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Algebraic and FFT-Based Methods for Discrete-Time Matrix Convolutions with Applications to Semi-Markov Models

This paper develops algebraic and FFT-accelerated methods for computing matrix-valued discrete-time convolutions and their inverses, applying these efficient algorithms to solve Markov renewal equations and evaluate semi-Markov reliability functions with significant runtime reductions while maintaining high accuracy.

Original authors: L. Kordalis, S. Trevezas

Published 2026-06-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: L. Kordalis, S. Trevezas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the future of a complex machine, like a factory assembly line or a computer network. This machine moves between different "states" (e.g., working, degraded, broken). In the old, simple way of modeling this (called a Markov chain), the machine has a "short memory": it decides its next move based only on where it is right now, completely forgetting how long it has been there.

But real life isn't that simple. A machine might run for a long time before breaking, or break down very quickly. To model this, we need Semi-Markov models, which remember how long the system has been in a state. However, doing the math for these models is like trying to solve a massive puzzle where every piece depends on every other piece that came before it.

Here is what this paper does, broken down into simple concepts:

1. The Problem: The "Math Traffic Jam"

To figure out the reliability of these systems (how likely they are to keep working), mathematicians use something called a convolution. Think of convolution as a way of "smearing" or "mixing" history together to predict the future.

If you have a sequence of events (like a machine working for 1 hour, then 2 hours, then 5 hours), calculating the future state requires mixing all those past hours together.

The Old Way: The paper says the traditional method is like trying to mix a giant bowl of soup by stirring it one grain of rice at a time. It works, but it takes forever. If you want to simulate a long time period, the computer gets stuck in a "traffic jam" of calculations, taking hours or even days to finish.

2. The Solution: The "Fast Fourier Transform" (FFT)

The authors introduce a new, super-fast way to do this mixing. They use a mathematical tool called the Fast Fourier Transform (FFT).

The Analogy: Imagine you need to mix 1,000 ingredients. The old way is to mix them one by one. The FFT way is like putting all the ingredients into a high-speed blender. Instead of taking hours, it takes seconds.
The Magic: The paper shows how to translate the complex "mixing" of matrix numbers (grids of numbers representing the machine's states) into a format where the FFT blender can work its magic. This turns a task that takes hours into one that takes seconds.

3. The "Inverse" Puzzle

To solve the equations, you often need to do the opposite of mixing: you need to "un-mix" or find the inverse.

The Challenge: Finding this inverse is like trying to un-bake a cake to get the raw eggs and flour back. It's notoriously difficult and slow.
The Innovation: The authors didn't just use the blender; they invented two new, faster recipes for "un-baking":
1. Newton's Method: A clever iterative guess-and-check technique that zooms in on the answer quickly.
2. Gauss-Jordan Elimination: A systematic way of clearing out the "noise" in the equations, adapted specifically for this type of mixing.
- They combined these with the FFT blender to make the "un-mixing" process incredibly fast and accurate.

4. Bridging the Gap: Continuous vs. Discrete

Real-world time flows continuously (like a river), but computers think in steps (like a staircase).

The Problem: The paper deals with "Semi-Markov processes" (continuous time) but solves them using "Semi-Markov chains" (discrete steps).
The Trick: They developed a way to approximate the smooth, flowing river of time by taking very small, precise steps (discretization). They proved that if you take steps small enough and use their fast FFT blender, the result is almost identical to the exact, slow mathematical solution, but it runs thousands of times faster.

5. The Results: Speed Without Sacrificing Accuracy

The authors tested their new methods on two scenarios:

A Factory System: A machine that produces waste, has a buffer tank, and can shut down if the tank fills up. They modeled different types of "waiting times" (how long it takes to fill the tank).
- Result: Their new method calculated the results in 3 seconds, while the old method took over 3,000 seconds (about 50 minutes). The accuracy was nearly perfect.
A Cybersecurity Attack: A model of a "Trojan horse" attack where a computer goes from clean to infected to fraudulent.
- Result: Their fast approximations matched the results of "Monte Carlo simulations" (a method that runs thousands of random simulations to find the average) almost perfectly, but did so much faster.

Summary

In short, this paper is about speeding up the math used to predict how long complex systems will last before they break.

Before: You had to do the math slowly and painfully, limiting how complex or long-term a system you could study.
Now: The authors built a "mathematical turbocharger" (using FFT and new inversion tricks) that lets computers solve these problems in seconds instead of hours, without losing any accuracy. This allows engineers and scientists to model much more complex, real-world scenarios that were previously too difficult to compute.

