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Analysis of the Ill-Conditioning of the Discrete Inverse Laplace transform in Monte Carlo Simulations of Quantum Many-Body Systems
Phil-Alexander Hofmann, Thomas Chuna, Tobias Dornheim +1
Analytic continuation of imaginary-time quantum Monte Carlo data to real-frequency spectra requires the inversion of a severely ill-posed two-sided Laplace transform and arises nat…
Mitigating Numerical Stiffness in Least-Squares Formulations of Elliptic PDEs for Physics-Informed Neural Networks
Phil-Alexander Hofmann, Michael Hecht
We present theoretical insights into residual loss formulations of physics-informed neural networks (PINNs) for learning solutions of partial differential equations (PDEs)…
Interpolation in Polynomial Spaces of p-Degree
Phil-Alexander Hofmann, Damar Wicaksono, Michael Hecht
We recently introduced the Fast Newton Transform (FNT), an hierarchical algorithm for performing multivariate Newton interpolation in arbitrary downward closed polynomial spaces of…
Accelerating Multivariate Newton Interpolation in Downward Closed Polynomial Spaces
Phil-Alexander Hofmann, Michael Hecht
We introduce the fast Newton transform (FNT), a multivariate Newton interpolation algorithm for downward closed polynomial spaces in quasi-tensorial grids. The FNT computes the New…
Multivariate Newton Interpolation in Downward Closed Spaces Reaches the Optimal Geometric Approximation Rates for Bos--Levenberg--Trefethen Functions
Michael Hecht, Phil-Alexander Hofmann, Damar Wicaksono +5
We extend the univariate Newton interpolation algorithm to arbitrary spatial dimensions and for any choice of downward-closed polynomial space, while preserving its quadratic runti…
Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
Juan-Esteban Suarez Cardona, Phil-Alexander Hofmann, Michael Hecht
We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials. Based on this polynomial surrogate model (PSM), we real…