imaging and vision is difficult to overestimate. For instance, the impulse
response of an optical system can be defined in terms of diffraction
integrals, that are in turn Fourier transforms of a function on a disk.
There are several popular competing approaches used to calculate
diffraction integrals, such as the extended Nijboer-Zernike (ENZ) theory.
In this talk, an alternative efficient method of computation of two
dimensional Fourier-type integrals based on approximation of the integrand
by Gaussian radial basis functions is discussed. Its outcome is a rapidly
converging series expansion for the integrals, allowing for their accurate
calculation. The proposed method yields a reliable and fast scheme for
simultaneous evaluation of such kind of integrals for several values of the
defocus parameter, as required in the characterization of the through-focus
the need of explicit parameterization. A key component of this approach is a volume integral which is identical to the integral over the interface. I will show results applying this approach to simulate interfaces that evolve according to Mullins-Sekerka dynamics used in certain phase transition problems. I will also discuss our latest results in generalization of this approach to summation of unstructured point clouds and regularization of hyper-singular integrals.
from reactant to product states is a challenging problem due to the time
scale separation. In this talk, we will discuss some recent progress in
mathematical theory of transition paths. In particular, we identify and
characterize the stochastic process corresponds to transition paths. The
study of transition path process helps to understand the transition
mechanism and provides a framework to design and analyze numerical
approaches for rare event sampling and simulation.
radially symmetric solutions to the Swift--Hohenberg equation is
explored both numerically through continuation and analytically through
the use of geometric blow-up techniques. The bifurcation structure for
these solutions is elucidated by formally treating the dimension as a
continuous parameter in the equations. This reveals a family of
solutions with an anomalous amplitude scaling that is far larger than
expected from a formal scaling in the far field. One key advantage of
the geometric blow-up techniques is that a priori knowledge of this
scaling is unnecessary as it naturally emerges from the construction.
The stability of these patterned states will also be discussed.
Reference Moody T. Chu
, Nonnegative Inverse Eigenvalue and Singular Value Problems, SIAM J. Numer. Anal (1992). Wei Ma and Zheng-J. Bai, A regularized directional derivative-based Newton method for inverse singular value problems, Inverse Problems (2012).
First we study a nonlinear eigenvalue problem and apply Quasi-Newton methods to this.
In many cases they turn to behave better than the Pulay mixer, which widely used in physics community.
Second we reformulate the problem as a minimization problem on a Stiefel manifold.
One that formed from mxn matrices with orthonormal columns.
Then for Quasi-Newton techniques one needs to transfer the secant conditions to the new tangent space, when moving on the manifold. We also consider nonlinear conjugate gradients in this setting.
This minimization approach seems to work well especially for metals, which are known to be hard.
Third (if time permits) we add temperature (the first two are for ground state). This means that we need to include entropy in the energy and optimize also with respect to occupation numbers.
Joint work with Kurt Baarman and Ville Havu.
"Structure-preserving numerical integration of ordinary and partial
differential equations " and is aimed to present both classical and
more recent results regarding the numerical treatment of nonlinear
differential equations, both for deterministic and stochastic problems.
The perspective is that of introducing numerical methods which act as
structure-preserving integrators, with special emphasys to numerically
retaining dissipativity properties possessed by the problem.