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Efficient localised model reduction for multiscale PDEs via Grassmannian interpolation

Christian Alber, Markus Bachmayr, Robert Scheichl, Huqing Yang

arXiv:2610.08657v1math.NAcs.NA

TL;DR

Multiscale PDEs with heterogeneous coefficients and high-dimensional parameter spaces are costly to solve repeatedly. The paper interpolates parameter-dependent optimal local spaces on the Grassmann manifold using sparse grids within MS-GFEM. The resulting method has nearly exponential spatial and parametric convergence under the stated analysis.

  • Problem

    Strong coefficient heterogeneity requires fine spatial discretizations, while high-dimensional parameter domains require repeated solves, making standard methods prohibitively costly.

    Source: §1 · p. 1

  • Method

    The method interpolates parameter-dependent optimal local MS-GFEM spaces as Grassmannian points, using tangent-space sparse-grid interpolation and operator-harmonic extension of interpolated boundary trace spaces.

    Source: §1 · p. 2

  • Results

    The combined estimates yield nearly exponential convergence in spatial approximation and parametric interpolation error.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Choosing the MS-GFEM subdomain partition controls the dimensionality of the local interpolation problems.

    Source: §1 · p. 2

    The analysis assumes that each oversampling-domain coefficient depends on only a few parameters.

    Source: §2.3 · p. 4

Abstract

from arXiv · show

Multiscale, parameter-dependent partial differential equations (PDEs) pose severe computational challenges due to strong coefficient heterogeneity and high-dimensional parameter spaces. We develop a geometric interpolation approach within the multiscale generalized finite element method (MS-GFEM) that targets the most expensive component: computing parameter-dependent optimal local approximation spaces. Leveraging the spatial localization of MS-GFEM and assuming local parameter dependence, we decompose the global problem into parametrically low-dimensional local subproblems. The optimal subspaces for each parameter are identified as points on a Grassmann manifold and approximated via Grassmann interpolation on sparse grids, which preserves the geometric structure of these spaces while efficiently handling high-dimensional parameter spaces. The resulting localized model reduction method inherits the nearly exponential spatial convergence of MS-GFEM and the parametric convergence rates of sparse grids. Numerical experiments for elliptic problems confirm the theoretical convergence results.

1. Introduction

Strong coefficient heterogeneity and high-dimensional parameter spaces make repeated solves for multiscale PDEs prohibitively costly. The paper interpolates parameter-dependent optimal local spaces on the Grassmann manifold, combining MS-GFEM localization with sparse grids.

1. Introduction

  • MS-GFEM’s small oversampling regions leave local spaces dependent on few parameters, while greedy reduced spaces typically guarantee accuracy only on finite training sets.High-dimensional training sets can become prohibitively expensive, and the resulting reduced local spaces may be impractically large.Source: §1 · pp. 1–2
  • The method interpolates the map from parameters to optimal local spaces, treating each leading eigenspace as a Grassmannian point and mapping through the tangent space and back.MS-GFEM localization makes the local subproblems depend on only a few parameters.Source: §1 · p. 2
  • Because interpolation of parameter-dependent operator-harmonic spaces need not preserve that space, the analysis interpolates boundary trace spaces and then applies operator-harmonic extension.The resulting local approximation error is bounded using the geodesic distance between interpolated and optimal subspaces.Source: §1 · p. 2
  • Combining MS-GFEM spatial estimates with sparse-grid convergence yields nearly exponential convergence in both spatial approximation and parametric interpolation error.The local interpolation dimension can be controlled by choosing the MS-GFEM subdomain partition appropriately.Source: §1 · p. 2

2. Parametric regularity of the optimal local MS-GFEM spaces

This section establishes real-analytic parameter dependence of the optimal local MS-GFEM spaces as Grassmannian-valued maps, under a spectral gap assumption.

