Open Access
Add to BibSonomy
Abstract
As a viable supplement to the fifth generation wireless communication, visible light communications (VLC) with affluent spectrum resources can cater to the ever-increasing high speed data transmission demand. However, the nonlinear characteristics of light emitting diode (LED) can distort the transmitted signal in the VLC link, which damages the communication quality. To mitigate the nonlinear impairments, a reproducing kernel Hilbert space post-distortion scheme is proposed in this paper, which is based on kernel recursive least squares (KRLS) with adaptive kernel width. In this kernel based method, the kernel width will affect the approximation ability of the model. Therefore, in the recursive process of KRLS, Gauss-Newton (GN) algorithm is adopted to update the kernel width. In addition, combined with the enhanced novelty criterion (ENC), the KRLS-GN post-distorter learns the sparse dictionary adaptively according to the input data, which is beneficial to complete the linearization under the limited memory budget constraints. The performance of the proposed KRLS-GN-ENC scheme is verified by simulations, and the results show that KRLS-GN-ENC can achieve a significant improvement over KRLS-ENC. Compared with the schemes based on classical polynomial filtering, KRLS-GN-ENC exhibits better nonlinear compensation performance and faster convergence speed.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the unprecedented advancement of high quality communication services, the demand for wireless data transmission rate is growing exponentially [1]. Traditional radio frequency systems are facing the challenge of spectrum congestion, which is insufficient for the demand of higher rate services [2]. To ensure the availability of spectrum, visible light communication (VLC) system has gradually attracted the attention of academia and industry with rich spectrum resources, high security and no electromagnetic interference [3]. Despite the above-mentioned advantages, the performance of VLC system is limited by the nonlinearity of light emitting diode (LED). The inherent nonlinearity will cause in-band and out-of-band distortions [4], thus seriously degrading the system quality and causing adjacent channel interference. In addition, the existence of nonlinearity will impair the power gain [5], resulting in the fact that the optical power of LED has to be highly restricted to the linear region to avoid the nonlinearity, which limits the coverage. Therefore, the compensation for nonlinear impairments is particularly critical for VLC systems.
Digital pre-distortion (DPD) is one of the commonly used linearization methods to suppress the nonlinearity of LED at the transmitter [6,7]. DPD intends to introduce a digital filter with the inverse response of LED, which is cascaded with LED to ensure the linear amplification of signal. To achieve linearization, DPD requires additional feedback physical circuits to estimate the inverse response [8], undoubtedly increasing the complexity and cost of implementation. Furthermore, DPD relies on the assumption of perfect feedback from the transmitter, which is unrealistic [9], so post-distortion technology is needed.
Owing to the simple structure and easy implementation, polynomial series model is widely used as the VLC post-distorter [10–12]. However, the abrupt truncation of polynomial series based approach with finite order term degrades its generalization performance, inevitably introducing modeling error [13]. To circumvent this drawback, the post-distortion schemes based on reproducing kernel Hilbert space (RKHS) have been proposed [14,15], which have been found to be more effective than the classical polynomial filtering schemes. The commonly used kernel adaptive algorithms include kernel least mean square (KLMS) [16], kernel recursive least mean square (KRLS) and the variants [17]. By using reproducing kernel to map the datasets from the low-dimensional input space to the high-dimensional RKHS, the kernel adaptive algorithms have the ability to model the nonlinearity and to find the solution for the convex nonlinear optimization problems [14,18]. In addition, combined with online dictionary based sparse techniques, low computational complexity can be achieved by employing the sparsification criterion, such as novelty criterion (NC), approximate linear correlation (ALD) and consistency criterion [15,19], to find the proper dictionary size.
For the RKHS based post-distortion schemes, an appropriate kernel width is crucial, which seriously affects the linearization performance [20]. If the value of kernel width is too large, all the data would be similar, then the system would degenerate into linear regression. While with small widths, the data would be different, resulting in overfitting. Due to the simplicity of implementation, empirical rule techniques [14], such as Silverman rule and Scott rule, are widely employed to determine the kernel width. However, statistical estimation errors may occur when calculating interquartile range or unobserved variance, resulting in poor kernel width initialization [21].
