164 research outputs found
Efficient algorithms for solving the p-Laplacian in polynomial time
The -Laplacian is a nonlinear partial differential equation, parametrized
by . We provide new numerical algorithms, based on the
barrier method, for solving the -Laplacian numerically in Newton iterations for all , where is the number of
grid points. We confirm our estimates with numerical experiments.Comment: 28 pages, 3 figure
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs
Riemannian Acceleration with Preconditioning for symmetric eigenvalue problems
In this paper, we propose a Riemannian Acceleration with Preconditioning
(RAP) for symmetric eigenvalue problems, which is one of the most important
geodesically convex optimization problem on Riemannian manifold, and obtain the
acceleration. Firstly, the preconditioning for symmetric eigenvalue problems
from the Riemannian manifold viewpoint is discussed. In order to obtain the
local geodesic convexity, we develop the leading angle to measure the quality
of the preconditioner for symmetric eigenvalue problems. A new Riemannian
acceleration, called Locally Optimal Riemannian Accelerated Gradient (LORAG)
method, is proposed to overcome the local geodesic convexity for symmetric
eigenvalue problems. With similar techniques for RAGD and analysis of local
convex optimization in Euclidean space, we analyze the convergence of LORAG.
Incorporating the local geodesic convexity of symmetric eigenvalue problems
under preconditioning with the LORAG, we propose the Riemannian Acceleration
with Preconditioning (RAP) and prove its acceleration. Additionally, when the
Schwarz preconditioner, especially the overlapping or non-overlapping domain
decomposition method, is applied for elliptic eigenvalue problems, we also
obtain the rate of convergence as , where is a constant
independent of the mesh sizes and the eigenvalue gap,
, is
the parameter from the stable decomposition, and
are the smallest two eigenvalues of the elliptic operator. Numerical results
show the power of Riemannian acceleration and preconditioning.Comment: Due to the limit in abstract of arXiv, the abstract here is shorter
than in PD
High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities
Solutions to many important partial differential equations satisfy bounds
constraints, but approximations computed by finite element or finite difference
methods typically fail to respect the same conditions. Chang and Nakshatrala
enforce such bounds in finite element methods through the solution of
variational inequalities rather than linear variational problems. Here, we
provide a theoretical justification for this method, including higher-order
discretizations. We prove an abstract best approximation result for the linear
variational inequality and estimates showing that bounds-constrained
polynomials provide comparable approximation power to standard spaces. For any
unconstrained approximation to a function, there exists a constrained
approximation which is comparable in the norm. In practice, one
cannot efficiently represent and manipulate the entire family of
bounds-constrained polynomials, but applying bounds constraints to the
coefficients of a polynomial in the Bernstein basis guarantees those
constraints on the polynomial. Although our theoretical results do not
guaruntee high accuracy for this subset of bounds-constrained polynomials,
numerical results indicate optimal orders of accuracy for smooth solutions and
sharp resolution of features in convection-diffusion problems, all subject to
bounds constraints
Quantum gravity: unification of principles and interactions, and promises of spectral geometry
Quantum gravity was born as that branch of modern theoretical physics that
tries to unify its guiding principles, i.e., quantum mechanics and general
relativity. Nowadays it is providing new insight into the unification of all
fundamental interactions, while giving rise to new developments in modern
mathematics. It is however unclear whether it will ever become a falsifiable
physical theory, since it deals with Planck-scale physics. Reviewing a wide
range of spectral geometry from index theory to spectral triples, we hope to
dismiss the general opinion that the mere mathematical complexity of the
unification programme will obstruct that programme.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
The numerical solution of partial differential equations on high-dimensional
domains gives rise to computationally challenging linear systems. When using
standard discretization techniques, the size of the linear system grows
exponentially with the number of dimensions, making the use of classic
iterative solvers infeasible. During the last few years, low-rank tensor
approaches have been developed that allow to mitigate this curse of
dimensionality by exploiting the underlying structure of the linear operator.
In this work, we focus on tensors represented in the Tucker and tensor train
formats. We propose two preconditioned gradient methods on the corresponding
low-rank tensor manifolds: A Riemannian version of the preconditioned
Richardson method as well as an approximate Newton scheme based on the
Riemannian Hessian. For the latter, considerable attention is given to the
efficient solution of the resulting Newton equation. In numerical experiments,
we compare the efficiency of our Riemannian algorithms with other established
tensor-based approaches such as a truncated preconditioned Richardson method
and the alternating linear scheme. The results show that our approximate
Riemannian Newton scheme is significantly faster in cases when the application
of the linear operator is expensive.Comment: 24 pages, 8 figure
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