2.1. Problem setting.

  • The analysis focuses on second-order elliptic diffusion equations, and extensions to other PDEs are deferred pending suitable assumptions such as holomorphic parameter dependence.Source: §2.1 · p. 3

2.2. GFEM.

  • GFEM uses problem-adapted local ansatz spaces, and its global error is controlled by the maximum local error, enabling independent construction and interpolation on each subdomain.Source: §2.2 · p. 3

2.3. MS-GFEM for parametric problems.

  • Assuming coefficients on each oversampling domain depend on only d_loc parameters, MS-GFEM defines optimal local spaces from dominant eigenspaces and obtains nearly exponential local-error decay.The dominant eigenspaces minimize the Kolmogorov n-width of the local operator.Source: §2.3 · pp. 4–5

2.4. Holomorphy of the operator T and of the spectral projection.

  • The operator T extends holomorphically to a neighborhood of the real parameter domain, and its dominant rank-n spectral projection is holomorphic assuming a spectral gap.Source: §2.4 · pp. 7–8

2.5. The parameter-to-subspace map on the Grassmannian.

  • Viewing each optimal local space as a Grassmannian point, the parameter-to-subspace map is real analytic, providing the regularity required for interpolation.Source: §2 · p. 3, §2.5 · p. 8
  • The Stiefel and Grassmann manifolds, together with their Riemannian metrics, are real analytic.The cited geometric constructions and the orthogonal-group action have analytic analogues.Source: §2.5 · p. 8
  • On the connected real domain DR containing [−1, 1]^dloc, the spaces V∂_{n,R} and V_{n,R} are real analytic in the parameter.The construction fixes a scalar product on the boundary space before defining the Grassmannian metric.Source: §2.5 · p. 8
  • A local analytic frame for V∂_{n,R} can be obtained by applying the real-analytic spectral projection Πn to a fixed reference basis.The analogous argument for V_{n,R} uses the analyticity of E0_z in z.Source: §2.5 · pp. 8–9

3. Sparse grid Grassmannian interpolation of the optimal local spaces

This section develops and analyzes sparse-grid Grassmannian interpolation for parameter-dependent optimal local spaces, including tangent-space interpolation mapped back to the manifold. It establishes interpolation regularity and traces interpolation error through local and global approximation, analyzing two variants with different local-solve and error-bound robustness tradeoffs.

3. Sparse grid Grassmannian interpolation of the optimal local spaces

  • The method approximates expensive optimal-space maps by applying the Riemannian logarithm, interpolating in the tangent space, and mapping back with the exponential.The logarithm must be defined for the spaces across the parameter domain.Source: §3 · p. 9

3.1. Sparse grid interpolation.

  • Anisotropic Smolyak sparse grids reduce evaluations versus full tensor grids, allocating more nodes to slower-converging directions in locally anisotropic parameter problems.For local problems, the subexponential error regime requires refinement above a threshold of at least d_loc/log(2); the algebraic estimate may be more relevant in practice.Source: §3.1 · pp. 10–11

3.2. Grassmannian interpolation error.

  • The analysis extends the Grassmannian interpolation result of [41] to infinite-dimensional spaces and transfers sparse-grid error to interpolated-space and local approximation error.The tangent-space map is real analytic and admits a holomorphic extension to a complex parameter region.Source: §3 · p. 9, §3.2 · p. 13
  • The logarithmic Grassmannian representation is real analytic with a holomorphic extension, but the extension region is not quantitatively characterized because the Riemannian logarithm is only real analytic.The region may be smaller than the coefficient holomorphy region.Source: §3.2 · p. 14

3.3.1. Variant I.

  • Variant I bounds local approximation error through the (n + 1)-th eigenvalue and the angle between optimal and interpolated eigenspaces, improving using oblique-projection ideas [3].Source: §3.3.1 · p. 14

3.3. Local approximation error.

  • The global GFEM error bound combines interpolation errors with the nearly exponentially decaying local eigenvalues, assuming exact computation of local particular functions and A(z)-harmonic extensions.Source: §3.3.1 · p. 15

3.3.1. Variant I.

  • Variant I interpolates boundary eigenspaces then computes A(z)-harmonic extensions, whereas interpolating the original spaces may leave them outside the required A(z)-harmonic subspace.This variant therefore requires local problems after interpolation for each parameter.Source: §3.3.1 · p. 15

3.3.2. Variant II.

  • Variant II avoids local solves after interpolation but projects interpolated spaces onto A(z)-harmonic functions; its error bound requires interpolation error to decay rapidly, unlike Variant I’s fixed-error bound.The Variant II bound does not yield the corresponding optimal-eigenspace estimate for fixed bounded interpolation error.Source: §3.3.2 · pp. 17–19

4. Numerical experiments

Across the experiments, interpolated-space global energy errors decrease toward the MS-GFEM error, with Variant I reaching comparable accuracy using fewer interpolation points than Variant II. Local interpolation quality depends on the spectral gap at the chosen eigenfunction cutoff, motivating adaptive local dimensions.

