Reduced order modeling for elliptic problems with high contrast diffusion coefficients

The paper addresses reduced order modeling for parametrized elliptic PDEs where the diffusion coefficient is piecewise constant over a fixed partition and allowed to vary over arbitrarily large ranges, violating the standard Uniform Ellipticity Assumption (UEA). In many-query forward simulations and in inverse problems (state and parameter estimation), one seeks low-dimensional spaces that approximate the solution manifold efficiently. Classical approaches (polynomial expansions and reduced basis methods) achieve sub-exponential rates under UEA due to analytic regularity of the parameter-to-solution map. However, for high-contrast settings where the parameter domain is Y=(0,∞)^d with piecewise constant coefficients on subdomains, solutions exhibit homogeneity u(ty)=t^{-1}u(y) and the solution manifold is unbounded, making uniform absolute error bounds and finite n-widths impossible. The core research question is how to build reduced spaces that yield uniform, contrast-independent approximation guarantees in relative error across all parameters, including regimes with very high contrast, and to extend such guarantees to Galerkin projections and inverse problems. The study also explores the role of limit (stiff-inclusion) solutions as some parameters tend to infinity and their incorporation into reduced spaces to achieve robustness.

Prior work on reduced-order modeling of parametrized PDEs under UEA shows sub-exponential convergence rates for polynomial approximations and reduced basis (RB) methods, with greedy algorithms providing near-optimal spaces and sometimes outperforming polynomial approaches (e.g., [5–7,10–12,17–19,21,28,29,32–35]). Analytic regularity under affine parameter dependence leads to power series truncations with rigorous error estimates (e.g., [6,32]). For high-dimensional settings, anisotropic assumptions can sustain convergence even for large or infinite parameter sets ([11,17,18]). High-contrast elliptic problems have been extensively studied for robust preconditioning and a posteriori error estimation ([2–4,9,20]), but reduced-order methods with contrast-robust guarantees had not been established. Library-width concepts and localized approaches (e.g., local RB or hp strategies) have been investigated ([23–26,31]). The paper builds on homogenization insights for stiff/soft inclusions ([22]) and functional-analytic tools such as Stein-type extensions for Lipschitz domains ([1,30,36]).

