Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Games (1503.03467v5)

Abstract: We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough ($L^\infty$) coefficients with rigorous a-priori accuracy and performance estimates. The method is discovered through a decision/game theory formulation of the problems of (1) identifying restriction and interpolation operators (2) recovering a signal from incomplete measurements based on norm constraints on its image under a linear operator (3) gambling on the value of the solution of the PDE based on a hierarchy of nested measurements of its solution or source term. The resulting elementary gambles form a hierarchy of (deterministic) basis functions of $H^1_0(\Omega)$ (gamblets) that (1) are orthogonal across subscales/subbands with respect to the scalar product induced by the energy norm of the PDE (2) enable sparse compression of the solution space in $H^1_0(\Omega)$ (3) induce an orthogonal multiresolution operator decomposition. The operating diagram of the multigrid method is that of an inverted pyramid in which gamblets are computed locally (by virtue of their exponential decay), hierarchically (from fine to coarse scales) and the PDE is decomposed into a hierarchy of independent linear systems with uniformly bounded condition numbers. The resulting algorithm is parallelizable both in space (via localization) and in bandwith/subscale (subscales can be computed independently from each other). Although the method is deterministic it has a natural Bayesian interpretation under the measure of probability emerging (as a mixed strategy) from the information game formulation and multiresolution approximations form a martingale with respect to the filtration induced by the hierarchy of nested measurements.

• The paper introduces novel orthogonal basis functions called "gamblets," derived from framing multigrid operator discovery as a hierarchical information game.
• This multiresolution decomposition enables efficient, near-linear complexity computation of solutions for PDEs with rough coefficients through localized computation and parallelization.
• The framework exhibits remarkable robustness against high contrasts and rough coefficients, offering a powerful method for practical applications where these issues are prevalent.

Multigrid with Rough Coefficients and Multiresolution Operator Decomposition

The paper presents a comprehensive and theoretically grounded framework for solving partial differential equations (PDEs) with rough ($L^\infty$) coefficients using a multigrid/multiresolution approach. The method combines elements of game theory, signal processing, and uncertainty quantification to develop a new algorithm capable of handling challenges associated with irregular PDE coefficients. The core innovation lies in reformulating the problem of discovering effective multigrid operators as a hierarchical information game, leading to the derivation of orthogonal basis functions termed "gamblets."

Key Contributions

1. Gamblets and Hierarchical Information Gaming: The method introduces a novel perspective by treating the identification of restriction and interpolation operators as a game-theoretic challenge. The resulting gamblets are deterministic basis functions characterized by:
   • Orthogonality across subscales with respect to the energy norm scalar product.
   • Sparse approximation of the solution space in $H^1_0(\Omega)$.
   • Formation of an orthogonal multiresolution decomposition of operators.
2. Multiresolution Orthogonal Decomposition: The paper constructs an inverted pyramid framework, where gamblets are computed locally and hierarchically from fine to coarse scales. This allows the eigenvalue problem to be decomposed into smaller, independent subproblems with uniformly bounded condition numbers, enabling efficient numerical solvability.
3. Bayesian Interpretation: Although the algorithm is deterministic, it possesses a natural Bayesian interpretation. The multiresolution approximations act as a martingale with respect to a hierarchy of nested measurements, which underpins the theoretical foundation of the method.
4. Complexity and Parallelizability: The proposed algorithm achieves near-linear complexity due to its ability to leverage both spatial and subscale parallelization. This contrasts with classical approaches where high coefficient variability often significantly affects convergence and performance.
5. Practical Robustness: The gamblet framework exhibits remarkable robustness against the degradation of performance caused by high contrasts and rough coefficients, making it suitable for practical applications where these issues are prevalent.
6. Localized Computation: The exponential decay of gamblets permits localized computation without a significant loss of accuracy. This property is essential for reducing computational costs while preserving the fidelity of solutions, notably circumventing the need for global eigendecomposition.

Implications and Future Directions

The implications of this research are profound for computational mathematics and engineering, especially in fields where solving PDEs with rough coefficients is imperative. The development of game-theoretically inspired computational frameworks opens avenues for:

• Hybrid Methods: Integrating uncertainty quantification with classical deterministic algorithms to enhance robustness and performance in high-contrast environments.
• Adaptive Techniques: Further exploration of adaptive refinement strategies could improve both efficiency and adaptability in dynamically varying scenarios.
• AI and Machine Learning: As these fields increasingly incorporate PDEs into modeling tasks, the multiresolution techniques presented here may contribute to improving the accuracy and generalizability of predictive models.

Future explorations might extend this method to more generalized PDEs, enhance adaptive capabilities, and refine parallelization schemes to handle expansive, complex simulations efficiently. Additionally, research into integrating machine learning frameworks could leverage these hierarchical decompositions for advanced data-driven estimation and inference tasks.