Seminar Topics

Background:

DNA data storage has emerged as a compelling medium for archival data, offering extraordinary density and durability. In practice, data is encoded into millions of short synthetic DNA strands stored in a liquid container. When retrieved, the strands are read by nanopore sequencers in random order and a random number of times, and are corrupted by substitution and deletion errors. Weinberger and Merhav [4] formally characterized the capacity of this channel, accounting for the Poisson-distributed read counts and the permutation of strands.

The classical approach to protect data in this setting is a concatenation scheme: a per-strand inner code handles sequence errors, and an outer Reed–Solomon code handles strand-level erasures. A fundamental limitation of this approach is that current DNA synthesis technology restricts strand lengths to roughly 200–300 nucleotides—a regime in which inner codes suffer a significant finite-block-length rate penalty.

Modern Relevance:

Wang and Guruswami [1] proposed Geno-Weaving to overcome this bottleneck. Rather than coding along each strand, the key idea is to code across strands: a single block code protects the same nucleotide position across all strands simultaneously. Since the number of strands (millions) is orders of magnitude larger than the strand length, this orthogonal direction provides a vastly larger effective block length, eliminating the finite-length penalty. The scheme uses polar codes and provably achieves the capacity of the DNA storage channel. Lin, Wang, and Guruswami [2] extended this framework to deletion errors, demonstrating—theoretically and through simulations—that a Geno-Weaving design for substitution channels already performs well on deletion channels at realistic rates of 0.1%–10%, outperforming even optimal concatenation schemes.

Seminar Focus:

This seminar explores the Geno-Weaving framework and its extension to deletion errors. The goal is to develop a deep understanding of the channel model, the capacity-achieving code construction, and the block-length advantage over concatenation. The student will work through the encoding and decoding procedures in detail, develop illustrative examples, and carry out a quantitative comparison of Geno-Weaving's code rates against those of concatenation schemes for different deletion probabilities.

Expected Outcomes:

- Channel Model & Capacity: A clear explanation of the DNA storage channel, including Poissonization and the three challenge factors (noise, permutation, and sampling), with worked examples.

- Code Construction: Step-by-step illustrations of the Geno-Weaving encoding and decoding, including the push/pull synchronization mechanism for deletion errors.

- Rate Comparison: Plots comparing Geno-Weaving's achievable rates against concatenation schemes across a range of deletion probabilities, with a discussion of the block-length gain.

References:

[1] H.-P. Wang and V. Guruswami, "Geno-Weaving: A Framework for Low-Complexity Capacity-Achieving DNA Data Storage," *IEEE Journal on Selected Areas in Information Theory*, vol. 6, pp. 383–393, 2025, doi: [10.1109/JSAIT.2025.3610643](https://doi.org/10.1109/JSAIT.2025.3610643).

[2] Y.-T. Lin, H.-P. Wang, and V. Guruswami, "Block Length Gain for Nanopore Channels," *arXiv preprint arXiv:2511.18027*, Nov. 2025.

[3] L. Organick, S. D. Ang, Y.-J. Chen, R. Lopez, S. Yekhanin, K. Makarychev, et al., "Random Access in Large-Scale DNA Data Storage," *Nature Biotechnology*, vol. 36, no. 3, pp. 242–248, Mar. 2018, doi: [10.1038/nbt.4079](https://doi.org/10.1038/nbt.4079).

[4] N. Weinberger and N. Merhav, "The DNA Storage Channel: Capacity and Error Probability Bounds," *IEEE Transactions on Information Theory*, vol. 68, no. 9, pp. 5657–5700, Sep. 2022, doi: [10.1109/TIT.2022.3176371](https://doi.org/10.1109/TIT.2022.3176371).

Prerequisites

A matrix code is a subspace of Fq^{m x n}. Given two K-dimensional matrix codes C and D, the Matrix Code Equivalence (MCE) problem asks whether there exists invertible matrices P \in Fq^{m x m} and Q \in Fq^{n x n} such that C = PDQ where the latter notation means {PAQ : A \in D}. In other words, we want to determine if C and D are equivalent up the left- and right-scrambling by invertible matrices. There exist any many solvers for this problem, though all the solvers run in exponential time.

