2,327 research outputs found
A Duality Approach to Error Estimation for Variational Inequalities
Motivated by problems in contact mechanics, we propose a duality approach for
computing approximations and associated a posteriori error bounds to solutions
of variational inequalities of the first kind. The proposed approach improves
upon existing methods introduced in the context of the reduced basis method in
two ways. First, it provides sharp a posteriori error bounds which mimic the
rate of convergence of the RB approximation. Second, it enables a full
offline-online computational decomposition in which the online cost is
completely independent of the dimension of the original (high-dimensional)
problem. Numerical results comparing the performance of the proposed and
existing approaches illustrate the superiority of the duality approach in cases
where the dimension of the full problem is high.Comment: 21 pages, 8 figure
Reduced basis isogeometric mortar approximations for eigenvalue problems in vibroacoustics
We simulate the vibration of a violin bridge in a multi-query context using
reduced basis techniques. The mathematical model is based on an eigenvalue
problem for the orthotropic linear elasticity equation. In addition to the nine
material parameters, a geometrical thickness parameter is considered. This
parameter enters as a 10th material parameter into the system by a mapping onto
a parameter independent reference domain. The detailed simulation is carried
out by isogeometric mortar methods. Weakly coupled patch-wise tensorial
structured isogeometric elements are of special interest for complex geometries
with piecewise smooth but curvilinear boundaries. To obtain locality in the
detailed system, we use the saddle point approach and do not apply static
condensation techniques. However within the reduced basis context, it is
natural to eliminate the Lagrange multiplier and formulate a reduced eigenvalue
problem for a symmetric positive definite matrix. The selection of the
snapshots is controlled by a multi-query greedy strategy taking into account an
error indicator allowing for multiple eigenvalues
Reduced basis methods for pricing options with the Black-Scholes and Heston model
In this paper, we present a reduced basis method for pricing European and
American options based on the Black-Scholes and Heston model. To tackle each
model numerically, we formulate the problem in terms of a time dependent
variational equality or inequality. We apply a suitable reduced basis approach
for both types of options. The characteristic ingredients used in the method
are a combined POD-Greedy and Angle-Greedy procedure for the construction of
the primal and dual reduced spaces. Analytically, we prove the reproduction
property of the reduced scheme and derive a posteriori error estimators.
Numerical examples are provided, illustrating the approximation quality and
convergence of our approach for the different option pricing models. Also, we
investigate the reliability and effectivity of the error estimators.Comment: 25 pages, 27 figure
Simultaneous Reduced Basis Approximation of Parameterized Elliptic Eigenvalue Problems
The focus is on a model reduction framework for parameterized elliptic
eigenvalue problems by a reduced basis method. In contrast to the standard
single output case, one is interested in approximating several outputs
simultaneously, namely a certain number of the smallest eigenvalues. For a fast
and reliable evaluation of these input-output relations, we analyze a
posteriori error estimators for eigenvalues. Moreover, we present different
greedy strategies and study systematically their performance. Special attention
needs to be paid to multiple eigenvalues whose appearance is
parameter-dependent. Our methods are of particular interest for applications in
vibro-acoustics
A new differentiation, shape of the unit ball and perimeter measure
We present a new blow-up method that allows for establishing the first
general formula to compute the perimeter measure with respect to the spherical
Hausdorff measure in noncommutative nilpotent groups. This result leads us to
an unexpected relationship between the area formula with respect to a distance
and the profile of its corresponding unit ball.Comment: 17 page
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