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A Tree-Based Localized Reduced Basis Method for Parametrized Parabolic PDEs

Mohamed Barakat, Diane Guignard

arXiv:2609.36141v1math.NAcs.NA

TL;DR

Global reduced spaces can struggle with nonlinear solution features, and time-dependent problems must capture both parameter and temporal variation. The paper develops a tree-localized POD-Greedy method with a certified estimator; its equivalence to approximation error supports convergence analysis and mode selection.

  • Problem

    A single global linear reduced space may need many basis functions for solution manifolds with strong nonlinear features, while time-dependent problems require capturing parameter dependence and temporal evolution.

    Source: §1 · pp. 1–2

  • Method

    The method adaptively partitions parameter space with a hierarchical tree and constructs local reduced spaces using POD-Greedy, with an inf-sup-based certified error estimator.

    Source: §1 · pp. 2–3, §3.3 · p. 14

  • Results

    466 (N=16) versus 1641 (N=9) library entries on DII, with maximum RB dimensions of 45 and 50, respectively; the estimator is equivalent to approximation error and supports convergence analysis.

    Source: §1 · p. 3, §5 · p. 26

  • Takeaways & Limitations

    The POD-mode selection strategy is designed to maintain optimal convergence without unnecessarily increasing local reduced-space dimensions.

    Source: §1 · p. 3

    The convergence analysis assumes affine data extend holomorphically to an open complex neighborhood where the bilinear form remains uniformly continuous and coercive.

    Source: §4 · p. 15

Abstract

from arXiv · show

We propose a localized reduced basis method for parametrized parabolic partial differential equations based on an adaptive partitioning of the parameter domain using hierarchical binary trees. A POD-Greedy procedure is employed to construct reduced spaces associated with each parameter subdomain. A certified \emph{a posteriori} error estimator is derived from an inf-sup stability analysis and shown to be equivalent to the underlying approximation error. This equivalence, together with suitable regularity assumptions on the parameter-to-solution map, enables a rigorous convergence analysis and is used to derive a strategy for selecting the number of POD modes to retain in each local reduced space. Numerical experiments for a parametrized convection--diffusion problem demonstrate the effectiveness of the proposed approach and confirm the predicted convergence behavior.

1 Introduction

The introduction motivates localized reduced spaces for parabolic problems whose temporal and parameter dependence challenge global approximation. It presents a tree-based extension for parabolic PDEs, with an error estimator and convergence analysis guiding local-space construction.

1 Introduction

  • For time-dependent problems, reduced spaces must capture both parameter dependence and temporal evolution, while direct space-time greedy sampling may stagnate.POD-Greedy compresses temporal trajectories for a selected parameter before choosing the next parameter using a certified error estimator.Source: §1 · pp. 1–2
  • Earlier parabolic residual estimators provide rigorous upper bounds for stepwise L2 error but lack an explicit discrete-time-derivative contribution and a corresponding efficiency lower bound.The introduction contrasts this limitation with the proposed estimator’s equivalence to approximation error.Source: §1 · pp. 2–3
  • A global linear reduced space can require many basis functions when solution manifolds have strong nonlinear features, such as moving structures, limiting online efficiency.This limitation is associated with slowly decaying Kolmogorov N-widths in settings including transport-dominated flows and wave propagation.Source: §1 · p. 2
  • The proposed method extends tree-based localized reduced bases from elliptic to parabolic PDEs, combining parameter-subdomain spaces with POD-Greedy construction and a criterion for retaining POD modes.The number of modes retained affects each local basis dimension in addition to the number of greedily selected parameter samples.Source: §1 · pp. 2–3
  • The estimator is derived from inf-sup stability and shown equivalent to approximation error; with suitable parameter-map smoothness, this supports rigorous convergence analysis and POD-mode selection.The analysis characterizes the number of parameter subdomains needed for prescribed accuracy while avoiding unnecessary increases in local-space dimension.Source: §1 · p. 3

2 Problem statement

The problem statement formulates a parameterized parabolic evolution equation and its fully discrete approximation under stability and data-structure assumptions. Preliminary continuity, inf-sup, and energy estimates establish stability needed for subsequent reduced-basis analysis.

2 Problem statement

  • The model seeks a parameter-dependent parabolic solution with an initial condition, control function, uniformly continuous and coercive bilinear form, and uniformly continuous linear functional.The solution is placed in L2(0,T;V) ∩ H1(0,T;V′), with V densely embedded in H.Source: §2 · p. 3
  • The fully discrete problem uses backward Euler time stepping and a conforming finite-element space for spatial discretization.Time is divided into M equal subintervals of length ∆t = T/M.Source: §2 · p. 3
  • Affine parameter representations of the bilinear form, forcing, and initial condition support offline–online decomposition; Empirical Interpolation can approximate unavailable affine decompositions.The affine components are parameter-independent, while their coefficient functions depend on the parameter.Source: §2 · p. 4

2.1 Preliminary results

  • Continuity and a discrete inf-sup condition yield well-posedness and support quasi-optimal reduced-basis error analysis, while a discrete energy estimate is established.The analysis also derives a parameter-uniform estimate using uniform stability bounds.Source: §2.1 · pp. 5, 7

3 The Reduced Basis Method

The reduced basis method combines POD-Greedy sampling with stability and error-estimation tools to construct local reduced spaces over an adaptive binary-tree partition. The analysis establishes quasi-optimality and a residual-based estimator, while the localized algorithm refines subdomains until their estimated errors meet a prescribed tolerance.