The security of certain cryptographic schemes in the current NIST competition rely on the MCE problem and therefore it is important to understand the true complexity of solving it. The task of the student is to understand and explain how the existing solvers for MCE work. In particular, they should they do so for at least the following listed solvers in the literature.

[1] A. Couvreur and C. Levrat, “Highway to Hull: An Algorithm for Solving the General Matrix Code Equivalence Problem,” Advances in Cryptology – CRYPTO 2025, pp. 253–283, 2025, doi: 10.1007/978-3-032-01855-7_9.
[2] A. K. Narayanan, Y. Qiao, and G. Tang, “Algorithms for Matrix Code and Alternating Trilinear Form Equivalences via New Isomorphism Invariants,” Advances in Cryptology – EUROCRYPT 2024, pp. 160–187, 2024, doi: 10.1007/978-3-031-58734-4_6.
[3] W. Beullens, “Graph-Theoretic Algorithms for the Alternating Trilinear Form Equivalence Problem,” Advances in Cryptology – CRYPTO 2023, pp. 101–126, 2023, doi: 10.1007/978-3-031-38548-3_4.

Prerequisites

The document exchange problem considers the scenario where two parties have slightly different versions of the same file, and one of them wants to update their version to match the other. Instead of sending the whole file, the idea is to share only a small description of the changes. Substring edits, where changes happen locally within the file, are a natural way to model these differences. This seminar will examine how coding theory can be applied to enhance the efficiency of document exchange under substring edits.

Post-quantum cryptography (PQC) has been an active area of research since the seminal work of Shor. 
Indeed, most public-key cryptography currently deployed (such as RSA and elliptic-curve-based schemes) is vulnerable to quantum adversaries. 
This applies in particular to public-key encryption (PKE) schemes.  
An attacker could record encrypted traffic and later decrypt it once a sufficiently capable quantum computer becomes available - a strategy known as "harvest now, decrypt later."

Post-quantum secure alternatives can be constructed from hard problems on codes and lattices. The recently standardized Kyber and HQC follow an encryption-based approach, while, e.g., NewHope is based on a key reconciliation mechanism:
Alice and Bob obtain noise variants of a common secret, and error correction removes this noise, allowing them to agree on the same shared key.

This project will survey constructions for exchanging a key in code-based and lattice-based cryptography. These constructions are to be categorized based on key properties, such as underlying metric, bandwidth requirements, and underlying assumptions.
An overview of the key techniques is to be developed, with particular focus on the question of whether they can be transferred from codes to lattices and vice versa. The project requires reading and understanding several references; a good starting point can be the following works:

Aguilar-Melchor, Carlos, et al. "Efficient encryption from random quasi-cyclic codes." IEEE Transactions on Information Theory 64.5 (2018): 3927-3943.

Bos, Joppe, et al. "CRYSTALS-Kyber: a CCA-secure module-lattice-based KEM." 2018 IEEE European symposium on security and privacy (EuroS&P). IEEE, 2018.

Alkim, Erdem, et al. "Post-quantum Key Exchange — A new hope." 25th USENIX security symposium (USENIX Security 16). 2016.

Information theory and combinatorics are deeply intertwined. Beyond the use of combinatorics in coding theory and compression, there are many -sometimes surprising- connections.
One such connection is the use of graph entropy in combinatorial existence proofs.

This seminar topic is about explaining the proof technique introduced in [1] and [2] and applied in [3]. The goal is a tutorial-style paper with the focus on clear exposition through well chosen worked examples and visualizations.

[1] M. Fredman, and J. Komlós, On the Size of Separating Systems and Perfect Hash Functions, SIAM J. Alg. Disc. Meth., 5 (1984), pp. 61-68.

[2] J. Körner, Fredman-Komlós bounds and information theory, SIAM J. on Algebraic and Discrete Meth., 4(7), (1986), pp. 560–570.

[3] N. Alon, E. Fachini, and J. Körner, Locally Thin Set Families, Combinatorics, Probability and Computing, vol. 9 (Nov. 2000), pp. 481–488.

Ultra-Reliable Low-Latency Communication (URLLC) is a key requirement of future wireless systems such as 6G. URLLC requires very high reliability (BLER ≈ 10e-5 – 10e-7) and extremely low latency (<1 ms), which places strong constraints on channel coding and decoding algorithms.

