1,181 research outputs found
A Posteriori Error Analysis of Fluid-Stucture Interactions: Time Dependent Error
A posteriori error analysis is a technique to quantify the error in
particular simulations of a numerical approximation method. In this article, we
use such an approach to analyze how various error components propagate in
certain moving boundary problems. We study quasi-steady state simulations where
slowly moving boundaries remain in mechanical equilibrium with a surrounding
fluid. Such problems can be numerically approximated with the Method of
Regularized Stokelets(MRS), a popular method used for studying viscous
fluid-structure interactions, especially in biological applications. Our
approach to monitoring the regularization error of the MRS is novel, along with
the derivation of linearized adjoint equations to the governing equations of
the MRS with a elastic elements. Our main numerical results provide a clear
illustration of how the error evolves over time in several MRS simulations.Comment: 14 pages, 6 figure
A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations
We prove sharp, computable error estimates for the propagation of errors in
the numerical solution of ordinary differential equations. The new estimates
extend previous estimates of the influence of data errors and discretisation
errors with a new term accounting for the propagation of numerical round-off
errors, showing that the accumulated round-off error is inversely proportional
to the square root of the step size. As a consequence, the numeric precision
eventually sets the limit for the pointwise computability of accurate solutions
of any ODE. The theoretical results are supported by numerically computed
solutions and error estimates for the Lorenz system and the van der Pol
oscillator
Enhancing adaptive sparse grid approximations and improving refinement strategies using adjoint-based a posteriori error estimates
In this paper we present an algorithm for adaptive sparse grid approximations
of quantities of interest computed from discretized partial differential
equations. We use adjoint-based a posteriori error estimates of the physical
discretization error and the interpolation error in the sparse grid to enhance
the sparse grid approximation and to drive adaptivity of the sparse grid.
Utilizing these error estimates provides significantly more accurate functional
values for random samples of the sparse grid approximation. We also demonstrate
that alternative refinement strategies based upon a posteriori error estimates
can lead to further increases in accuracy in the approximation over traditional
hierarchical surplus based strategies. Throughout this paper we also provide
and test a framework for balancing the physical discretization error with the
stochastic interpolation error of the enhanced sparse grid approximation
Estimating Global Errors in Time Stepping
This study introduces new time-stepping strategies with built-in global error
estimators. The new methods propagate the defect along with the numerical
solution much like solving for the correction or Zadunaisky's procedure;
however, the proposed approach allows for overlapped internal computations and,
therefore, represents a generalization of the classical numerical schemes for
solving differential equations with global error estimation. The resulting
algorithms can be effectively represented as general linear methods. We present
a few explicit self-starting schemes akin to Runge-Kutta methods with global
error estimation and illustrate the theoretical considerations on several
examples
Automated goal-oriented error control I: stationary variational problems
This article presents a general and novel approach to the automation of
goal-oriented error control in the solution of nonlinear stationary finite
element variational problems. The approach is based on automated linearization
to obtain the linearized dual problem, automated derivation and evaluation of a
posteriori error estimates, and automated adaptive mesh refinement to control
the error in a given goal functional to within a given tolerance. Numerical
examples representing a variety of different discretizations of linear and
nonlinear partial differential equations are presented, including Poisson's
equation, a mixed formulation of linear elasticity, and the incompressible
Navier-Stokes equations.Comment: 21 page
Adaptive Finite Element Solution of Multiscale PDE-ODE Systems
We consider adaptive finite element methods for solving a multiscale system
consisting of a macroscale model comprising a system of reaction-diffusion
partial differential equations coupled to a microscale model comprising a
system of nonlinear ordinary differential equations. A motivating example is
modeling the electrical activity of the heart taking into account the chemistry
inside cells in the heart. Such multiscale models pose extremely
computationally challenging problems due to the multiple scales in time and
space that are involved.
