American Institute of Mathematical Sciences

February  2020, 25(2): 715-732. doi: 10.3934/dcdsb.2019263

Boundary value problem: Weak solutions induced by fuzzy partitions

 Institute for Research and Applications of Fuzzy Modelling, NSC IT4Innovations, University of Ostrava, 30. dubna 22,701 03 Ostrava 1, Czech Republic

* Corresponding author: Irina Perfilieva

Received  January 2019 Revised  August 2019 Published  November 2019

Fund Project: This research was supported by the Czech Ministry of Education, Youth and Sports, project OP VVV (AI-Met4AI): No. CZ.02.1.01/0.0/0.0/17-049/0008414. Additional support was given by the Grant Agency of the Czech Republic, project 18-06915S.

The aim of this paper is to propose a new methodology in the construction of spaces of test functions used in a weak formulation of the Boundary Value Problem. The proposed construction is based on the so called "two dimensional" approach where at first, we select a partition of a domain and second, a dimension of an approximating functional subspace on each partition element. The main advantage consists in the independent selection of the key parameters, aiming at achieving a requested quality of approximation with a reasonable complexity. We give theoretical justification and illustration on examples that confirm our methodology.

Citation: Linh Nguyen, Irina Perfilieva, Michal Holčapek. Boundary value problem: Weak solutions induced by fuzzy partitions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 715-732. doi: 10.3934/dcdsb.2019263
References:
 [1] K. W. Anthony, Advanced Real Analysis, Birkhäuser, 2005. Google Scholar [2] I. Babuška, U. Banerjee and J. E. Osborn, Survey of meshless and generalized finite element methods: A unified approach, Acta Numer., 12 (2003), 1-125.  doi: 10.1017/S0962492902000090.  Google Scholar [3] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar [5] K. W. Cassel, Variational Methods With Applications in Science and Engineering, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139136860.  Google Scholar [6] D. Čena, Cubic spline wavelets with four vanishing moments on the interval and their applications to option pricing under Kou model, Int. J. Wavelets Multiresolut. Inf. Process., 17 (2019), 1850061, 27 pp. doi: 10.1142/S0219691318500613.  Google Scholar [7] P. Ciarlet, The Finite Element Method for Elliptic Problems, Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. doi: 10.1137/1.9780898719208.  Google Scholar [8] B. C. Cuong, N. C. Luong and H. V. Long, Approximation properties of fuzzy systems for multi-variables functions, PanAmer. Math. J., 20 (2010), 97-113.   Google Scholar [9] C. A. J. Fletcher, Computational Galerkin Methods, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. doi: 10.1007/978-3-642-85949-6.  Google Scholar [10] M. Holčapek, I. Perfilieva, V. Novák and V. Kreinovich, Necessary and sufficient conditions for generalized uniform fuzzy partitions, Fuzzy Sets and Systems, 277 (2015), 97-121.  doi: 10.1016/j.fss.2014.10.017.  Google Scholar [11] K. Höllig, R. Ulrich and W. Joachim, Weighted extended B-spline approximation of Dirichlet problems, SIAM J. Numer. Anal., 39 (2001), 442-462.  doi: 10.1137/S0036142900373208.  Google Scholar [12] H. V. Long, A note on the rates of uniform approximation of fuzzy systems, Internat. J. Comput. Intelligence Systems, 4 (2011), 712-727.  doi: 10.2991/ijcis.2011.4.4.24.  Google Scholar [13] J. M. Melenk, On approximation in meshless methods, in Frontiers of Numerical Analysis, Universitext, Springer, Berlin, 2005, 65–141. doi: 10.1007/3-540-28884-8_2.  Google Scholar [14] L. Nguyen, I. Perfilieva and M. Holčapek, Weak boundary value problem: Fuzzy partition in Galerkin method, World Scientific Proceedings Series on Computer Engineering and Information Science, (2018), 1478–1485. doi: 10.1142/9789813273238_0184.  Google Scholar [15] I. Perfilieva, Fuzzy transforms: Theory and applications, Fuzzy Sets and Systems, 157 (2006), 993-1023.  doi: 10.1016/j.fss.2005.11.012.  Google Scholar [16] I. Perfilieva, A. P. Singh and S. P. Tiwari, On the relationship among $F$-transform, fuzzy rough set and fuzzy topology, Soft Computing, 21 (2017), 3513-3523.  doi: 10.1007/s00500-017-2559-x.  Google Scholar [17] I. Perfilieva, M. Daňková and B. Bede, Towards a higher degree $F$-transform, Fuzzy Sets and Systems, 180 (2011), 3-19.  doi: 10.1016/j.fss.2010.11.002.  Google Scholar [18] I. Perfilieva and P. Vlašánek, F-transform and discrete convolution, Proc. of the EUSFLAT Conf., (2015), 1054–1059. Google Scholar [19] D. B. Reddy, Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements, Texts in Applied Mathematics, 27, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0575-3.  Google Scholar [20] K. Rektorys, Variational Methods in Mathematics, Science and Engineering, D. Reidel Publishing Co., Dordrecht-Boston, MA, 1977. doi: 10.1007/978-94-011-6450-4.  Google Scholar [21] J. Volker, Finite Element Methods for Incompressible Flow Problems, Springer Series in Computational Mathematics, 51, Springer, Cham, 2016. doi: 10.1007/978-3-319-45750-5.  Google Scholar [22] J. G. Wang and G. Liu, A point interpolation meshless method based on radial basis functions, Internat. J. Numerical Methods in Engineering, 54 (2002), 1623-1648.  doi: 10.1002/nme.489.  Google Scholar