Short block-length channel codes are particularly important in URLLC scenarios. However, short codes often suffer from degraded error-rate performance, making the design of efficient codes and decoders challenging. 

Among the promising candidates for short block-length coding are extended BCH (eBCH) codes and polar codes, both of which can achieve near-optimal performance when decoded with appropriate algorithms.

The student task is to implement an OSD decoding algorithms for an eBCH code and SCL decoder for polar code with parameters (n,k) = (128,64). The 5G toolbox can be used. Using these implementations, the student will perform Monte Carlo simulations to evaluate the error-rate performance of the considered coding schemes. The communication system should consist of binary phase shift keying (BPSK) modulation over an additive white Gaussian noise (AWGN) channel. The main performance metric to be evaluated is the block error rate (BLER) as a function of the signal-to-noise ratio. The student should generate BLER curves for both coding schemes and compare their performance in the short block-length regime. In addition to the error-rate performance, the student should also analyze the decoding latency. 

[1] M. Geiselhart, F. Krieg, J. Clausius, D. Tandler and S. ten Brink, "6G: A Welcome Chance to Unify Channel Coding?

[2] W. Saad, M. Bennis and M. Chen, "A Vision of 6G Wireless Systems: Applications, Trends, Technologies, and Open Research Problems,"

[3] C. Yue, V. Miloslavskaya, M. Shirvanimoghaddam, B. Vucetic and Y. Li, "Efficient Decoders for Short Block Length Codes in 6G URLLC,"

Prerequisites

Federated Learning is vulnerable to adversarial clients that may submit poisonous model updates to the central server. As a countermeasure, many robust aggregation rules have been proposed, aiming to limit the effect of such malicious gradients on the aggregated global model. One of the commonly studied techniques, gradient clipping, bounds the norm of updates prior to aggregation.

The authors of [1] propose an adaptive gradient clipping method as a pre-aggregation step. They show theoretically that adaptive clipping preserves the robustness of the subsequent robust aggregation method, and empirically improves robustness significantly. In this work, static gradient-clipping (with a fixed norm threshold) cannot ensure robustness. On the other hand, the work in [2] shows that averaging normalized gradients (with norm 1) can achieve robustness in federated optimization, particularly under certain assumptions about data heterogeneity across clients.

The goal of this seminar is to analyze the differences in the convergence analyses presented in [1] and [2], and to investigate whether the convergence results from [2] can be extended to more general heterogeneity models, such as those considered in [1].

[1] Allouah, Youssef et al. “Adaptive Gradient Clipping for Robust Federated Learning”. In ICLR, 2025

[2] Zuo, Shiyuan et al. “Efficient Federated Learning against Byzantine Attacks and Data Heterogeneity via Aggregating Normalized Gradients”. In NeurIPS, 2025.

Prerequisites

The duality principle is a powerful concept extensively used in various scientific disciplines, such as Lagrangian duality in optimization theory. In information theory, however, there are many different types of 'duality', which vary in formality. Starting with Shannon's observations on a source-channel duality [1], other notions of duality were later introduced, including functional duality [2] and operational duality [3]. The duality of Gaussian broadcast channels and Gaussian multiple-access channels [4] helped to compute the dirty-paper achievable rate region efficiently for the Gaussian multiple-input multiple-output broadchast channel [5]. The student will review and compare different fomulations of duality in the literature on information theory. Two case studies shall demonstrate how duality was applied to solve a new problem. Students interested in this topic should have a strong background in information theory and multi-user information theory.

[1] C. E. Shannon, “Coding theorems for a discrete source with a fidelity criterion,” IRE Int. Conv. Rec, pp. 142–163, Mar. 1959.
[2] S. Pradhan, J. Chou, and K. Ramchandran, “Duality between source coding and channel coding and its extension to the
side information case,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1181–1203, May 2003. DOI: 10.1109/TIT.2003.810622.
[3] A. Gupta and S. Verdu, “Operational duality between lossy compression and channel coding,” IEEE Trans. Inf. Theory, vol. 57, no. 6, pp. 3171–3179, Jun. 2011. DOI: 10.1109/TIT.2011.2136910.
[4] N. Jindal, S. Vishwanath, and A. Goldsmith, “On the duality of gaussian multiple-access and broadcast channels,” IEEE Trans. Inf. Theory, vol. 50, no. 5, pp. 768–783, May 2004. DOI: 10.1109/TIT.2004.826646.
[5] S. Vishwanath, N. Jindal, and A. Goldsmith, “Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2658–2668, Oct. 2003, doi: 10.1109/TIT.2003.817421.