Technical Summary: Algebraic and FFT-Based Methods for Discrete-Time Matrix Convolutions

Problem Statement
The paper addresses the computational challenges inherent in solving Markov renewal equations for semi-Markov models. While semi-Markov processes offer greater flexibility than standard Markov chains by allowing general sojourn time distributions, their analysis relies heavily on the computation of matrix-valued convolutions. In the discrete-time setting, these computations are essential for determining transition probabilities, reliability, and availability functions. However, existing approaches are typically limited to direct methods or theoretical generating functions, which suffer from prohibitive computational costs (quadratic complexity) when applied to large state spaces or long time horizons. Furthermore, the continuous-time case involves matrix Stieltjes convolutions, which generally lack closed-form solutions and require numerical approximation. The authors identify a gap in the theoretical development of the convolution product for matrix sequences and the lack of efficient algorithms to accelerate these computations.

Methodology
The authors propose a unified framework that bridges algebraic theory with numerical acceleration techniques:

Algebraic Framework: The paper establishes the discrete-time matrix convolution product $A * B$ within the ring of matrix-valued sequences. It treats these sequences as formal power series, allowing the convolution to be viewed as polynomial multiplication. Key algebraic properties are derived, including the existence and explicit representation of the convolutional inverse $A^{(-1)}$ , which is crucial for solving renewal equations of the form $M = L + q * M$ . The inverse is shown to exist if and only if the initial matrix $A(0)$ is nonsingular.
FFT-Based Acceleration: To overcome the $O(N^2)$ complexity of direct convolution, the authors employ the Fast Fourier Transform (FFT). By mapping the convolution to the frequency domain, the product of two sequences of length $N$ can be computed in $O(N \log N)$ time (specifically $O(s^2 N \log N + s^3 N)$ for matrix dimension $s$ ). The paper provides rigorous error analysis, separating truncation errors from floating-point errors introduced by the Cooley–Tukey FFT algorithm, demonstrating that relative errors remain of order $O(u \log N)$ .
Inversion Schemes: Computing the convolutional inverse is identified as a bottleneck. The paper proposes two accelerated inversion methods:
- Newton's Iteration: An iterative scheme adapted for polynomial matrices modulo truncation ideals, doubling the precision at each step.
- Gauss–Jordan Elimination: A novel adaptation of the Gauss–Jordan algorithm to the truncated power-series ring, accelerated by FFT-based convolution for matrix multiplications.
Continuous-Time Approximation: For continuous-time semi-Markov models involving matrix Stieltjes convolutions, the authors develop a discrete approximation strategy. This involves discretizing the Stieltjes integral using a mean-value approximation (and optionally Simpson-type extrapolation) to convert the continuous problem into a discrete matrix convolution problem, which is then solved using the aforementioned FFT-accelerated methods.

Key Contributions
The paper makes three primary contributions:

Theoretical Formulation: It formulates discrete-time Markov renewal equations as matrix-valued convolution equations and derives explicit and recursive representations for the convolutional inverse, filling a gap in the algebraic theory of matrix sequences.
Algorithmic Efficiency: It introduces FFT-based algorithms for computing both convolutions and their inverses. Specifically, it adapts Newton's method and Gauss–Jordan elimination to the convolution context, significantly reducing computational complexity from quadratic to quasi-linear.
Unified Discrete-Continuous Framework: It demonstrates how matrix Stieltjes convolutions (continuous-time) can be effectively approximated by discrete-time convolutions. This allows the same FFT-accelerated tools to be applied to continuous-time semi-Markov processes, particularly those with heavy-tailed sojourn time distributions where analytical solutions are unavailable.

Results
The proposed methods were validated through numerical experiments in two contexts:

Discrete-Time Semi-Markov Chains: Applied to a reliability model with three states and various sojourn time distributions (Discrete Lindley, Gamma, Weibull, and Yule). The results showed a dramatic reduction in runtime. For a truncation level of $N=2^{15}$ , the classical recursive method required approximately 3,200–3,700 seconds, whereas the FFT–Gauss–Jordan method completed the task in roughly 3.1–3.4 seconds. The FFT–Newton method was slightly slower (~11–12 seconds) but still vastly superior to direct methods. The relative errors compared to the classical recursive method were negligible (on the order of $10^{-11}$ to $10^{-15}$ ).
Continuous-Time Semi-Markov Processes: Applied to a four-state model representing a sequential Trojan attack. The discrete approximations of reliability and availability functions were compared against exact solutions (for the exponential/Markov case) and Monte Carlo simulations (for heavy-tailed Log-Gamma distributions). The proposed approximations showed close agreement with both benchmarks, confirming high accuracy despite the discretization and FFT approximations.

Significance
The paper claims that its framework provides a practical and efficient solution for the numerical evaluation of semi-Markov reliability and availability functions, which was previously limited by computational constraints. By reducing the computational cost, the methods enable the analysis of systems with large state spaces and long time horizons, and facilitate the use of smaller discretization steps for higher accuracy. The ability to handle heavy-tailed distributions without relying on Monte Carlo simulations (which can be computationally expensive and statistically noisy) is highlighted as a significant advantage. The authors position this work as a bridge between theoretical semi-Markov theory and practical numerical implementation, offering a flexible tool for applications in reliability engineering, actuarial science, and other fields involving complex time-dependent stochastic systems.