Approximating partial differential equations without boundary conditions

TL;DR

This work addresses the challenge of numerically solving elliptic PDEs when boundary conditions are unavailable by leveraging measurements of the solution and a model class constraining boundary traces in $H^s(\Gamma)$ with $s>\tfrac12$. The authors develop a near-optimal recovery framework in which the target solution is represented via Riesz representers on the boundary, and propose a fractional-diffusion-based approach to compute these representers without directly forming nonlocal $H^s(\Gamma)$ inner products. They present both a theoretical saddle-point algorithm and a practical sinc-quadrature FE method to approximate the Riesz representers, along with rigorous error analyses showing exponential decay in the sinc-quadrature parameter and algebraic FE-discretization errors, culminating in a near-optimal recovery bound $\|u-u_{k,h}\|_{H^1(\Omega)} \le R(K^s_{\boldsymbol\omega}) + O\big(e^{-{\pi^2}/{k}} + h^q\big)$. Numerical experiments on an L-shaped domain verify the theory, showing recovery improves with mesh refinement, higher boundary regularity, and more measurements, while highlighting practical aspects of Riesz representer approximation. The approach enables practical, boundary-condition–free PDE recovery with near-optimal guarantees, applicable to inverse-type problems where boundary data are incomplete but measurements are available.

Abstract

We consider the problem of numerically approximating the solutions to an elliptic partial differential equation (PDE) for which the boundary conditions are lacking. To alleviate this missing information, we assume to be given measurement functionals of the solution. In this context, a near optimal recovery algorithm based on the approximation of the Riesz representers of these functionals in some intermediate Hilbert spaces is proposed and analyzed in [Binev et al. 2024]. Inherent to this algorithm is the computation of $H^s$, $s>1/2$, inner products on the boundary of the computational domain. We take advantage of techniques borrowed from the analysis of fractional diffusion problems to design and analyze a fully practical near optimal algorithm not relying on the challenging computation of $H^s$ inner products.

Figures (5)

• Figure 1: Recovery of the smooth solution $u_1$ in \ref{['eqn:smooth']} with $m=33$ Gaussian measurements. Left: $s=0.55$ ($e_1=0.17372$), middle: $s=1.45$ ($e_1=0.06029$), right: $u_1$.
• Figure 2: Recovery of the non-smooth solution $u_2$ in \ref{['eqn:nonsmooth']} with $m=33$ Gaussian measurements. Left: $s=0.55$ ($e_2=0.12761$), middle: $s=7/6$ ($e_2=0.05098$), right: $u_2$.
• Figure 3: Recovery of the smooth solution $u_1$ in \ref{['eqn:smooth']} with $s=1$ and $m$ Gaussian measurements. Left: $m=3$ ($e_1=0.69813$), middle: $m=12$ ($e_1=0.26451$), right: $m=33$ ($e_1=0.08176$).
• Figure 4: Recovery of the non-smooth solution $u_2$ in \ref{['eqn:nonsmooth']} with $s=1$ and $m$ Gaussian measurements. Left: $m=3$ ($e_2=0.40164$), middle: $m=12$ ($e_2=0.11890$), right: $m=33$ ($e_2=0.05865$).
• Figure 5: Convergence of the Riesz representer for $n=2,...,6$. The $H^1(\Omega)$ error is computed using a reference discrete solution ($n=9$). Left: Gaussian measurement centered at $z=(-0.5,0.5)$; right: Gaussian measurement centered at $z=(-0.1,0.1)$.

Theorems & Definitions (23)

• lemma 1
• proof
• theorem 1
• proof
• corollary 1
• proof
• lemma 2
• proof
• corollary 2
• proof