Contribution: In this paper, we propose the KRLS based post-distortion with adaptive kernel width for the nonlinear VLC systems, which has fast convergence speed and excellent linearization performance. The main contributions are as follows:
- • In order to compensate for the nonlinearity caused by LED, an adaptive post-distorter based on KRLS algorithm is proposed. As a second-order estimation technique, KRLS updates the mean and covariance simultaneously, which provides improved convergence and bit error rate (BER) performance as compared with the existing stochastic gradient based first-order post-distorters in RKHS, such as KLMS [13,22].
- • Next, Gauss-Newton (GN) algorithm is adopted for KRLS to yield the KRLS-GN scheme, which optimizes the kernel width in an adaptive way, thus avoiding the estimation error introduced by the empirical rule techniques.
- • Further, without compromising the BER performance, the enhanced novelty criterion (ENC) formed by combining ALD and NC is employed for the online sparsification. The use of ENC reduces the computational complexity, and has been proved to be capable of achieving lower errors with smaller network sizes [19].
The rest of this paper is organized as follows. In section 2, the DCO-OFDM VLC system model is introduced, also presented are the nonlinear LED and VLC channel. Section 3 presents the proposed RKHS post-distortion scheme by combing KRLS with ENC sparsification, including the kernel width adaptive optimization. In Section 4, the performance of the proposed algorithm is verified by numerical simulations. Finally, the paper is concluded in section 5.
2. System model
The block diagram of the proposed RKHS based post-distortion scheme is shown in Fig. 1. Since VLC system still faces the problem of inter symbol interference (ISI) caused by multipath fading, orthogonal frequency division multiplexing (OFDM) is employed to combat ISI [23,24]. In order to ensure that the transmitted signal be positive, DC offset optical OFDM (DCO-OFDM) is considered in this paper, which can gain higher spectral efficiency [25].
2.1 DCO-OFDM
In Fig. 1, the data bits are operated by the quadrature amplitude modulation (QAM) to generate the complex signal ${\boldsymbol X} = [X(0), \ldots ,X(k), \ldots ,X(N - 1)]$, where N is the number of subcarriers. Then, to make the driving signal of LED be real, Hermitian symmetry is adopted as
2.2 Nonlinearity of LED
To describe the input-output relationship of LED, Rapp model [27,28] has been widely used, which considers the amplitude distortion and gets to smooth the transition between the linear and saturation operating regions of LED. The Rapp model can be described as
2.3 VLC channel model
Previous research has shown that the light emission of LED is Lambertian in nature [29], so the generalized Lambert radiation intensity can be used to simulate the VLC channel. The channel gain [30] can be described by
3. KRLS-GN-ENC post-distortion
3.1 KRLS algorithm
As an extension of recursive least squares (RLS) [12], KRLS adopts the Mercer theorem [32] to implement the implicit feature mapping $\varphi :{\mathrm{\mathbb{R}}^n} \to \mathrm{\mathbb{H}}$, where ${\mathrm{\mathbb{R}}^n}$ is the n-dimensional real space, and $\mathrm{\mathbb{H}}$ is the feature space with high dimension. The input data at instant i is denoted as $r(i)$, and its transformed version in the RKHS is ${\boldsymbol \varphi }(i) = {\boldsymbol \varphi }(r(i))$, where ${\boldsymbol \varphi }({\cdot} )$ is the mapping function. By minimizing the cost function $L({\boldsymbol \omega })$, KRLS calculates the weighting vector ${\boldsymbol \omega }$ to achieve the linearization in an adaptive way. At each iteration, ${\boldsymbol \omega }$ can be determined by solving
3.2 Adaptive kernel width
For the kernel based RKHS methods, Gaussian kernel has been widely used for the universal approximations [33], which can be described as
where $\upsilon$ is the kernel width, and b is the kernel center. It should be noted that the dictionary in RKHS scheme is referred to the centers set, which is usually selected from the observation data. In (11), to construct the $i \times i$ matrix ${\boldsymbol G}(i)$, i centers are chosen from the dictionary. The performance of the linearization scheme is largely dependent on the kernel width, and the value by the sampling way could not always yield the best property. Therefore, in this paper, GN algorithm [34] is applied to adaptively update the width, according to the characteristics of the nonlinear signal. During the i-th iteration, the optimal solution for the value of ${\upsilon _i}$ can be obtained by minimizing the cost function $L({\upsilon _{i - 1}})$ as
3.3 Online sparsification
KRLS-GN is efficient in calculating the weight by utilizing the recurrent method, however, the complexity still increases quadratically with the size of the dictionary, so sparsification is necessary for reducing the dictionary size, thus further reducing the complexity [35]. Meanwhile, the reduction in dictionary may lead to performance degradation, which should be avoided, so the adopted sparsification algorithm is critical. In this paper, we choose the online sparsification, which refers to the process of adding samples to a centers set called the sparse dictionary $\mathrm{{\cal C}}(i)$.