Prerequisites

Secure aggregation is a fundamental cryptographic technique used in distributed machine learning paradigms, such as federated learning, enabling a central server to compute the sum of users' inputs without learning anything about their individual data.

This seminar explores the fundamental information-theoretic limits of secure aggregation by analyzing two critical extensions to the standard model. The first paper [1] investigates a two-round communication protocol for handling unknown user dropouts, characterizing the optimal communication cost required. The second paper [2] extends the traditional client-server architecture to a hierarchical network (clients, relays, and server). It introduces a "relay security" constraint to keep intermediate nodes oblivious to user inputs and establishes the optimal trade-off between communication efficiency and key generation efficiency. 

[1]. Zhao, Yizhou, and Hua Sun. "Information theoretic secure aggregation with user dropouts." IEEE Transactions on Information Theory 68.11 (2022): 7471-7484.

[2]. Zhang, Xiang, et al. "Optimal communication and key rate region for hierarchical secure aggregation with user collusion." IEEE Transactions on Information Theory 72.2 (2025): 1030-1050.

Prerequisites

Two methods for non-linear quantized precoding in communication systems are discussed in references [1] and [2]. The student is expected to thoroughly read and understand one of the two papers, as this will be the seminar's primary focus. Afterward, a comparison between the two methods is needed.

[1] Wu, Zheyu, Junjie Ma, Ya-Feng Liu, and Bruno Clerckx. "An AMP-Based Asymptotic Analysis For Nonlinear One-Bit Precoding." arXiv preprint arXiv:2601.13214 (2026).

[2] Jacobsson, Sven, Giuseppe Durisi, Mikael Coldrey, Tom Goldstein, and Christoph Studer. "Quantized precoding for massive MU-MIMO." IEEE Transactions on Communications 65, no. 11 (2017): 4670-4684.

Modern communication systems increasingly employ probabilistic amplitude shaping (PAS) to close the gap to the Shannon capacity by generating non-uniform input distributions (typically approximating Maxwell–Boltzmann distributions). At the core of PAS lies the distribution matcher (DM), which maps uniformly distributed information bits into shaped sequences of amplitudes.

A central challenge is designing DMs that achieve low rate loss and low complexity.

Enumerative Sphere Shaping (ESS) addresses these challenges by restricting sequences to lie within a hypersphere in the energy domain. Instead of fixing symbol composition (as in CCDM), ESS selects sequences such that their total energy does not exceed a predefined radius. This approach results in competitive performance in the short blocklength regime.

The student will explore ESS [1][2], in depth including its theoretical foundation in information and coding theory, algorithmic implementation, storage and computational complexity, and comparisons with other shaping methods such as CCDM and shell mapping [3].

[1] https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=8850066

[2] https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=8895789&casa_token=sZch6uYNxusAAAAA:Oq9owJ8tNEA5puQv1Bai7enCei8nYMoPEwt9jO3q-9upB20dTKJPltXrbI7mPSKnsVs_nK_G

[3] https://www.mdpi.com/1099-4300/22/5/581

Prerequisites

Field of Research & Contents of the Paper

Time-varying phase noise is of interest in many communication scenarios, including high-frequency wireless channels and coherent optical communications. 

In [1], the authors introduce an iterative receiver for channels distorted by phase noise. They use strong correlations in the phase noise process for mitigation. A downside is the requirement for pilot insertion, effectively decreasing spectral efficiency. 

Expectation propagation (EP) [2][3, Ch. 10, Sec. 10.7] has been recently used to enhance the receiver design, thereby mitigating the need for pilot insertion [4]. 

The task of this seminar paper is to

1. summarize the system model used in [1],[4],
2. analyze the approach, benefits, and drawbacks in [1],
3. explain the principles of EP, and
4. compare how EP modifies the receiver in [4] relative to [1]. 

A successful submission provides intuition for tasks 1-3 and a thorough analysis of the differences between [1] and [4] for task 4. 