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References:
 [1] K. W. Anthony, Advanced Real Analysis, Birkhäuser, 2005. Google Scholar [2] I. Babuška, U. Banerjee and J. E. Osborn, Survey of meshless and generalized finite element methods: A unified approach, Acta Numer., 12 (2003), 1-125.  doi: 10.1017/S0962492902000090.  Google Scholar [3] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar [5] K. W. Cassel, Variational Methods With Applications in Science and Engineering, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139136860.  Google Scholar [6] D. Čena, Cubic spline wavelets with four vanishing moments on the interval and their applications to option pricing under Kou model, Int. J. Wavelets Multiresolut. Inf. Process., 17 (2019), 1850061, 27 pp. doi: 10.1142/S0219691318500613.  Google Scholar [7] P. Ciarlet, The Finite Element Method for Elliptic Problems, Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. doi: 10.1137/1.9780898719208.  Google Scholar [8] B. C. Cuong, N. C. Luong and H. V. Long, Approximation properties of fuzzy systems for multi-variables functions, PanAmer. Math. J., 20 (2010), 97-113.   Google Scholar [9] C. A. J. Fletcher, Computational Galerkin Methods, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. doi: 10.1007/978-3-642-85949-6.  Google Scholar [10] M. Holčapek, I. Perfilieva, V. Novák and V. Kreinovich, Necessary and sufficient conditions for generalized uniform fuzzy partitions, Fuzzy Sets and Systems, 277 (2015), 97-121.  doi: 10.1016/j.fss.2014.10.017.  Google Scholar [11] K. Höllig, R. Ulrich and W. Joachim, Weighted extended B-spline approximation of Dirichlet problems, SIAM J. Numer. Anal., 39 (2001), 442-462.  doi: 10.1137/S0036142900373208.  Google Scholar [12] H. V. Long, A note on the rates of uniform approximation of fuzzy systems, Internat. J. Comput. Intelligence Systems, 4 (2011), 712-727.  doi: 10.2991/ijcis.2011.4.4.24.  Google Scholar [13] J. M. Melenk, On approximation in meshless methods, in Frontiers of Numerical Analysis, Universitext, Springer, Berlin, 2005, 65–141. doi: 10.1007/3-540-28884-8_2.  Google Scholar [14] L. Nguyen, I. Perfilieva and M. Holčapek, Weak boundary value problem: Fuzzy partition in Galerkin method, World Scientific Proceedings Series on Computer Engineering and Information Science, (2018), 1478–1485. doi: 10.1142/9789813273238_0184.  Google Scholar [15] I. Perfilieva, Fuzzy transforms: Theory and applications, Fuzzy Sets and Systems, 157 (2006), 993-1023.  doi: 10.1016/j.fss.2005.11.012.  Google Scholar [16] I. Perfilieva, A. P. Singh and S. P. Tiwari, On the relationship among $F$-transform, fuzzy rough set and fuzzy topology, Soft Computing, 21 (2017), 3513-3523.  