 425, 170-188 (2018) |
| [22] | Qiu, L.; Wang, F. J.; J., L., A meshless singular boundary method for transient heat conduction problems in layered materials, Comput. Math. Appl., 78, 3544-3562 (2019) · Zbl 1443.65251 |
| [23] | Sarler, B.; Vertnik, R., Meshfree explicit local radial basis function collocation method for diffusion problems, Comput. Math. Appl., 51, 1269-1282 (2006) · Zbl 1168.41003 |
| [24] | Boudjaj, L.; Naji, A.; Ghafrani, F., Solving biharmonic equation as an optimal control problem using localized radial basis functions collocation method, Eng. Anal. Bound. Elem., 107, 208-217 (2019) · Zbl 1464.65195 |
| [25] | You, X. Y.; Li, W.; Chai, Y. B., A truly meshfree method for solving acoustic problems using local weak form and radial basis functions, Appl. Math. Comput., 365, 124694 (2020) · Zbl 1433.65318 |
| [26] | Alves, C. J.S.; Martins, N. F.M.; Valtchev, S. S., Trefftz methods with cracklets and their relation to BEM and MFS, Eng. Anal. Bound. Elem., 95, 93-104 (2018) · Zbl 1403.65186 |
| [27] | Xi, Q.; Fu, Z. J.; Rabczuk, T., An efficient boundary collocation scheme for transient thermal analysis in large-size-ratio functionally graded materials under heat source load, Comput. Mech., 64, 1221-1235 (2019) · Zbl 1464.74418 |
| [28] | Fu, Z. J.; Yang, L. W.; Zhu, H. Q.; Xu, W. Z., A semi-analytical collocation Trefftz scheme for solving multi-term time fractional diffusion-wave equations, Eng. Anal. Bound. Elem., 98, 137-146 (2019) · Zbl 1404.65193 |
| [29] | Liu, C. S., A modified Trefftz method for two-dimensional Laplace equation considering the domain’s characteristic length, Comput. Model. Eng. Sci., 21, 53 (2007) · Zbl 1232.65157 |
| [30] | Liu, C. S.; Atluri, S. N., Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method, Eng. Anal. Bound. Elem., 37, 74-83 (2013) · Zbl 1352.65569 |
| [31] | Liu, Y. C.; Fan, C. M.; Yeih, W. C.; Ku, C. Y., Numerical solutions of two-dimensional Laplace and biharmonic equations by the localized Trefftz method, Comput. Math. Appl. (2019) |
| [32] | Fan, C. M.; Liu, Y. C.; Yeih, W. C.; Ku, C. Y., Localized Trefftz method for solving two-dimensional Laplace, Helmholtz and biharmonic equations, The 9th Conference on Trefftz Methods and the 5th Conference on Method of Fundamental Solutions, Lisbon, Portugal, July 29-31, 2019 (2019) |
| [33] | Fan, C. M.; Huang, Y. K.; Chen, C. S.; Kuo, S. R., Localized method of fundamental solutions for solving two-dimensional laplace and biharmonic equations, Eng. Anal. Bound. Elem., 101, 188-197 (2019) · Zbl 1464.65267 |
| [34] | Gu, Y.; Fan, C. M.; Xu, R. P., Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems, Appl. Math. Lett., 93, 8-14 (2019) · Zbl 1458.74143 |
| [35] | Qu, W. Z.; Fan, C. M.; Li, X. L., Analysis of an augmented moving least squares approximation and the associated localized method of fundamental solutions, Comput. Math. Appl., 80, 13-30 (2020) · Zbl 1446.65195 |
| [36] | Wang, Y.; Rudy, Y., Application of the method of fundamental solutions to potential-based inverse electrocardiography, Ann. Biomed. Eng., 34, 1272-1288 (2006) |
| [37] | Kalinin, A.; Potyagaylo, D.; Kalinin, V., Solving the inverse problem of electrocardiography on the endocardium using a single layer source, Front. Physiol., 10, 58 (2019) |
| [38] | Ramanathan, C.; Ghanem, R. N.; Jia, P.; Ryu, K.; Rudy, Y., Noninvasive electrocardiographic imaging for cardiac electrophysiology and arrhythmia, Nat. Med., 10, 422-428 (2004) |
| [39] | Wang, Y.; Cuculich, P. S.; Zhang, J.; Desouza, K. A.; Vijayakumar, R.; Chen, J.; Faddis, M. N.; Lindsay, B. D.; Smith, T. W.; Rudy, Y., Noninvasive electroanatomic mapping of human ventricular arrhythmias with electrocardiographic imaging, Sci. Transl. Med., 3, 98ra84 (2011) |
| [40] | Ku, C. Y.; Kuo, C. L.; Fan, C. M.; Liu, C. S.; Guan, P. C., Numerical solution of three-dimensional Laplacian problems using the multiple scale Trefftz method, Eng. Anal. Bound. Elem., 50, 157-168 (2015) · Zbl 1403.65164 |
| [41] | Lv, H.; Hao, F.; Wang, Y.; Chen, C. S., The MFS versus the Trefftz method for the Laplace equation in 3D, Eng. Anal. Bound. Elem., 83, 133-140 (2017) · Zbl 1403.65259 |
| [42] | Karageorghis, A.; Lesnic, D.; Marin, L., The method of fundamental solutions for an inverse boundary value problem in static thermo-elasticity, Comput. Struct., 135, 32-39 (2014) |
| [43] | Wang, F. J.; Fan, C. M.; Hua, Q. S.; Gu, Y., Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations, Appl. Math. Comput., 364, 124658 (2020) · Zbl 1433.65338 |
| [44] | https://www.mathworks.com/help/matlab/ref/rand.html. |
| [45] | http://graphics.stanford.edu/data/3dscanrep/. |
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