High-Dimensional Function Approximation and Reduced-Order Modeling

The field of high-dimensional function approximation and reduced-order modeling is witnessing significant advancements, driven by the need to efficiently process and analyze complex data. Researchers are developing innovative methods to sample high-dimensional functions, reducing computational complexity and enabling the application of these techniques to large-scale problems. Another key area of focus is the development of parallel algorithms for reduced-order modeling, which can scale to thousands of processors and facilitate the analysis of massive datasets. Furthermore, data-driven model order reduction techniques are being explored, which can preserve essential dynamic characteristics of complex systems while simplifying their representation. Noteworthy papers include:

  • The development of sampling formulas for high-dimensional functions in reproducing kernel Hilbert spaces, which can significantly reduce computational complexity.
  • The introduction of a novel framework for data-driven model order reduction of T-product-based dynamical systems, offering significant memory and computational savings.
  • The presentation of a rigorous mathematical framework for expressing high-order derivatives of functional tree tensor networks, enabling the construction of order conditions for Runge-Kutta methods.

Sources

Irregular Sampling of High-Dimensional Functions in Reproducing Kernel Hilbert Spaces

A parallel implementation of reduced-order modeling of large-scale systems

Data-driven model order reduction for T-Product-Based dynamical systems

Derivatives of tree tensor networks and its applications in Runge--Kutta methods

Quasitubal Tensor Algebra Over Separable Hilbert Spaces

Numerical Derivatives, Projection Coefficients, and Truncation Errors in Analytic Hilbert Space With Gaussian Measure

On Runge-Kutta methods of order 10

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