- Problem setting: Consider −div(a(y)∇u(y))=f on a domain Ω⊂R^m with homogeneous Dirichlet boundary conditions. The diffusion is piecewise constant: a(y)|_{Ω_j}=y_j with parameters y∈Y=(0,∞]^d. Solutions are in H^1_0(Ω) and satisfy the homogeneity u(ty)=t^{-1}u(y). - Extended manifold and normalization: Define limit spaces V_S={v∈H^1_0(Ω): ∇v|_{Ω_j}=0 for j∈S} and, for fixed finite y_{S^c}, define stiff-inclusion limit solutions u_S(y_{S^c})∈V_S as limits of u(y) when y_j→∞ for j∈S. Extend the parameter set to Y=(0,∞]^d and the manifold accordingly, setting u(∞,…,∞)=0. Restrict attention to the coercive regime B={u(y): y∈[1,∞]^d} and to the normalized subset N={u(y): y∈[1,∞]^d, min_j y_j=1}. - Compactness and framing: Prove B (hence N) is compact in H^1_0(Ω) via convergence to limit solutions as some parameters go to infinity, and establish two-sided bounds c_f ≤ ||u(y)||_{H^1_0} ≤ C_f on N, where c_f:=min_j ||f||_{H^{-1}(Ω_j)}>0 under mild data conditions. - Piecewise polynomial construction in parameter space: Partition Y via dyadic rectangles R_ℓ=∏_j [2^{ℓ_j}, 2^{ℓ_j+1}] for ℓ∈{0,…,L}^d, with faces touching infinity when ℓ_j=L. For each rectangle, build local polynomial approximations of total degree k to the parameter-to-solution map. • Inner rectangles (all ℓ_j<L): Center at ȳ and use a Neumann-series-based expansion of (I+B(ỹ))^{-1} where B(ỹ)=∑_j ỹ_j A^{-1}_{ȳ}A_j, yielding truncations with errors O(3^{-k}) in both the parameter-weighted energy norm and H^1_0. • Infinite rectangles (some ℓ_j=L): Approximate u_S(y_{S^c}) polynomially in the active variables y_{S^c}, and bound ||u(y)−u_S(y_{S^c})||_{H^1_0} by geometric arguments. Under a Lipschitz partition assumption (all unions of subdomains have Lipschitz boundaries), construct extension operators to prove a quantitative estimate ||u−u_S||_{H^1_0} ≤ C_0 C_f max_{j∈S} y_j^{-1}, giving O(2^{-L}) on rectangles with y_j≥2^L. - Balance k and L: Choose L=L_k≈αk+γ (with α=ln3/ln2) so that polynomial truncation error O(3^{-k}) and stiff-limit error O(2^{-L}) balance. - Global reduced space: For the normalized manifold N, form V_n as the sum of local spaces V_{ℓ,k} over E_k={0,…,L_k}^d\{1,…,L_k}^d. Use dimension bounds dim(V_{ℓ,k})≲(k+d−1 choose d−1) and cardinality estimates of E_k to relate n to k, yielding n≲C (k+1)^{2d−2}. - Galerkin projection: Show that for contrast-robust relative error control with Galerkin projection P^y_{V_n}, V_n must include functions from limit spaces V_S. The constructed V_{ℓ,k} for infinite rectangles are subspaces of V_S, ensuring robustness. For geometries of disjoint inclusions, derive the same sub-exponential rates for Galerkin errors as for orthogonal projections. - Inverse problems: • State estimation via PBDW: With measurement functionals in H^{-1}(Ω), apply the PBDW framework using V_n, resulting in recovery errors controlled by µ_n times the best-approximation error, hence inheriting the sub-exponential rate. • Parameter estimation (inverse diffusivity): Given a PBDW estimator v* = ∑_{i=1}^n c_i u_i, exploit −Δu|_{Ω_j}=f/y_j and define an estimator for inverse diffusivity by (1/y^*)_j := ∑_{i=1}^n c_i (1/y^{(i)}_j). Prove a relative error bound for 1/y^* in ℓ_∞ proportional to the state recovery error.

- Uniform relative-error approximation without UEA: There exist linear spaces V_n such that for all y∈(0,∞]^d, ||u(y)−P_{V_n}u(y)||_{H^1_0} ≤ ε_n ||u(y)||_{H^1_0}, with ε_n→0 independently of contrast (Theorem 2.8). - Sub-exponential rates via piecewise polynomials: Under a Lipschitz partition and choosing L≈αk, the global space V_n built from local degree-k polynomials on dyadic rectangles achieves ||u−P_{V_n}u||_{H^1_0} ≤ C exp(−c n^{1/(2d−2)}) on the normalized manifold N, hence a similar relative-error bound on the full manifold M by homogeneity (Theorem 3.7). The Kolmogorov n-width d_n(N) obeys the same sub-exponential rate. - Quantitative stiff-limit convergence: For any S and y, ||u(y)−u_S(y_{S^c})||_{H^1_0} ≤ C_0 C_f max_{j∈S} y_j^{-1}, giving O(2^{-L}) on infinite rectangles (Lemma 3.5). This bound hinges on extension operators for Lipschitz unions and reveals a geometry-dependent constant C_0, which can be large in general but improves for disjoint inclusions. - Galerkin robustness requires limit-space content: If V_n∩V_S={0} for some S, there exist parameters with Galerkin relative-error bounded away from zero (Proposition 4.1). The constructed spaces include V_S components, and for disjoint inclusions yield Galerkin error bounds matching the projection rates: ||u−P^y_{V_n}u||_{H^1_0} ≤ C exp(−c n^{1/(2d−2)}) ||u||_{H^1_0} (Theorem 4.2). - State estimation (PBDW): Using V_n, both the background approximation v*∈V_n and the PBDW estimator u* satisfy max{||u−v*||_{H^1_0}, ||u−u*||_{H^1_0}} ≤ C µ_n exp(−c n^{1/(2d−2)}) ||u||_{H^1_0} (Proposition 4.4). - Parameter estimation (inverse diffusivity): With the estimator (1/y^*)_j=∑_i c_i (1/y^{(i)}_j), one obtains ||1/y^* − 1/y||_∞ ≤ (C_f/c_f) C µ_n exp(−c n^{1/(2d−2)}) ||1/y||_∞ (Proposition 4.5). - Numerical evidence: Greedy parameter selection (especially Greedy Galerkin) captures limit solutions and achieves rapid (often near-exponential) decay of relative Galerkin errors; increasing parameter dimension degrades the decay rate (curse of dimensionality). Even for non-Lipschitz partitions, numerics show exponential-like decay, suggesting geometric assumptions may be conservative.

