Phil-Alexander Hofmann

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 realise a variational formulation, solving a vast class of linear and non-linear partial differential equations (PDEs). The PSMs are as flexible as the physics-informed neural nets (PINNs) and provide an alternative for addressing inverse PDE problems, such as PDE-parameter inference. In contrast to PINNs, the PSMs result in a convex optimisation problem for a vast class of PDEs, including all linear ones, in which case the PSM-approximate is efficiently computable due to the exponential convergence rate of the underlying variational gradient descent. As a practical consequence prominent PDE problems were resolved by the PSMs without High Performance Computing (HPC) on a local machine. This gain in efficiency is complemented by an increase of approximation power, outperforming PINN alternatives in both accuracy and runtime. Beyond the empirical evidence we give here, the translation of classic PDE theory in terms of the Sobolev space approximates suggests the PSMs to be universally applicable to well-posed, regular forward and inverse PDE problems.

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 Newton coefficients directly, without relying on embeddings into enclosing tensor-product spaces. For a downward closed index set $A \subset \mathbb N_0^m$, the FNT achieves a time complexity of $\mathcal O(m \overline n |A|)$, where $\overline n$ is the mean of the coordinate-wise maximal polynomial degrees $n_1, \ldots, n_m$ across the $m$ spatial dimensions. In the univariate case, the FNT renders the classic Newton divided difference scheme (DDS). In the multivariate case, however, it improves on the quadratic complexity $\mathcal O(|A|^2)$ of the DDS and on the cost $\mathcal{O}(m \overline n (n_1+1) \cdots (n_m+1))$ of the tensorial interpolation. For sufficiently regular functions, Newton interpolation in Euclidean-degree downward closed polynomial spaces is known to deliver approximation rates equal to those of the tensor product interpolation. Thus, the acceleration power of the FNT comes from requiring substantially fewer degrees of freedom while reaching the same approximation quality as the tensorial interpolation. The inverse transformation has the same time complexity and enables fast evaluation and differentiation of Newton interpolants in quasi-tensorial grids.