doi: 10.1007/s00500-017-2559-x.  Google Scholar [17] I. Perfilieva, M. Daňková and B. Bede, Towards a higher degree $F$-transform, Fuzzy Sets and Systems, 180 (2011), 3-19.  doi: 10.1016/j.fss.2010.11.002.  Google Scholar [18] I. Perfilieva and P. Vlašánek, F-transform and discrete convolution, Proc. of the EUSFLAT Conf., (2015), 1054–1059. Google Scholar [19] D. B. Reddy, Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements, Texts in Applied Mathematics, 27, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0575-3.  Google Scholar [20] K. Rektorys, Variational Methods in Mathematics, Science and Engineering, D. Reidel Publishing Co., Dordrecht-Boston, MA, 1977. doi: 10.1007/978-94-011-6450-4.  Google Scholar [21] J. Volker, Finite Element Methods for Incompressible Flow Problems, Springer Series in Computational Mathematics, 51, Springer, Cham, 2016. doi: 10.1007/978-3-319-45750-5.  Google Scholar [22] J. G. Wang and G. Liu, A point interpolation meshless method based on radial basis functions, Internat. J. Numerical Methods in Engineering, 54 (2002), 1623-1648.  doi: 10.1002/nme.489.  Google Scholar
The approximation quality with respect to the degree of polynomials ($m$) and the number of basic functions ($N$)
 Example 1 $\bf{m}$ $\setminus$ $\bf{N}$ $4$ $8$ $12$ $16$ 1 $4.24\times10^{-3}$ $3.34\times10^{-4}$ $8.61\times10^{-5}$ $3.39\times10^{-5}$ 2 $2.34\times10^{-5}$ $7.92\times10^{-7}$ $1.30\times10^{-7}$ $3.70\times10^{-8}$ 3 $1.44\times10^{-7}$ $2.09\times10^{-9}$ $2.17\times10^{-10}$ $4.55\times10^{-11}$ 4 $1.10\times10^{-9}$ $1.27\times10^{-11}$ $1.24\times10^{-11}$ $9.41\times10^{-12}$ Example 2 $\bf{m}$ $\setminus$ $\bf{N}$ $4$ $8$ $12$ $16$ 1 $7.30\times10^{-3}$ $1.64\times10^{-3}$ $8.83\times10^{-4}$ $6.24\times10^{-4}$ 2 $4.21\times10^{-3}$ $1.52\times10^{-3}$ $8.80\times10^{-4}$ $6.23\times10^{-4}$ 3 $2.23\times10^{-3}$ $7.87\times10^{-4}$ $4.86\times10^{-4}$ $3.54\times10^{-4}$ 4 $2.13\times10^{-3}$ $7.85\times10^{-4}$ $4.75\times10^{-4}$ $3.52\times10^{-4}$ Example 3 $\bf{m}$ $\setminus$ $\bf{N}$ $4$ $8$ $12$ $16$ 1 $1.0308$ $0.7448$ $0.5683$ $0.2451$ 2 $0.8802$ $0.5818$ $0.1846$ $9.83\times10^{-2}$ 3 $0.8501$ $0.3572$ $9.60\times10^{-2}$ $2.03\times10^{-2}$ 4 $0.8338$ $0.2113$ $3.67\times10^{-2}$ $7.41\times10^{-3}$ Example 4 $\bf{m}$ $\setminus$ $\bf{N}$ $4$ $8$ $12$ $16$ 1 $2.07\times10^{-2}$ $1.71\times10^{-3}$ $4.36\times10^{-4}$ $1.72\times10^{-4}$ 2 $1.40\times10^{-3}$ $4.99\times10^{-5}$ $8.26\times10^{-6}$ $2.37\times10^{-6}$ 3 $7.52\times10^{-5}$ $1.15\times10^{-6}$ $1.21\times10^{-7}$ $2.49\times10^{-8}$ 4 $3.28\times10^{-6}$ $2.15\times10^{-8}$ $1.44\times10^{-9}$ $2.16\times10^{-10}$
