Resolving computational bottlenecks in complex wave interactions
Whether tracking an earthquake’s shockwave through the Earth or tracing the forces pulling apart a DNA strand, energy encounters complex real-world boundaries. In applied mathematics and physics, capturing these events requires solving sophisticated wave physics and boundary-value problems. For over a century, the primary tools used to crack these equations have relied on factorisation(opens in new window) methods, Wiener-Hopf(opens in new window) techniques and Riemann-Hilbert problems. Rooted in complex analysis and operator theory, these techniques decompose a complex analytic function into a product or sum of simpler, independent components, transforming seemingly impossible wave physics into solvable equations. Funded under the Marie Skłodowska-Curie Actions programme, the EffectFact project set out to resolve a critical computational bottleneck: while these techniques work flawlessly for simple, single-variable (scalar) equations, they break down when applied to the complex, multivariable matrix systems required by modern science and technology.
Solving the matrix Wiener-Hopf problem
“To overcome this long-standing issue, we moved beyond abstract theory to develop effective, constructive and numerically reliable matrix factorisation techniques for matrix Wiener-Hopf problems that remain stable even when structural invariants are unknown a priori,” notes project coordinator Gennady Mishuris. Project researchers developed new algorithms for matrix polynomial factorisation and translated the related approaches into usable software by creating the ExactMPF package. They also created analytical tools proving whether an unknown set of partial indices is stable under external perturbations. The importance of these breakthroughs becomes clear when looking at the mathematical backdrop. “While single-variable scalar equations are long understood, serving as a cornerstone for everything from fluid mechanics and diffraction theory to financial modelling, the real world rarely operates in isolation,” explains Mishuris. “Modern applications are dominated instead by coupled, multi-field systems whose interconnected variables must be expressed as complex matrix-valued functions.”
Why traditional matrix factorisation fails
In the late 1950s, mathematicians Israel Gohberg and Mark Krein proved that matrix factorisation exists, but their theory was non-constructive, providing no explicit calculation procedure. The theory relied on hidden structural invariants (partial indices). Under the Gohberg-Krein-Bojarski stability criterion, this factorisation is highly unstable under small perturbations unless these indices satisfy restrictive conditions. Given that real-world physical data inherently carry noise, this instability created computational barriers. Attempting blind factorisation without knowing the partial indices caused traditional algorithms to diverge, producing incorrect solutions. This prevented advancements in quantum systems, metamaterials and biomechanics, which depend on coupled equations that cannot be reduced to scalar forms.
New solutions for practical engineering uses
By delivering ExactMPF and resolving these stability flaws, EffectFact members leveraged their algorithms to solve real-world physical equations. In spectral factorisation, they introduced new numerical methods to handle singular and noisy matrix systems. Regarding wave propagation and diffraction, researchers deployed iterative Wiener-Hopf approaches to evaluate acoustic-wave scattering by lattice arrays and to trace crack propagation through discrete media. In metamaterials and microstructured media, they investigated gyro-elastic and discrete systems, uncovering novel mechanisms for directional wave control, edge resonances and wave localisation – essential for next-generation vibration isolation. In fracture mechanics, project work resolved multiscale problems including crack propagation, hydraulic fracturing, toughness homogenisation and material failure in heterogeneous media. This boosts understanding of how structural damage initiates. Ultimately, EffectFact advanced biomedical modelling by exploring joint mechanics, tissue interactions, cellular indentation and biocompatible scaffold structures. “Rather than producing isolated analytical data, we sought to establish practical, cross-disciplinary methodologies that researchers can directly apply to diverse scientific and industrial challenges,” concludes Mishuris.
