20,773 research outputs found
New Error Bounds for Approximations from Projected Linear Equations
Joint Technical Report of U.H. and M.I.T.
Technical Report C-2008-43
Dept. Computer Science
University of Helsinki
and LIDS Report 2797
Dept. EECS
M.I.T.
July 2008; revised July 2009We consider linear fixed point equations and their approximations by projection on a low dimensional subspace. We derive new bounds on the approximation error of the solution, which are expressed in terms of low dimensional matrices and can be computed by simulation. When the fixed point mapping is a contraction, as is typically the case in Markov decision processes (MDP), one of our bounds is always sharper than the standard contraction-based bounds, and another one is often sharper. The former bound is also tight in a worst-case sense. Our bounds also apply to the non-contraction case, including policy evaluation in MDP with nonstandard projections that enhance exploration. There are no error bounds currently available for this case to our knowledge
Stability under Galerkin truncation of A-stable Runge--Kutta discretizations in time
We consider semilinear evolution equations for which the linear part is
normal and generates a strongly continuous semigroup and the nonlinear part is
sufficiently smooth on a scale of Hilbert spaces. We approximate their semiflow
by an implicit, A-stable Runge--Kutta discretization in time and a spectral
Galerkin truncation in space. We show regularity of the Galerkin-truncated
semiflow and its time-discretization on open sets of initial values with bounds
that are uniform in the spatial resolution and the initial value. We also prove
convergence of the space-time discretization without any condition that couples
the time step to the spatial resolution. Then we estimate the Galerkin
truncation error for the semiflow of the evolution equation, its Runge--Kutta
discretization, and their respective derivatives, showing how the order of the
Galerkin truncation error depends on the smoothness of the initial data. Our
results apply, in particular, to the semilinear wave equation and to the
nonlinear Schr\"odinger equation
Residual, restarting and Richardson iteration for the matrix exponential
A well-known problem in computing some matrix functions iteratively is a lack of a clear, commonly accepted residual notion. An important matrix function for which this is the case is the matrix exponential. Assume, the matrix exponential of a given matrix times a given vector has to be computed. We interpret the sought after vector as a value of a vector function satisfying the linear system of ordinary differential equations (ODE), whose coefficients form the given matrix. The residual is then defined with respect to the initial-value problem for this ODE system. The residual introduced in this way can be seen as a backward error. We show how the residual can efficiently be computed within several iterative methods for the matrix exponential. This completely resolves the question of reliable stopping criteria for these methods. Furthermore, we show that the residual concept can be used to construct new residual-based iterative methods. In particular, a variant of the Richardson method for the new residual appears to provide an efficient way to restart Krylov subspace methods for evaluating the matrix exponential.\u
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations
This work derives a residual-based a posteriori error estimator for reduced
models learned with non-intrusive model reduction from data of high-dimensional
systems governed by linear parabolic partial differential equations with
control inputs. It is shown that quantities that are necessary for the error
estimator can be either obtained exactly as the solutions of least-squares
problems in a non-intrusive way from data such as initial conditions, control
inputs, and high-dimensional solution trajectories or bounded in a
probabilistic sense. The computational procedure follows an offline/online
decomposition. In the offline (training) phase, the high-dimensional system is
judiciously solved in a black-box fashion to generate data and to set up the
error estimator. In the online phase, the estimator is used to bound the error
of the reduced-model predictions for new initial conditions and new control
inputs without recourse to the high-dimensional system. Numerical results
demonstrate the workflow of the proposed approach from data to reduced models
to certified predictions
POD model order reduction with space-adapted snapshots for incompressible flows
We consider model order reduction based on proper orthogonal decomposition
(POD) for unsteady incompressible Navier-Stokes problems, assuming that the
snapshots are given by spatially adapted finite element solutions. We propose
two approaches of deriving stable POD-Galerkin reduced-order models for this
context. In the first approach, the pressure term and the continuity equation
are eliminated by imposing a weak incompressibility constraint with respect to
a pressure reference space. In the second approach, we derive an inf-sup stable
velocity-pressure reduced-order model by enriching the velocity reduced space
with supremizers computed on a velocity reference space. For problems with
inhomogeneous Dirichlet conditions, we show how suitable lifting functions can
be obtained from standard adaptive finite element computations. We provide a
numerical comparison of the considered methods for a regularized lid-driven
cavity problem
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