4.1.1. Experiments for one parameter.

  • In the one-parameter cross-point problem, n = 7 has a uniform gap and all βk remain below 10^-5; n = 5 lacks a gap at z = 0 and has larger angles and global error.The figure compares the spectrum, β angles, and relative energy errors for interpolated and optimal spaces.Source: §4.1.1 · p. 20, Figure 1 · p. 21

4.1.2. Experiments for two parameters.

  • In the two-parameter cross-point problem, relative energy errors and Variant-II angles βH z and θz shift toward smaller values as interpolation points increase.The plotted statistics use a log10 transformation.Source: §4.1.2 · p. 20, Figure 2 · p. 21

4.2. Affine case with many parameters.

  • With eight global parameters, each local interpolation problem depends on at most two; the gap threshold τ = 0.08 is empirical, not theoretically derived.The authors leave a robust general selection rule for τ to future work.Source: §4.2 · p. 22
  • Although subdomains 7 and 13 each depend on two parameters, subdomain 13 has a uniform gap at the 15th cutoff while subdomain 7 does not.The experiments therefore use 15 nonconstant eigenfunctions for subdomain 13 and adaptively choose the dimension for subdomain 7.Source: §4.2 · p. 22, Figure 5 · p. 23, Figure 6 · p. 23
  • For subdomain 7, adaptive spectral-gap selection yields decaying β angles, whereas the non-adaptive cutoff bn = 15 shows no visible decay.The missing uniform gap at that cutoff is associated with poor interpolation quality.Source: §4.2 · pp. 22–23, Figure 7 · p. 24
  • With more interpolation points, global energy error decreases toward the MS-GFEM error, and Variant I reaches comparable accuracy with fewer points than Variant II.Because adaptive spaces retain at most n nonconstant eigenfunctions, their error may remain above the MS-GFEM error even with many interpolation points.Source: §4.2 · p. 23, Figure 8 · p. 24

5. Technical results

Section 5 establishes technical properties used in the paper’s analysis: completeness of a local function space, holomorphic extension of T, and geometric results for Grassmann manifolds. It also proves analyticity of the exponential and logarithm maps and states a triangle-comparison theorem for Hilbert manifolds.

5. Technical results

  • The local zero-mean boundary space is shown to be complete in the energy norm, with the proof using the extended Poincaré–Steklov inequality.The argument identifies the limit as the harmonic extension of its trace.Source: §5 · pp. 24–25

5.1. Holomorphy of T.

  • T is extended holomorphically by reformulating it with a fixed reference inner product, since its original parameter-dependent adjoints are unavailable for complex parameters.The construction uses holomorphic change-of-inner-product operators and agrees with the original definition for real parameters.Source: §5.1 · pp. 26–27

5.2. Principal angles under linear isomorphisms.

  • A proposition establishes bounds relating principal angles between subspaces to those between their images under a linear isomorphism.The supplied statement does not specify the bounds’ exact form.Source: §5.2 · p. 27

5.3. Analyticity of the exponential map.

  • The exponential map on the Grassmann manifold is proved real analytic, and the logarithm map is real analytic as well.The proof proceeds through analyticity of the geodesic vector field and its flow.Source: §5.3 · pp. 28–29

5.4. Toponogov’s theorem for Hilbert manifolds.

  • A Hilbert-manifold triangle-comparison theorem states that, under curvature bounds and the stated minimal-geodesic conditions, a comparison triangle exists in a constant-curvature surface.The comparison triangle has the same side lengths and angle inequalities as specified in the theorem.Source: §5.4 · p. 29
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