Finite Element Approximation of Data-Driven Problems in Conductivity

Abstract: This paper is concerned with the finite element discretization of the data driven approach according to arXiv:1510.04232 for the solution of PDEs with a material law arising from measurement data. To simplify the setting, we focus on a scalar diffusion problem instead of a problem in elasticity. It is proven that the data convergence analysis from arXiv:1708.02880 carries over to the finite element discretization as long as $H(\mathrm{div})$-conforming finite elements such as the Raviart-Thomas element are used. As a corollary, minimizers of the discretized problems converge in data in the sense of arXiv:1708.02880, as the mesh size tends to zero and the approximation of the local material data set gets more and more accurate. We moreover present several heuristics for the solution of the discretized data driven problems, which is equivalent to a quadratic semi-assignment problem and therefore NP-hard. We test these heuristics by means of two examples and it turns out that the "classical" alternating projection method according to arXiv:1510.04232 is superior w.r.t. the ratio of accuracy and computational time.

• The paper introduces an H(div)-conforming finite element discretization that ensures variational convergence for data-driven conductivity problems.
• It rigorously establishes error bounds and convergence rates through mesh refinement and densification of measured data.
• The study analyzes NP-hard quadratic semi-assignment problems using projection-based and heuristic algorithms to achieve computational tractability.

Finite Element Approximation of Data-Driven Problems in Conductivity

Introduction and Problem Formulation

The study addresses the finite element discretization of data-driven variational problems in elliptic conductivity, building on the direct data-driven approach originally proposed by Kirchdoerfer and Ortiz. In the classical paradigm, material models are constructed from empirical data, inducing modeling errors due to simplifications and noisy measurements. The data-driven approach circumvents the explicit construction of constitutive models by leveraging the measured data set directly, formulating the solution as a best-fit problem between a physical equilibrium constraint and the measured material responses.

For stationary conductivity, the governing PDE is $-\operatorname{div} \kappa(\nabla u) = f$, and the material law is expressed only via a finite measurement set $$\mathcal{D}^{\mathrm{loc}} \subset \mathbb{R}^d \times \mathbb{R}^d$$. The data-driven variational problem is then framed as a minimization of the distance in $L^2$ between the equilibrium set and the (non-convex, discrete) set of admissible material states derived from data, resulting in a problem of the form: $$\min_{(y,z)} \|y-z\|^2 \quad \text{s.t.}\ y \in \mathcal{D},\ z \in \mathcal{E}$$ where $\mathcal{E}$ is the equilibrium set and $\mathcal{D}$ encodes the availability of the discrete material data.

Discretization Strategy and $H(\operatorname{div})$-Conformity

The main technical focus is the numerical treatment of this mixed-integer, possibly ill-posed variational problem. The authors show that when discretizing the equilibrium set using $H(\operatorname{div})$-conforming finite elements, in particular Raviart-Thomas spaces, the properties essential for the variational convergence of the data-driven approach persist at the discrete level. The analysis includes:

• Discretization of $\mathcal{E}$ using Raviart-Thomas spaces $Q_h \subset H(\operatorname{div})$ for fluxes and conforming $U_h\subset H^1_0$ for potentials, satisfying the inf-sup (LBB) stability condition and providing adequate sequence approximation properties.
• Proofs that any sequence of discrete minimizers for the discrete problem converges (in a data-driven topology) to minimizers of the continuous data-driven problem as the mesh is refined and the material data is densified.
• Rigorous qualification of the necessary regularity and approximation properties of these finite element spaces, including explicit interpolation operators and best approximation estimates.

The necessity of $H(\operatorname{div})$-conformity is underlined by demonstrating the failure of the data-topology convergence when standard non-conforming or $L^2$-based spaces are used. The analysis provides concrete examples in which nonconforming discretization leads to a fundamental inconsistency: sequences of approximations fail to converge in the data topology unless conformity is enforced.

