What question did this study set out to answer?

The aim is to construct orthogonal bases on distorted geometries using a VMP-induced inner product.

May 4, 2026Open Access

VMP Product and the Numerical Exploration of New Basis Functions

Key Points

The aim is to construct orthogonal bases on distorted geometries using a VMP-induced inner product.
Introduced a VMP-induced inner product for orthogonalization.
Applied Gram-Schmidt process to enhance Gram matrix conditioning and reduce mode coupling.
Used a homotopy to continuously recover classical L2 orthogonal systems.
Improved orthogonal basis formations demonstrated through reproducible numerical examples.
Achieved nearly block-diagonal structure post-orthogonalization.
Demonstrated effectiveness in various distorted, non-orthogonal geometries.

Abstract

We introduce a VMP-induced inner product (built on a matrix-twisted cross-product view) and use it with Gram–Schmidt to construct orthogonal bases on distorted, non-orthogonal geometries. The approach improves Gram matrix conditioning, reduces mode coupling, and yields nearly block-diagonal structure after orthogonalization. A simple homotopy lets these VMP-adapted bases continuously recover the classical L2 orthogonal systems (e.g., Legendre/Fourier). We illustrate the method with reproducible numerical examples.

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Cite This Study

Tamás Bódis (Thu,) studied this question.

synapsesocial.com/papers/69f837423ed186a739981561 https://doi.org/https://doi.org/10.5281/zenodo.19987699

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