For the KRLS-GN post-distortion, we employ the ENC criterion by combining ALD with NC to accomplish the sparsification of the dictionary [14]. First, we need to check the distance between the linear spans of the new input and the sparse dictionary in the RKHS space. If it is smaller than threshold ${\delta _1}$, the input sample is consistent with the local dictionary, so it should be discarded. Otherwise, the approximated model error $e(i)$ is then calculated to determine whether it exceeds threshold ${\delta _2}$. Only when $e(i)$ is greater than ${\delta _2}$, the input will be accepted as the new center and added to the dictionary. When combined with the ENC sparse technique, the overall complexity of KRLS-GN can be reduced from $O({N^2})$ to $O({c^2})$, where c is the size of the dictionary. Since the size by ENC is far smaller than N, so a reduced complexity is promised. The KRLS-GN-ENC scheme is summarized in Algorithm 1.
The proposed KRLS-GN-ENC algorithm aims to learn a sparse dictionary from the observation data, so as to obtain the corresponding low-dimensional weighting vector ${\boldsymbol \alpha }\textrm{(}N\textrm{)}$ and kernel width ${\upsilon _N}$, to reduce the complexity. With the sparse dictionary ${\mathrm{{\cal C}}_N}$, the output of the proposed post-distorter on the n-th time slot can be written as
4. Simulation results
4.1 Simulation setup
In this section, the performance of the proposed KRLS-GN-ENC post-distortion is verified by numerical simulations. For DCO-OFDM, 64-QAM modulation is considered, where the size of IFFT is 1024. In the Rapp model, ${V_{\max }}$ is chosen as 0.5 V, ${V_{on}}$ is set to be 0.2 V, and the knee factor $\kappa$ is set as 2. For the VLC channel, ${S_{PD}}$ is 1cm2, D is 2.3 m. While $\gamma$ and $\theta$ are both set as ${0^ \circ }$, $T(\gamma )$ and $g(\gamma )$ are both 1, ${\gamma _c}$ is ${60^ \circ }$, and ${\theta _{{1 / 2}}}$ is ${\pi / 4}$. In the kernel based post-distortion schemes, $\lambda$ is set to be 10−4, and the kernel width is initialized to 10. In the figures, the x-axis is the signal-to-noise ratio (SNR), denoted by ${{{E_b}} / {{n_0}}}$ in dB, and the y-axis is BER. Power back-off (BO) is an index to specify the operating point of LED, where smaller BO usually indicates the severer nonlinearity. As for the transmit signal, smaller BO means higher power, so it makes more sense to exert the LED with comparatively smaller BOs.
In Fig. 3, the value of BO is 0 dB, and the BER performance curves for the algorithms under different SNRs are exhibited in Fig. 3. “Slicer” stands for the conventional linear receiver without nonlinear processing, and “Bound” expresses the ideal performance bound of linear system, which can be regarded as the BER limit. “Poly-LS” and “Poly-RLS” represent the performance of polynomial series, where least square (LS) [11] and RLS [12] are adopted to find the weights, respectively, and the polynomial order is 9. To evaluate the property of ENC, the KRLS-GN-NC scheme is also compared, where the distance between input and local dictionary is calculated in the original space by NC, i. e. $di{s_2} = \min ||{c_j} - r(n)||$. In the simulations, the thresholds of ENC are set as 10−6 and 0.01, while the ones for NC are 0.1 and 0.01, respectively.
4.2 Simulation results
From Fig. 3, it is clear that when combined with the adaptive width, the performance of KRLS-GN is obviously improved compared with KRLS, which is also better than the polynomial series schemes. When the BER is $2 \times {10^{ - 5}}$, the SNR required by KRLS-ENC is about 23 dB, while the SNR of KRLS-GN-ENC is only 21 dB, thus demonstrating a 2 dB gain in SNR. When compared with the Poly-RLS scheme, KRLS-based RKHS schemes can achieve better properties, where the best performance can be obtained by the GN aided adaptive width.