 Example 1 $\bf{m}$ $\setminus$ $\bf{N}$ $4$ $8$ $12$ $16$ 1 $4.24\times10^{-3}$ $3.34\times10^{-4}$ $8.61\times10^{-5}$ $3.39\times10^{-5}$ 2 $2.34\times10^{-5}$ $7.92\times10^{-7}$ $1.30\times10^{-7}$ $3.70\times10^{-8}$ 3 $1.44\times10^{-7}$ $2.09\times10^{-9}$ $2.17\times10^{-10}$ $4.55\times10^{-11}$ 4 $1.10\times10^{-9}$ $1.27\times10^{-11}$ $1.24\times10^{-11}$ $9.41\times10^{-12}$ Example 2 $\bf{m}$ $\setminus$ $\bf{N}$ $4$ $8$ $12$ $16$ 1 $7.30\times10^{-3}$ $1.64\times10^{-3}$ $8.83\times10^{-4}$ $6.24\times10^{-4}$ 2 $4.21\times10^{-3}$ $1.52\times10^{-3}$ $8.80\times10^{-4}$ $6.23\times10^{-4}$ 3 $2.23\times10^{-3}$ $7.87\times10^{-4}$ $4.86\times10^{-4}$ $3.54\times10^{-4}$ 4 $2.13\times10^{-3}$ $7.85\times10^{-4}$ $4.75\times10^{-4}$ $3.52\times10^{-4}$ Example 3 $\bf{m}$ $\setminus$ $\bf{N}$ $4$ $8$ $12$ $16$ 1 $1.0308$ $0.7448$ $0.5683$ $0.2451$ 2 $0.8802$ $0.5818$ $0.1846$ $9.83\times10^{-2}$ 3 $0.8501$ $0.3572$ $9.60\times10^{-2}$ $2.03\times10^{-2}$ 4 $0.8338$ $0.2113$ $3.67\times10^{-2}$ $7.41\times10^{-3}$ Example 4 $\bf{m}$ $\setminus$ $\bf{N}$ $4$ $8$ $12$ $16$ 1 $2.07\times10^{-2}$ $1.71\times10^{-3}$ $4.36\times10^{-4}$ $1.72\times10^{-4}$ 2 $1.40\times10^{-3}$ $4.99\times10^{-5}$ $8.26\times10^{-6}$ $2.37\times10^{-6}$ 3 $7.52\times10^{-5}$ $1.15\times10^{-6}$ $1.21\times10^{-7}$ $2.49\times10^{-8}$ 4 $3.28\times10^{-6}$ $2.15\times10^{-8}$ $1.44\times10^{-9}$ $2.16\times10^{-10}$
The approximation quality with respect to the degree of polynomials ($m$) and the number of basic functions ($N$)
 Example 3 $\bf{m}$ $\setminus$ $\bf{N}$ $32$ $64$ $128$ $256$ 1 $4.40\times10^{-2}$ $5.62\times10^{-3}$ $7.07\times10^{-4}$ $8.78\times10^{-5}$ 2 $6.20\times10^{-3}$ $4.13\times10^{-4}$ $2.54\times10^{-5}$ $1.57\times10^{-6}$ 3 $9.25\times10^{-4}$ $2.64\times10^{-5}$ $8.08\times10^{-7}$ $2.24\times10^{-8}$ 4 $9.20\times10^{-5}$ $1.55\times10^{-6}$ $2.23\times10^{-8}$ $5.54\times10^{-10}$
 Example 3 $\bf{m}$ $\setminus$ $\bf{N}$ $32$ $64$ $128$ $256$ 1 $4.40\times10^{-2}$ $5.62\times10^{-3}$ $7.07\times10^{-4}$ $8.78\times10^{-5}$ 2 $6.20\times10^{-3}$ $4.13\times10^{-4}$ $2.54\times10^{-5}$ $1.57\times10^{-6}$ 3 $9.25\times10^{-4}$ $2.64\times10^{-5}$ $8.08\times10^{-7}$ $2.24\times10^{-8}$ 4 $9.20\times10^{-5}$ $1.55\times10^{-6}$ $2.23\times10^{-8}$ $5.54\times10^{-10}$
The comparison of the proposed method (FPP) and the piecewise linear FEM based on the approximation errors and the convergence rates
