Weighted balanced truncation method for approximating kernel functions by exponentials

TL;DR

This work introduces Weighted Balanced Truncation (WBT) as a general method to compress sum-of-exponentials representations of kernel functions by incorporating a weight function into the balanced truncation process. By formulating the SOE as a linear dynamical system and computing weighted Gramians, WBT produces a reduced $P$-term SOE that maintains accuracy over a target interval, with TLBT recovered as a special case. Numerical results on smooth Ewald-splitting and inverse-power/Coulomb kernels show substantial accuracy gains—often exceeding 4 digits—over classical model reduction, especially for long-range kernels, while maintaining uniform error distribution. The method is implemented in the VP-WBT software, enabling kernel approximation with customizable weights and providing a practical tool for fast convolution, molecular dynamics, and related applications in physics and engineering.

Abstract

Kernel approximation with exponentials is useful in many problems with convolution quadrature and particle interactions such as integral-differential equations, molecular dynamics and machine learning. This paper proposes a weighted balanced truncation to construct an optimal model reduction method for compressing the number of exponentials in the sum-of-exponentials approximation of kernel functions. This method shows great promise in approximating long-range kernels, achieving over 4 digits of accuracy improvement for the Ewald-splitting and inverse power kernels in comparison with the classical balanced truncation. Numerical results demonstrate its excellent performance and attractive features for practical applications.

Figures (6)

• Figure 1: Maximum errors of SOE approximations of the Ewald splitting kernel on $[0,10]$ with respect to the reduction term $P$, computed using both the classical MR and the WBT with different Ewald splitting parameter $\Lambda=10,50$ and $100$.
• Figure 2: (a) The maximum errors in the SOE approximation using the HSVD method with respect to $P$, where the marked points indicate the highest achievable accuracy. (b-d) The compression accuracy versus the number of terms for the highest accuracy case using the HSVD and WBT methods.
• Figure 3: The error distribution of the SOE approximation to the Ewald splitting kernel at $\Lambda = 100$, obtained by three different methods with $P=27$. The red dashed horizontal line represents the maximum error ($1.1\times 10^{-9}$) of the WBT with $w(r) = 1/\sqrt{r+10^{-4}}$.
• Figure 4: The error distribution of the WBT for the Coulomb kernel with different weight functions. The approximation interval is $[1,1024]$ with $P=15$. The red dashed horizontal line represents the maximum error ($3.0\times 10^{-8}$) of $w(r) = 1/\sqrt{r+10}$.
• Figure 5: Maximum errors of SOE approximations of power function kernel on $[1,1024]$ with respect to the reduction term $P$. (a) Coulomb kernel reduced by different model reduction techniques, (b) Singular power functions with different $\alpha$ reduced by the WBT. The green dashed line represents the precision of $1.0\times 10^{-15}$.