To show the linearization effect of the proposed algorithm more intuitively, the comparisons of 64-QAM constellations before and after the post-distorters are illustrated when SNR is 20 dB. It can be clearly observed from Fig. 4 that compared with Poly-RLS, the constellations of the kernel-based post-distortion schemes are distributed more compactly, so the smaller Euclidean distance is promised. The constellation comparisons between Poly-LS, KRLS-ENC and KRLS-GN-ENC are depicted in Fig. 5, where the constellations of Poly-LS get more compact than that of KRLS-ENC, but the most compact constellations are with KRLS-GN-ENC, which can match the BER property shown in Fig. 3.
To further demonstrate the property of the proposed post-distorter, simulations are carried out with BO being 2 dB, and the performance curves are presented in Fig. 6. From the figure, we can see that the BER performance of KRLS-based RKHS schemes can still surpass Poly-LS and Poly-RLS, and KRLS-ENC performs better than KRLS-NC, which is similar to KRLS-GN-ENC and KRLS-GN-NC. Numerical results show that with the reduced dictionary size, the performance of ENC is exposed, which indicates that ENC is more suitable for small networks. Furthermore, the dictionary size and BER of the kernel based schemes are counted, as is shown in Table 1, where SNR is 20 dB. From the results, it is clear that the dictionary size of ENC directed KRLS can be reduced by 12.5% than NC sparsification. Meanwhile, the dictionary size of KRLS-GN-ENC is about 22.1% lower than KRLS-GN-NC, however, the BER property of KRLS-GN-ENC is better than the latter one.
Table 1. Dictionary Size and BER of Different Kernel Based Schemes
As is shown in Fig. 7, the mean squared error (MSE) performance of the schemes are presented, where SNR is 20 dB. It is clear that kernel based schemes have faster convergence speed and better MSE properties than the polynomial series scheme, where Poly-RLS is adopted to be compared.
In order to explore the linearization performance limit of KRLS-GN-ENC algorithm, we have also carried out the simulations with different BOs, and the results are shown in Fig. 8. The performance of the RKHS schemes can be improved with increased BOs, which implies the relatively weaker nonlinearity. The curves labelling the 4 dB and 6 dB BOs tend to coincide, but there is still gap between RKHS and the linear bound, so performance loss still exists.
To further testify the robustness of proposed KRLS-GN-ENC post-distortion scheme, the polynomial kernel ${\kappa _P}(a,b) = {(ab + 1)^\upsilon }$ in [36] is employed, and the results are shown in Fig. 9. The polynomial-kernel directed KRLS-GN-ENC can still outperform Poly-LS and Poly-RLS, so there is no dependency of our scheme on the selection of specific kernel functions.
5. Conclusion
Aiming at mitigating the nonlinear distortion caused by LED in the VLC systems, the KRLS-GN RKHS post-distortions is proposed in this paper, which adopts Gauss-Newton algorithm to update the kernel width. Due to the investigation of the ENC sparse method, the computational complexity of the proposed scheme can be reduced without impairing the BER performance. In order to evaluate the linearization performance of the KRLS-GN-ENC scheme, we adopt the polynomial series based nonlinear compensation schemes for comparisons, and perform the simulations in VLC systems. The numerical results show that the proposed scheme can achieve an obvious BER performance improvement over KRLS-ENC, and the new one can also provide faster convergence speed than the polynomial series based schemes.