Data-Topology and Convergence Analysis

The concept of "data convergence" is central. It formalizes convergence in a topology intermediate between the weak and strong topologies on $Z \times Z$, so that weak convergence of each component plus strong convergence of the difference suffices. The authors rigorously adapt the "data convergence" framework (cf. [CMO2018]) to the context incorporating a discretized equilibrium set. The main results are:

• Well-posedness and existence: For the discretized problem, existence (but not uniqueness) of discrete minimizers is obtained under only mild compactness of sets of discrete admissible data.
• Variational convergence: Under mesh refinement and improved accuracy of measurement data (quantified via decay of the Hausdorff distance of the data sets), minimizers of the discretized problem converge (up to subsequences) to minimizers of the continuous limit problem in the data topology.
• Approximation of the material data set: The construction and analysis incorporate piecewise-constant approximations of the material data set and quantify the requisite approximation properties (fine and uniform approximability with explicit error bounds).
• The data closure of the solution set is characterized, generalizing the closure results for nonlinear (and possibly nonconvex) material data.

Optimization and Algorithmic Aspects

At the discrete level, the data-driven problem reduces, due to the discrete nature of $\mathcal{D}_k$, to a quadratic semi-assignment problem (QSAP), known to be NP-hard. The paper presents and analyzes several approaches:

• Projection/alternating minimization heuristics: The widely used alternating projection algorithm is understood as a proximal gradient-type method, involving projections onto the equilibrium and data sets, with theoretical discussion of its convergence limitations in the nonconvex (data-driven) setting.
• Douglas-Rachford and proximal-gradient variants: The method is extended by inclusion of adjustable step-sizes, and also with Douglas-Rachford splitting, with performance compared numerically.
• Local search with model order reduction: A heuristic exploiting the QSAP structure—a local search restricted to K-nearest neighbors in the measured data, with fast approximation of projections via proper orthogonal decomposition (POD) reduced models—is proposed as a tractable means of seeking improved discrete minimizers.

Numerical experiments reveal that while the alternating projection algorithm retains a favorable performance/accuracy profile, all methods are constrained by computational bottlenecks inherent to the size of the assignment problem when the data set and mesh become large.

Numerical Results

Numerical validation is provided for both linear (Fourier-type) and strongly nonlinear monotone (arctangent-based) material laws, with a range of mesh sizes, data densities, and noise levels. Key observations include:

• Under mesh refinement and data set densification (and in the vanishing noise limit), the error of the minimizers with respect to classical finite element solutions (using the true law) converges linearly or quadratically (in appropriate norms), as evidenced by computed experimental orders of convergence.
• The performance of the projection algorithm, in terms of the ratio of solution accuracy to computational time, is superior to that of the more involved local search or other projection-based methods.
• When using $H(\operatorname{div})$-conforming elements and uniform data approximation, the observed accuracy in both $L^2$- and $H^1_0$-based errors is comparable to direct finite element solutions based on known constitutive laws, provided the data set densely samples the local material behavior.

Implications and Outlook

Practically, the results provide theoretical justification for the use of $H(\operatorname{div})$-conforming finite elements in data-driven computation for PDEs in conductivity (scalar diffusion). The framework ensures the convergence of the finite element/data approximation, elevating the data-driven paradigm from a heuristic to a mathematically justified methodology—at least in the scalar, monotonic setting with dense data and sufficiently regular solutions. The generalization to vectorial (elasticity) problems is acknowledged as a major open challenge due to the complexity of constructing suitable conforming elements.

Theoretically, the results extend the scope of Γ-convergence and data-topology methods to discrete variational models subject to nonconvex, finite material data. The insight that the discrete assignment problem prevents efficient global minimization (NP-hardness) frames the practical need for more robust heuristics or relaxation methods with explicit error control.

Open problems include:

• Extension to vectorial problems in elasticity, where conforming discretization is far less trivial.
• Design of heuristics or randomized algorithms for large-scale QSAPs with quantifiable suboptimality bounds.
• Rigorous analysis of convergence properties of projection-based iterative methods in the nonconvex, data-driven setting.
• Incorporation of noisy and inconsistent data, including potential learning-based regularizations, with guaranteed convergence.

Conclusion

The paper delivers a comprehensive and rigorous framework for the finite element approximation of data-driven problems in scalar conductivity, unifying discretization, variational convergence, and algorithmic tractability. The necessity of $H(\operatorname{div})$-conforming elements is unambiguously demonstrated. While efficient exact solution of the discrete problem is out of computational reach for high-dimensional data, projection-based heuristics, particularly the alternating projection method, offer competitive accuracy, verified in nontrivial numerical settings. The methodology establishes a robust baseline for ongoing development of data-driven computational mechanics, with multiple open avenues for theoretical and practical enhancement.