∂tC + 1 2 σS∂SSC + rS∂SC − rC = 0, C(S, T ) = (S −K) (1) where r is the interest rate and σ the volatility. While in the original model σ is constant, it is common practice to calibrate (adjust) σ to allow (??) to reproduce market observations Ck at S0, t0, namely known calls with maturities Tk and strikes Kk, k=1..K. The calibration problem (also called the “smile”) is quite stiff. Avellaneda ...

In this paper we introduce the Localized Reduced Basis Multiscale (LRBMS) method for parameter dependent heterogeneous elliptic multiscale problems. The LRBMS method brings together ideas from both Reduced Basis methods to efficiently solve parametrized problems and from multiscale methods in order to deal with complex heterogeneities and large domains. Experiments on 2D and real world 3D data ...

We use asymptotically optimal adaptive numerical methods (here specifically a wavelet scheme) for snapshot computations within the offline phase of the Reduced Basis Method (RBM). The resulting discretizations for each snapshot (i.e., parameter-dependent) do not permit the standard RB ‘truth space’, but allow for error estimation of the RB approximation with respect to the exact solution of the...

In [3] it was shown that by writing the solution of the Black-Scholes partial differential equation on a small set of basis functions the computing time can be dramatically reduced. In this study we explore the generalization of the technique to basket options.

in this paper, countable compactness and the lindel¨of propertyare defined for l-fuzzy sets, where l is a complete de morgan algebra. theydon’t rely on the structure of the basis lattice l and no distributivity is requiredin l. a fuzzy compact l-set is countably compact and has the lindel¨ofproperty. an l-set having the lindel¨of property is countably compact if andonly if it is fuzzy compact. ...

Lattice-based reformulation techniques have been used successfully both theoretically and computationally. One such reformulation is obtained from the lattice kerZ(A) = {x ∈ Z | Ax = 0}. Some of the hard instances in the literature that have been successfully tackled by lattice-based techniques, such as market split and certain classes of knapsack instances, have randomly generated input A. The...

This article presents a reduced order method for simulation and control of uid ows. The major advantage of this method over others such as nite element, nite diierence or spectral method is that it has fewer degrees of freedom. The present methodology's feasibility for ow control is demonstrated on two boundary control problems. The rst one is a velocity tracking problem in cavity ow and the se...

In this paper we introduce a two-step Certified Reduced Basis (RB) method. In the first step we construct from an expensive finite element “truth” discretization of dimension N an intermediate RB model of dimension N N . In the second step we construct from this intermediate RB model a derived RB (DRB) model of dimension M ≤ N . The construction of the DRB model is effected at cost O(N) and in ...

A (multi-)wavelet expansion is used to derive a rigorous bound for the (dual) norm Reduced Basis residual. We show theoretically and numerically that the error estimator is online efficient, reliable and rigorous. It allows to control the exact error (not only with respect to a “truth” discretization).