7,220 research outputs found
An integral method for solving nonlinear eigenvalue problems
We propose a numerical method for computing all eigenvalues (and the
corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that
lie within a given contour in the complex plane. The method uses complex
integrals of the resolvent operator, applied to at least column vectors,
where is the number of eigenvalues inside the contour. The theorem of
Keldysh is employed to show that the original nonlinear eigenvalue problem
reduces to a linear eigenvalue problem of dimension .
No initial approximations of eigenvalues and eigenvectors are needed. The
method is particularly suitable for moderately large eigenvalue problems where
is much smaller than the matrix dimension. We also give an extension of the
method to the case where is larger than the matrix dimension. The
quadrature errors caused by the trapezoid sum are discussed for the case of
analytic closed contours. Using well known techniques it is shown that the
error decays exponentially with an exponent given by the product of the number
of quadrature points and the minimal distance of the eigenvalues to the
contour
Transition probability of Brownian motion in the octant and its application to default modeling
We derive a semi-analytic formula for the transition probability of
three-dimensional Brownian motion in the positive octant with absorption at the
boundaries. Separation of variables in spherical coordinates leads to an
eigenvalue problem for the resulting boundary value problem in the two angular
components. The main theoretical result is a solution to the original problem
expressed as an expansion into special functions and an eigenvalue which has to
be chosen to allow a matching of the boundary condition. We discuss and test
several computational methods to solve a finite-dimensional approximation to
this nonlinear eigenvalue problem. Finally, we apply our results to the
computation of default probabilities and credit valuation adjustments in a
structural credit model with mutual liabilities
A spectral projection method for transmission eigenvalues
In this paper, we consider a nonlinear integral eigenvalue problem, which is
a reformulation of the transmission eigenvalue problem arising in the inverse
scattering theory. The boundary element method is employed for discretization,
which leads to a generalized matrix eigenvalue problem. We propose a novel
method based on the spectral projection. The method probes a given region on
the complex plane using contour integrals and decides if the region contains
eigenvalue(s) or not. It is particularly suitable to test if zero is an
eigenvalue of the generalized eigenvalue problem, which in turn implies that
the associated wavenumber is a transmission eigenvalue. Effectiveness and
efficiency of the new method are demonstrated by numerical examples.Comment: The paper has been accepted for publication in SCIENCE CHINA
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