The work directly addresses the challenge of constructing reduced models for elliptic PDEs with arbitrarily high contrast, where standard UEA-based theory does not apply and uniform absolute-error bounds fail due to homogeneity and unboundedness of the solution manifold. By extending the manifold to include stiff-limit solutions u_S and leveraging compactness on a normalized subset, the authors reframe the target to uniform relative-error bounds that are contrast-independent. The piecewise polynomial strategy across dyadic parameter rectangles, together with quantitative control of convergence to limit solutions (via extension operators under Lipschitz partitions), leads to sub-exponential approximation rates in the reduced dimension. The inclusion of V_S components is shown to be essential for Galerkin robustness, and the same rates are proven for Galerkin projections in disjoint-inclusion geometries. The framework further extends to inverse problems: PBDW-based state estimation inherits the approximation rates, and a tailored estimator for inverse diffusivity yields relative error bounds, tying parameter recovery accuracy to the state approximation quality. The results parallel classical UEA settings in rate type but reveal explicit dependence on partition geometry and dimension, confirming the curse of dimensionality. Numerics corroborate theory, emphasize the importance of greedy selection and inclusion of limit solutions, and suggest that geometric assumptions may be relaxable in practice.

The paper develops a contrast-robust reduced-order modeling framework for elliptic PDEs with piecewise constant, arbitrarily large diffusion parameters. Key contributions include: (i) extension to stiff-limit solutions and compactness on a normalized manifold enabling uniform relative-error approximation; (ii) a piecewise polynomial construction over dyadic parameter partitions achieving sub-exponential decay of errors with reduced dimension; (iii) rigorous Galerkin error bounds with the same rates when reduced spaces incorporate limit-space functions (notably for disjoint inclusions); and (iv) extensions to state (PBDW) and parameter (inverse diffusivity) estimation with contrast-robust recovery guarantees. The approach reveals the necessity of limit components for Galerkin robustness and quantifies geometry dependence through extension operators. Future research directions include removing or weakening geometric assumptions (beyond Lipschitz partitions and disjoint inclusions), tightening constants (e.g., mitigating C_0 and dimensional blow-up), exploring optimal reduced basis selection that may further improve rates, and extending the methodology to more general coefficients or boundary conditions.

- Geometric assumptions: Key quantitative estimates (e.g., ||u−u_S|| bounds) require a Lipschitz partition, with sharp Galerkin results proved for disjoint inclusions; extending theory to general partitions is open. - Dimensional dependence: The sub-exponential rate exp(−c n^{1/(2d−2)}) deteriorates with increasing d (curse of dimensionality); constants (notably C_0) may grow rapidly with d. - Data condition: The lower framing constant c_f>0 requires ||f||_{H^{-1}(Ω_j)}>0 for at least one subdomain; otherwise, lower bounds fail and relative-error framing may break. - Parameter estimation: The provided bound is for inverse diffusivity and may be unsatisfactory when some y_j≈1; a uniform bound for |y^*_j−y_j| is not established. - Implementation aspects: Construction uses piecewise polynomials with separate spaces per rectangle; although a global space is formed, practical deployment may benefit from localized or library approaches and careful basis conditioning. - Numerical constants: Error bounds contain geometry-dependent constants and may be pessimistic compared to observed performance; optimality of rates for RB spaces in high-contrast settings is conjectured but not fully analyzed.