7 research outputs found
Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints
The proximal Galerkin finite element method is a high-order, low iteration
complexity, nonlinear numerical method that preserves the geometric and
algebraic structure of bound constraints in infinite-dimensional function
spaces. This paper introduces the proximal Galerkin method and applies it to
solve free boundary problems, enforce discrete maximum principles, and develop
scalable, mesh-independent algorithms for optimal design. The paper leads to a
derivation of the latent variable proximal point (LVPP) algorithm: an
unconditionally stable alternative to the interior point method. LVPP is an
infinite-dimensional optimization algorithm that may be viewed as having an
adaptive barrier function that is updated with a new informative prior at each
(outer loop) optimization iteration. One of the main benefits of this algorithm
is witnessed when analyzing the classical obstacle problem. Therein, we find
that the original variational inequality can be replaced by a sequence of
semilinear partial differential equations (PDEs) that are readily discretized
and solved with, e.g., high-order finite elements. Throughout this work, we
arrive at several unexpected contributions that may be of independent interest.
These include (1) a semilinear PDE we refer to as the entropic Poisson
equation; (2) an algebraic/geometric connection between high-order
positivity-preserving discretizations and certain infinite-dimensional Lie
groups; and (3) a gradient-based, bound-preserving algorithm for two-field
density-based topology optimization. The complete latent variable proximal
Galerkin methodology combines ideas from nonlinear programming, functional
analysis, tropical algebra, and differential geometry and can potentially lead
to new synergies among these areas as well as within variational and numerical
analysis
Snapshot-Based Methods and Algorithms
An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science
Model Order Reduction
An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science
Model Order Reduction
An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science