We describe a mathematically consistent approach to couple the microscale and
macroscale models based on introducing an intermediate "coupling scale". Since
the ordinary differential equations are defined on a much finer spatial scale
than the finite element discretization for the partial differential equation,
we introduce a Monte Carlo approach to sampling the fine scale ordinary
differential equations. We derive goal-oriented a posteriori error estimates
for quantities of interest computed from the solution of the multiscale model
using adjoint problems and computable residuals. We distinguish the errors in
time and space for the partial differential equation and the ordinary
differential equations separately and include errors due to the transfer of the
solutions between the equations. The estimate also includes terms reflecting
the sampling of the microscale model. Based on the accurate error estimates, we
devise an adaptive solution method using a "blockwise" approach. The method and
estimates are illustrated using a realistic problem.Comment: 25 page
A scalable matrix-free spectral element approach for unsteady PDE constrained optimization using PETSc/TAO
We provide a new approach for the efficient matrix-free application of the
transpose of the Jacobian for the spectral element method for the adjoint based
solution of partial differential equation (PDE) constrained optimization. This
results in optimizations of nonlinear PDEs using explicit integrators where the
integration of the adjoint problem is not more expensive than the forward
simulation. Solving PDE constrained optimization problems entails combining
expertise from multiple areas, including simulation, computation of
derivatives, and optimization. The Portable, Extensible Toolkit for Scientific
computation (PETSc) together with its companion package, the Toolkit for
Advanced Optimization (TAO), is an integrated numerical software library that
contains an algorithmic/software stack for solving linear systems, nonlinear
systems, ordinary differential equations, differential algebraic equations, and
large-scale optimization problems and, as such, is an ideal tool for performing
PDE-constrained optimization. This paper describes an efficient approach in
which the software stack provided by PETSc/TAO can be used for large-scale
nonlinear time-dependent problems. While time integration can involve a range
of high-order methods, both implicit and explicit. The PDE-constrained
optimization algorithm used is gradient-based and seamlessly integrated with
the simulation of the physical problem
A posteriori error estimation in a finite element method for reconstruction of dielectric permittivity
We present a posteriori error estimates for finite element approximations in
a minimization approach to a coefficient inverse problem. The problem is that
of reconstructing the dielectric permittivity , , from
boundary measurements of the electric field. The electric field is related to
the permittivity via Maxwell's equations. The reconstruction procedure is based
on minimization of a Tikhonov functional where the permittivity, the electric
field and a Lagrangian multiplier function are approximated by peicewise
polynomials. Our main result is an estimate for the difference between the
computed coefficient and the true minimizer , in
terms of the computed functions.Comment: 17 pages, 1 figur
A-posteriori error estimates for inverse problems
Inverse problems use physical measurements along with a computational model
to estimate the parameters or state of a system of interest. Errors in
measurements and uncertainties in the computational model lead to inaccurate
estimates. This work develops a methodology to estimate the impact of different
errors on the variational solutions of inverse problems. The focus is on time
evolving systems described by ordinary differential equations, and on a
particular class of inverse problems, namely, data assimilation. The
computational algorithm uses first-order and second-order adjoint models. In a
deterministic setting the methodology provides a posteriori error estimates for
the inverse solution. In a probabilistic setting it provides an a posteriori
quantification of uncertainty in the inverse solution, given the uncertainties
in the model and data. Numerical experiments with the shallow water equations
in spherical coordinates illustrate the use of the proposed error estimation
machinery in both deterministic and probabilistic settings.Comment: Contains a total of 51 page
New bounding techniques for goal-oriented error estimation applied to linear problems
The paper deals with the accuracy of guaranteed error bounds on outputs of
interest computed from approximate methods such as the finite element method. A
considerable improvement is introduced for linear problems thanks to new
bounding techniques based on Saint-Venant's principle. The main breakthrough of
these optimized bounding techniques is the use of properties of homothetic
domains which enables to cleverly derive guaranteed and accurate boundings of
contributions to the global error estimate over a local region of the domain.
Performances of these techniques are illustrated through several numerical
experiments.Comment: 36 page
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