3 The Reduced Basis Method

  • The reduced solution is quasi-optimal, and equivalence of the dual norms supports a uniform stability estimate in the high-fidelity norm.The quasi-optimality estimate bounds the reduced-basis error by the best approximation from the reduced space, up to a constant factor.Source: §3 · pp. 9–10

3.1 POD/Greedy sampling

  • The POD-Greedy procedure selects parameters by maximizing an a posteriori error estimator and adds dominant POD modes from the resulting trajectory errors.The mode count is chosen using POD eigenvalues and the subdomain’s longest-edge length.Source: §3.1 · p. 11, §3.3 · p. 14

3.2 Error estimator

  • A residual-based a posteriori estimator uses the reduced-basis residual dual norm and a computable lower bound for the discrete inf–sup constant.The paper identifies the Successive Constraint Method as an offline–online procedure for constructing and evaluating that lower bound.Source: §3.2 · pp. 12–13

3.3 Localized RB method

  • Local reduced spaces are built on an adaptive binary-tree partition, recursively bisected when their maximum training-set error estimate exceeds the prescribed tolerance.For dimensions greater than one, the method splits along the longest edge to maintain a balanced partition.Source: §3.3 · p. 14

4 Convergence analysis

The convergence analysis uses holomorphic parameter dependence and stability to bound local approximation errors, then establishes conditional convergence guarantees for the tree-partitioned method. Its practical guidance separates greedy-space error from POD trajectory-compression error and adapts POD retention and tolerances accordingly.

4 Convergence analysis

  • Holomorphic extensions of the problem data and uniform coercivity support bounds on Taylor coefficients and local polynomial approximation of each subdomain’s solution manifold.The analysis then relates this approximation to the subdomain’s Kolmogorov N-width.Source: §4 · pp. 15, 17, 18

4.1 Convergence of the localized RB method

  • Theorem 4.1 bounds the number of generated subdomains K(ε) with a constant independent of ε, assuming quasi-uniform partitioning and enough POD modes to satisfy the stated error estimate.The partition must also satisfy the paper’s parameter-regularity assumption.Source: §4 · p. 15, §4.1 · p. 19

4.2 Practical takeaways

  • The approximation error has greedy-space and POD-compression contributions; when compression error is dominated by greedy error, the estimator has asymptotic decay OThe supplied passage does not state the remainder of the decay rate.Source: §4.2 · p. 19
  • Controlling POD compression at the greedy-selected parameter samples suffices to control it over the full parameter domain, and its error is of the same asymptotic order as the greedy approximation error.The number of retained modes is selected to meet the POD truncation condition at each iteration.Source: §4.2 · p. 21
  • Using each subdomain’s local side length for the POD tolerance avoids unnecessary global refinement, while Copt trades online basis size against offline cost and storage.An excessively large Copt can delay the asymptotic regime; the quasi-uniformity condition used in the theorem is not enforced in practice.Source: §4.1 · p. 19, §4.2 · pp. 21–22

5 Numerical results

Experiments on a parametrized time-dependent convection–diffusion problem test the predicted convergence, compare online costs, and examine how Copt affects offline and online performance. The results show convergence consistent with theory and trade-offs between reduced online cost, offline effort, and storage.

5 Numerical results

  • For D = DI, observed convergence rates align with Theorem 4.1 for N = 1, 2, and 3.Figure 1 plots the number of subdomains against tolerance.Source: §5 · p. 23, Figure 1 · p. 23
  • For D = DI, N = 2 achieves the tolerance with smaller maximum reduced-basis spaces than N = 3, at the cost of a larger library and storage requirements.Both nonlinear models improve online cost over the linear reference, which reaches tolerance with a reduced-basis space of size 53.Source: Figure 1 · p. 23, §5 · p. 24, Figure 2 · p. 24
  • For D = DII, observed convergence rates are consistent with Theorem 4.1 for N = 4, 9, and 16.Figure 3 plots the number of subdomains against tolerance.Source: §5 · p. 24, Figure 3 · p. 25
  • For N = 9, the nonlinear model reaches the tolerance with a reduced-basis dimension approximately one quarter that of the linear model.The linear reference reaches tolerance at dimension 170.Source: §5 · p. 24, Figure 4 · p. 25
  • ζ = 50 requires a finer partition and higher offline cost than ζ = 5, but improves the online stage while achieving the prescribed tolerance.The paper attributes the online improvement to a smaller maximum reduced-basis dimension.Source: §5 · p. 26, Figure 5 · p. 26
  • With N = 9 and ζ = 5, the maximum reduced-basis dimension is 50 and the library has 1641 entries; with N = 16 and ζ = 50, they are 45 and 466, respectively.The online costs are comparable, while the second configuration has a more efficient offline stage and substantially lower storage requirements.Source: §5 · p. 26
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