[1] G. Colavolpe, A. Barbieri and G. Caire, "Algorithms for iterative decoding in the presence of strong phase noise," in IEEE Journal on Selected Areas in Communications, 2005.

[2] T. P. Minka, "Expectation Propagation for approximate Bayesian inference," in Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, 2001. 

[3] C.M. Bishop, "Pattern Recognition and Machine Learning," Springer, 2006.

[4] E. Conti, A. Vannucci, A. Piemontese, G. Colavolpe, "Phase Noise Detection via Expectation Propagation and Related Algorithms", arXiv, 2024.

Organization of the Supervision

• After you have successfully applied for the topic, we will hold a kick-off meeting to discuss all necessary details for the upcoming work. After that, the responsibility for the seminar paper is all on you. You can always contact me, ask questions, and request meetings, and I am happy to assist you along the way. However, I need you to be proactive.
• I can provide comments on your work (e.g., the paper's outline and some paragraphs you have already written) at any time you need them and guide you as you prepare your paper and presentation. Regarding final drafts, I comment on one complete version of your paper. Ideally, this would mean that you do some work (literature research, the paper's outline, the body, the introduction, and the conclusion), and we discuss your results after every step. Ultimately, you provide me with one final draft, and I will comment on it. The draft must be handed in at least 10 days prior to the paper submission deadline. This method is more successful and instructive than writing a paper a few days before the deadline.
• Beware that when I comment on your work, this is not the same as a correction. My comments are suggestions for improving the quality of the work, but I cannot spend the time searching for and correcting every mistake that exists or could be made (this would also not align with the seminar's purpose).
• I expect solid research. There should be two significant references. Those are papers you know in detail and that majorly impact your work. Of course, they are always accompanied by supporting references.

Prerequisites

Channel resolvability, introduced in [1] based on ideas from [2], measures how much needs to be transmitted to approximate a random variable at the channel output accurately.

Not only does it have deep ties to the capacity of a channel, but it is also an important tool in information theoretic secrecy, e.g., when communication should be hidden.

The goal of this topic is to understand and present the concept of resolvability, to outline it's relation to other channel properties such as capacity, and to illustrate its use in information theory by means of an exemplary application.

1. Han, Verdu. 1993. Approximation Theory of Output Statistics. https://doi.org/10.1109/18.256486
2. Wyner. 1975. The common information of two dependent random variables. https://doi.org/10.1109/TIT.1975.1055346

Prerequisites

Modern short-reach optical communication systems, such as those used in data centers, require high data rates together with low cost and energy consumption. To improve spectral efficiency and approach channel capacity, probabilistic constellation shaping (PCS) has emerged as a promising technique by adapting the probability distribution of transmitted symbols.

While PCS is well established for coherent systems, its application to intensity modulation/direct detection (IM/DD) systems is more challenging due to constraints such as signal non-negativity and the integration with forward error correction (FEC). In [1], the authors propose a practical PCS scheme for IM/DD systems by modifying the bit-to-symbol mapping, enabling a near-optimal distribution and demonstrating measurable performance gains.

However, many PCS approaches are derived under average power constraints. In practical IM/DD systems, hardware limitations such as the finite extinction ratio impose a peak power constraint, which can significantly alter the optimal input distribution. This aspect is investigated in [2], where probabilistic shaping is analyzed under peak power limitations.

Furthermore, recent studies such as [3] show that classical probabilistic shaping assumptions may not directly translate to IM/DD systems, highlighting the importance of system-specific constraints in shaping design.

The goal of this seminar is to understand the implementation of PCS in IM/DD systems as presented in [1], and to analyze how peak power constraints and system limitations affect shaping performance, as discussed in [2] and [3].

[1] Zonglong He, Tianwai Bo, and Hoon Kim, "Probabilistically shaped coded modulation for IM/DD system," Opt. Express 27, 12126-12136 (2019)

[2] T. Wiegart, F. Da Ros, M. P. Yankov, F. Steiner, S. Gaiarin and R. D. Wesel, "Probabilistically Shaped 4-PAM for Short-Reach IM/DD Links With a Peak Power Constraint," in Journal of Lightwave Technology, vol. 39, no. 2, pp. 400-405, 15 Jan.15, 2021, doi:10.1109/JLT.2020.3029371

[3] D. Che, J. Cho and X. Chen, "Does Probabilistic Constellation Shaping Benefit IM-DD Systems Without Optical Amplifiers?," in Journal of Lightwave Technology, vol. 39, no. 15, pp. 4997-5007, Aug.1, 2021, doi: 10.1109/JLT.2021.3083530.

