Fast numerical solution of parabolic integrodifferential equations with applications in finance
We numerically solve parabolic problems u_t+\cAu=0 in (0,T)*Ω, T<∞, where Ω⊂R is a
bounded interval and \cA is a strongly elliptic integrodifferential operator of order ρ∈0,2. A
discontinuous Galerkin (dG) discretization in time and a wavelet discretization in space are
used. The densely populated matrices in the corresponding linear systems of equations are
replaced by sparse ones using appropriate wavelet compression techniques. The linear
systems in each time step are solved by an incomplete GMRES iteration. Under these …
bounded interval and \cA is a strongly elliptic integrodifferential operator of order ρ∈0,2. A
discontinuous Galerkin (dG) discretization in time and a wavelet discretization in space are
used. The densely populated matrices in the corresponding linear systems of equations are
replaced by sparse ones using appropriate wavelet compression techniques. The linear
systems in each time step are solved by an incomplete GMRES iteration. Under these …
We numerically solve parabolic problems $u_t+\cA u=0$ in , , where is a bounded interval and $\cA$ is a strongly elliptic integrodifferential operator of order . A discontinuous Galerkin (dG) discretization in time and a wavelet discretization in space are used. The densely populated matrices in the corresponding linear systems of equations are replaced by sparse ones using appropriate wavelet compression techniques. The linear systems in each time step are solved by an incomplete GMRES iteration. Under these conditions, we show that the complexity of our algorithm is linear (up to some logarithmic terms) in the number of spatial degrees of freedom and present error estimates. Applications to purely discontinuous Lévy processes arising in finance are given.
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