Volumetric heat Kernel: Padé-Chebyshev approximation, convergence, and computation

• Discretization and spectrum-free computation of the volumetric heat kernel.
• Simple scale selection as compromise between approximation accuracy and smoothness.
• Higher approximation accuracy with respect to previous work.
• Convergence results as the polynomial degree increases.
• Volumetric heat kernel is independent of evaluation the Laplacian spectrum.

This paper proposes an accurate and computationally efficient solver of the heat equation (∂t+Δ)F(·,t)=0(∂t+Δ)F(·,t)=0, F(·,0)=fF(·,0)=f, on a volumetric domain, through the (r,r  )-degree Padé-Chebyshev rational approximation of the exponential representation F(·,t)=exp(−tΔ)fF(·,t)=exp(−tΔ)f of the solution. To this end, the heat diffusion problem is converted to a set of r   differential equations, which involve only the Laplace–Beltrami operator, and whose solution converges to F(·,t)F(·,t), as r→+∞r→+∞. The discrete heat equation is equivalent to r sparse, symmetric linear systems and is independent of the volume discretization as a tetrahedral mesh or a regular grid, the evaluation of the Laplacian spectrum, and the selection of a subset of eigenpairs. Our approach has a super-linear computational cost, is free of user-defined parameters, and has an approximation accuracy lower than 10−r. Finally, we propose a simple criterion to select the time value that provides the best compromise between approximation accuracy and smoothness of the solution.

Figure optionsDownload high-quality image (528 K)Download as PowerPoint slide