Funding
China Postdoctoral Science Foundation (2017M623129); Natural Science Foundation of Shaanxi Province (2019JM-532); National Natural Science Foundation of China (61941105).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. J. Zhang, H. Du, P. Zhang, J. Cheng, and L. Yang, “Performance analysis of 5 G mobile relay systems for high-speed trains,” IEEE J. Select. Areas Commun. 38(12), 2760–2772 (2020). [CrossRef]
2. C. Wang, X. You, X. Gao, X. Zhu, Z. Li, C. Zhang, H. Wang, Y. Huang, Y. Chen, H. Haas, J. S. Thompson, E. G. Larsson, M. D. Renzo, W. Tong, P. Zhu, X. Shen, H. V. Poor, and L. Hanzo, “On the road to 6G: Visions, requirements, key technologies and testbeds,” IEEE Commun. Surv. Tut. 25(2), 905–974 (2023). [CrossRef]
3. Y. Zhao, P. Zou, F. Hu, J. Zhang, S. Yu, and N. Chi, “Scalar superposed coded modulation in the multiple-input-single-output visible light communication system,” J. Lightwave Technol. 40(9), 2703–2709 (2022). [CrossRef]
4. N. Sharan and S. K. Ghorai, “PAPR reduction and non-linearity alleviation using hybrid of precoding and companding in a visible light communication (VLC) system,” Opt. Quantum Electron. 52(6), 304 (2020). [CrossRef]
5. P. Miao, G. Chen, K. Cumanan, Y. Yao, and J. A. Chambers, “Deep hybrid neural network-based channel equalization in visible light communication,” IEEE Commun. Lett. 26(7), 1593–1597 (2022). [CrossRef]
6. G. Xu, H. Yu, C. Hua, and T. Liu, “Chebyshev polynomial-LSTM model for 5 G millimeter-wave power amplifier linearization,” IEEE Microw. Wirel. Co. 32(6), 611–614 (2022). [CrossRef]
7. F. J. Escribano, J. Sáez-Landete, and A. Wagemakers, “Chaos-based multicarrier VLC modulator with compensation of LED nonlinearity,” IEEE Trans. Commun. 67(1), 590–598 (2019). [CrossRef]
8. C. Cheng, X. Li, Q. Xiang, J. Li, Y. Jin, Z. Wei, H. Y. Fu, and Y. Yang, “4-bit DAC based 6.9 Gb/s PAM-8 UOWC system using single-pixel mini-LED and digital pre-compensation,” Opt. Express 30(15), 28014–28023 (2022). [CrossRef]
9. T. Sasai, M. Nakamura, E. Yamazaki, A. Matsushita, S. Okamoto, K. Horikoshi, and Y. Kisaka, “Wiener-Hammerstein model and its learning for nonlinear digital pre-distortion of optical transmitters,” Opt. Express 28(21), 30952–30963 (2020). [CrossRef]
10. H. Qian, S. Yao, S. Cai, and T. Zhou, “Adaptive postdistortion for nonlinear LEDs in visible light communications,” IEEE Photonics J. 6(4), 1–8 (2014). [CrossRef]
11. K. Ying, Z. Yu, R. J. Baxley, H. Qian, G.-K. Chang, and G. T. Zhou, “Nonlinear distortion mitigation in visible light communications,” IEEE Wireless Commun. 22(2), 36–45 (2015). [CrossRef]
12. L. Zhang, R. Jiang, X. Tang, Z. Chen, J. Chen, and H. Wang, “A simplified post equalizer for mitigating the nonlinear distortion in SiPM based OFDM-VLC system,” IEEE Photonics J. 14(1), 1–7 (2022). [CrossRef]
13. R. Mitra, F. Miramirkhani, V. Bhatia, and M. Uysal, “Low complexity least minimum symbol error rate based post-distortion for vehicular VLC,” IEEE Trans. Veh. Technol. 69(10), 11800–11810 (2020). [CrossRef]
14. W. Liu, J. C. Principe, and S. Haykin, Kernel adaptive filtering: a comprehensive introduction, 57, (Wiley: Hoboken, NJ, USA, 2011).