 Example 1 # N FPP FEM Error Rate Error Rate $8$ $1.8\times10^{-3}$ _ $2.3\times10^{-2}$ _ $16$ $2.2\times10^{-4}$ 3.03 $4.9\times10^{-3}$ 2.23 $32$ $2.8\times10^{-5}$ 2.97 $1.1\times10^{-3}$ 2.16 $64$ $3.3\times10^{-6}$ 3.08 $2.8\times10^{-4}$ 1.97 $128$ $4.5\times10^{-7}$ 2.87 $6.8\times10^{-5}$ 2.04 $256$ $5.1\times10^{-8}$ 3.14 $1.7\times10^{-5}$ 2.00 Example 2 # N FPP FEM Error Rate Error Rate $8$ $7.3\times10^{-3}$ _ $2.2\times10^{-2}$ _ $16$ $1.6\times10^{-3}$ 2.19 $5.4\times10^{-3}$ 2.03 $32$ $6.2\times10^{-4}$ 1.37 $1.7\times10^{-3}$ 1.67 $64$ $3.0\times10^{-4}$ 1.05 $6.5\times10^{-4}$ 1.39 $128$ $1.5\times10^{-4}$ 1.00 $2.9\times10^{-4}$ 1.16 $256$ $8.6\times10^{-5}$ 8.03 $1.4\times10^{-4}$ 1.05 Example 3 # N FPP FEM Error Rate Error Rate $8$ $0.9785$ _ $1.0055$ _ $16$ $0.6878$ 5.09 $0.8559$ 2.32 $32$ $0.2447$ 1.49 $0.2764$ 1.63 $64$ $3.9\times10^{-2}$ 2.65 $7.1\times10^{-2}$ 1.96 $128$ $5.4\times10^{-3}$ 2.85 $1.8\times10^{-2}$ 1.98 $256$ $6.9\times10^{-4}$ 2.97 $4.4\times10^{-3}$ 2.03 Example 4 # N FPP FEM Error Rate Error Rate $8$ $1.1\times10^{-2}$ _ $4.9\times10^{-2}$ _ $16$ $1.7\times10^{-3}$ 2.69 $1.1\times10^{-2}$ 2.16 $32$ $1.7\times10^{-4}$ 3.32 $2.6\times10^{-3}$ 2.08 $64$ $1.9\times10^{-5}$ 3.16 $6.2\times10^{-4}$ 2.07 $128$ $2.4\times10^{-6}$ 2.98 $1.5\times10^{-4}$ 2.05 $256$ $3.1\times10^{-7}$ 2.95 $3.8\times10^{-5}$ 1.98
 Example 1 # N FPP FEM Error Rate Error Rate $8$ $1.8\times10^{-3}$ _ $2.3\times10^{-2}$ _ $16$ $2.2\times10^{-4}$ 3.03 $4.9\times10^{-3}$ 2.23 $32$ $2.8\times10^{-5}$ 2.97 $1.1\times10^{-3}$ 2.16 $64$ $3.3\times10^{-6}$ 3.08 $2.8\times10^{-4}$ 1.97 $128$ $4.5\times10^{-7}$ 2.87 $6.8\times10^{-5}$ 2.04 $256$ $5.1\times10^{-8}$ 3.14 $1.7\times10^{-5}$ 2.00 Example 2 # N FPP FEM Error Rate Error Rate $8$ $7.3\times10^{-3}$ _ $2.2\times10^{-2}$ _ $16$ $1.6\times10^{-3}$ 2.19 $5.4\times10^{-3}$ 2.03 $32$ $6.2\times10^{-4}$ 1.37 $1.7\times10^{-3}$ 1.67 $64$ $3.0\times10^{-4}$ 1.05 $6.5\times10^{-4}$ 1.39 $128$ $1.5\times10^{-4}$ 1.00 $2.9\times10^{-4}$ 1.16 $256$ $8.6\times10^{-5}$ 8.03 $1.4\times10^{-4}$ 1.05 Example 3 # N FPP FEM Error Rate Error Rate $8$ $0.9785$ _ $1.0055$ _ $16$ $0.6878$ 5.09 $0.8559$ 2.32 $32$ $0.2447$ 1.49 $0.2764$ 1.63 $64$ $3.9\times10^{-2}$ 2.65 $7.1\times10^{-2}$ 1.96 $128$ $5.4\times10^{-3}$ 2.85 $1.8\times10^{-2}$ 1.98 $256$ $6.9\times10^{-4}$ 2.97 $4.4\times10^{-3}$ 2.03 Example 4 # N FPP FEM Error Rate Error Rate $8$ $1.1\times10^{-2}$ _ $4.9\times10^{-2}$ _ $16$ $1.7\times10^{-3}$ 2.69 $1.1\times10^{-2}$ 2.16 $32$ $1.7\times10^{-4}$ 3.32 $2.6\times10^{-3}$ 2.08 $64$ $1.9\times10^{-5}$ 3.16 $6.2\times10^{-4}$ 2.07 $128$ $2.4\times10^{-6}$ 2.98 $1.5\times10^{-4}$ 2.05 $256$ $3.1\times10^{-7}$ 2.95 $3.8\times10^{-5}$ 1.98
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