Prerequisites

As there are continuously increasing throughput demands on fiber optical networks, many types of multiplexing systems have become a hot topic for research as there is a need to better utilize the bandwidth resources of the optical fiber and increase spectral efficiency, some of these allow of unique joint Digital Signal Processing. Carrier phase estimation is an important aspect of DSP-based receivers, through carrier phase estimation the effects of laser phase noise are mitigated, the effects of laser phase noise are in general more important as systems become more spectrally efficient. Traditionally carrier and phase recovery is done per channel basis using either blind or data aided methods, however some methods are able to exploit correlation between the channels and other such properties to perform joint carrier and phase recovery.

This topic focuses mainly on joint carrier and phase recovery algorithms and their performance. This topic looks at Frequency Comb-Based Transmission Systems and how in those systems the phase coherence between the lines can be exploited to simplify or increase the performance of digital carrier phase recovery algorithms and comparisons are made between different algorithms and different pilot distributions over the channels.

Papers:

1. Frequency Comb-Based WDM Transmission Systems Enabling Joint Signal Processing, Lars Lundberg et al

2. Joint Carrier Recovery for DSP Complexity Reduction in Frequency Comb-Based Superchannel Transceivers, Lars Lundberg et al

3. Pilot-Aided Joint-Channel Carrier-Phase Estimation in Space-Division Multiplexed Multicore Fiber Transmission, Arni F. Alfredsson et al                

4. Optimization of Pilot-Aided Joint Phase Recovery for Frequency Comb-Based Wideband Transmission, Gabriele Di Rosa et al

Interested student please contact Muhammad Ahmed Leghari for further discussion

Contact: ahmed.leghari@adtran.com

Photodetectors in optical communication systems measure only field intensity, the phase is lost. Phase retrieval methods aim to recover the full complex field from intensity-only measurements, avoiding the need for a local oscillator laser. This assignment covers two such approaches: the Kramers–Kronig (KK) receiver, which uses a single detector with a specially prepared transmit signal, and the Gerchberg–Saxton (GS) algorithm with a dispersive element, which requires two intensity measurements.

The student's task is to understand the working principles of both methods and summarize them in a report.

Possible literature

• Recent Advances in Coherent Optical Communications for Short-Reach: Phase Retrieval Methods, Karar et al. Photonics 2023
• Gerchberg–Saxton Algorithm Based Phase Correction in Optical Wireless Communication, Li et al. Phyiscal Comm 2017

Prerequisites

Next-generation optical access networks are evolving towards ultra-high bit rates (above 50 Gbps per wavelength) and extended fiber reach architectures. This trend will likely push the optoelectronics to their limits, thus requiring impairment compensation based on digital signal processing (DSP) techniques in the transceivers.

This seminar will focus on the methods in overcoming physical-layer limitations in future passive optical network (PON) systems presented in [1]. The student should analyze key impairments (e.g., chromatic dispersion, bandwidth limitations, and noise), discuss DSP-based mitigation techniques, and evaluate the trade-offs between performance, complexity, and cost in practical deployments. Also, a discussion on future of pon should be made by combining [1] and [2].

Literature to focus on:

[1] "Planning tools for next-generation DSP-based passive optical networks above 50G"

[2] "What is beyond 50G: future standards of optical access networks"

Prerequisites

Which is the right equation to solve within the domain of fiber optics?

It all boils down to the application scenario, to the interplay between the duration of the transmission pulse, the channel dynamic, and the desired level of accuracy.

The most accurate and computationally demanding model is provided by Maxwell's equations, while other equations (like Manakov equations and Schrödinger equations) can be obtained within certain assumptions.

Tasks of the student is to review the subject, pointing out typical scenarios, their relevant equation, the assumptions within which it has been derived, and a numerical method to solve it.

The student can refer to [Menyuk99] and references therein.

REFERENCES: 

[Menyuk99] Menyuk, C. R. "Application of multiple-length-scale methods to the study of optical fiber transmission." Journal of Engineering Mathematics 36.1 (1999): 113-136.

Prerequisites