15. R. Mitra, F. Miramirkhani, V. Bhatia, and M. Uysal, “Mixture-kernel based post-distortion in RKHS for time-varying VLC channels,” IEEE Trans. Veh. Technol. 68(2), 1564–1577 (2019). [CrossRef]
16. R. Mitra and B. Vimal, “Low complexity post-distorter for visible light communications,” IEEE Commun. Lett. 21(9), 1977–1980 (2017). [CrossRef]
17. J. D. A. Santos and G. A. Barreto, “An outlier-robust kernel RLS algorithm for nonlinear system identification,” Nonlinear Dyn. 90(3), 1707–1726 (2017). [CrossRef]
18. S. Jain, R. Mitra, and V. Bhatia, “KLMS-DFE based adaptive post-distorter for visible light communication,” Opt. Commun. 451, 353–360 (2019). [CrossRef]
19. F. Tan and X. Guan, “Research progress on intelligent system’s learning, optimization, and control—part II: online sparse kernel adaptive algorithm,” IEEE Trans. Syst., Man Cybern. 50(12), 5369–5385 (2020). [CrossRef]
20. B. Chen, J. Liang, N. Zheng, and J. C. Príncipe, “Kernel least mean square with adaptive kernel size,” Neurocomputing 191, 95–106 (2016). [CrossRef]
21. R. Mitra, V. Bhatia, S. Jain, and K. Choi, “Performance analysis of random Fourier features-based unsupervised multistage-clustering for VLC,” IEEE Commun. Lett. 25(8), 2659–2663 (2021). [CrossRef]
22. R. Mitra and V. Bhatia, “Adaptive sparse dictionary-based kernel minimum symbol error rate post-distortion for nonlinear LEDs in visible light communications,” IEEE Photonics J. 8(4), 1 (2016). [CrossRef]
23. R. Yang, S. Ma, Z. Xu, H. Li, X. Liu, X. Ling, X. Deng, Z. Xun, and S. Li, “Spectral and energy efficiency of DCO-OFDM in visible light communication systems with finite-alphabet inputs,” IEEE Trans. Wirel. Commun. 21(8), 6018–6032 (2022). [CrossRef]
24. J. M. G. Linnartz, X. Deng, and P. V. Voorthuisen, “Impact of dynamic LED non-linearity on DCO-OFDM optical wireless communication,” IEEE Commun. Lett. 25(10), 3335–3339 (2021). [CrossRef]
25. X. Ling, S. Li, P. Ge, J. Wang, N. Chi, and X. Gao, “Optimal DCO-OFDM signal shaping with double-sided clipping in visible light communications,” Opt. Express 28(21), 30391–30409 (2020). [CrossRef]
26. G. Stepniak and J. Siuzdak, “Influence of lighting LED design parameters on the dynamic nonlinear response,” J. Lightwave Technol. 40(4), 954–960 (2022). [CrossRef]
27. H. Li, J. Wei, and N. Jin, “Low-complexity tone reservation scheme using pre-generated peak-canceling signals,” IEEE Commun. Lett. 23(9), 1586–1589 (2019). [CrossRef]
28. M. Abd Elkarim, M. H. Aly, H. M. AbdelKader, and M. M. Elsherbini, “LED nonlinearity mitigation in LACO-OFDM optical communications based on adaptive predistortion and postdistortion techniques,” Appl. Opt. 60(24), 7279–7289 (2021). [CrossRef]
29. J. Wei, C. Gong, N. Huang, and Z. Xu, “Channel modeling and signal processing for array-based visible light communication system under link misalignment,” IEEE Photonics J. 14(2), 1–10 (2022). [CrossRef]
30. C. Wang, Y. Yang, Z. Yang, C. Feng, J. Cheng, and C. Guo, “Joint SIC-based precoding and sub-connected architecture design for MIMO VLC systems,” IEEE Trans. Commun. 71(2), 1044–1058 (2023). [CrossRef]
31. Y. Wang, J. Gui, Y. Yin, J. Wang, J. Sun, G. Gui, H. Gacanin, H. Sari, and F. Adachi, “Automatic modulation classification for MIMO systems via deep learning and zero-forcing equalization,” IEEE Trans. Veh. Technol. 69(5), 5688–5692 (2020). [CrossRef]
32. L. Yan, X. Duan, B. Liu, and J. Xu, “Gaussian processes and polynomial chaos expansion for regression problem: linkage via the RKHS and comparison via the KL divergence,” Entropy 20(3), 191 (2018). [CrossRef]
33. S. Jain, R. Mitra, and V. Bhatia, “Hybrid adaptive precoder and post-distorter for massive-MIMO VLC,” IEEE Commun. Lett. 24(1), 150–154 (2020). [CrossRef]
34. K. Mom, M. Langer, and B. Sixou, “Deep Gauss–Newton for phase retrieval,” Opt. Lett. 48(5), 1136–1139 (2023). [CrossRef]
35. B. Scholkopf and A. J. Smola, Learning with kernels: regularization, optimization, and beyond. MIT Press Cambridge, (2001).
36. F. Z. Geng and S. Qian, “An optimal reproducing kernel method for linear nonlocal boundary value problems,” Appl. Math. Lett. 77, 49–56 (2018). [CrossRef]
