127 research outputs found
Parallel Controllability Methods For the Helmholtz Equation
The Helmholtz equation is notoriously difficult to solve with standard
numerical methods, increasingly so, in fact, at higher frequencies.
Controllability methods instead transform the problem back to the time-domain,
where they seek the time-harmonic solution of the corresponding time-dependent
wave equation. Two different approaches are considered here based either on the
first or second-order formulation of the wave equation. Both are extended to
general boundary-value problems governed by the Helmholtz equation and lead to
robust and inherently parallel algorithms. Numerical results illustrate the
accuracy, convergence and strong scalability of controllability methods for the
solution of high frequency Helmholtz equations with up to a billion unknowns on
massively parallel architectures
Solving forward and inverse Helmholtz equations via controllability methods
Waves are useful for probing an unknown medium by illuminating it with a source.
To infer the characteristics of the medium from (boundary) measurements,
for instance, one typically formulates inverse scattering problems
in frequency domain as a PDE-constrained optimization problem.
Finding the medium, where the simulated wave field
matches the measured (real) wave field, the inverse problem
requires the repeated solutions of forward (Helmholtz) problems.
Typically, standard numerical methods, e.g. direct solvers or iterative methods,
are used to solve the forward problem.
However, large-scaled (or high-frequent) scattering problems
are known being competitive in computation and storage for standard methods.
Moreover, since the optimization problem is severely ill-posed
and has a large number of
local minima, the inverse problem requires additional regularization
akin to minimizing the total variation.
Finding a suitable regularization for the inverse problem is critical
to tackle the ill-posedness and to reduce the computational cost and storage requirement.
In my thesis, we first apply standard methods to forward problems.
Then, we consider the controllability method (CM)
for solving the forward problem: it
instead reformulates the problem in the time domain
and seeks the time-harmonic solution of the corresponding wave equation.
By iteratively reducing the mismatch between the solution at
initial time and after one period with the conjugate gradient (CG) method,
the CMCG method greatly speeds up the convergence to the time-harmonic
asymptotic limit. Moreover, each conjugate gradient iteration
solely relies on standard numerical algorithms,
which are inherently parallel and robust against higher frequencies.
Based on the original CM, introduced in 1994 by Bristeau et al.,
for sound-soft scattering problems, we extend the CMCG method to
general boundary-value problems governed by the Helmholtz equation.
Numerical results not only show the usefulness, robustness, and efficiency
of the CMCG method for solving the forward problem,
but also demonstrate remarkably accurate solutions.
Second, we formulate the PDE-constrained optimization
problem governed by the inverse scattering problem
to reconstruct the unknown medium.
Instead of a grid-based discrete representation combined with
standard Tikhonov-type regularization, the unknown medium is
projected to a small finite-dimensional subspace,
which is iteratively adapted using dynamic thresholding.
The adaptive (spectral) space is governed by solving
several Poisson-type eigenvalue problems.
To tackle the ill-posedness that the Newton-type optimization
method converges to a false local minimum,
we combine the adaptive spectral inversion (ASI) method with the frequency stepping strategy.
Numerical examples illustrate the usefulness of the ASI approach,
which not only efficiently and remarkably reduces the dimension of the
solution space, but also yields an accurate and robust method
Parallel Controllability Methods For the Helmholtz Equation
The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the time-harmonic solution of the corresponding time-dependent wave equation. Two different approaches are considered here based either on the first or second-order formulation of the wave equation. Both are extended to general boundary-value problems governed by the Helmholtz equation and lead to robust and inherently parallel algorithms. Numerical results illustrate the accuracy, convergence and strong scalability of controllability methods for the solution of high frequency Helmholtz equations with up to a billion unknowns on massively parallel architectures
Parallel Controllability Methods For the Helmholtz Equation
The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the time-harmonic solution of the corresponding time-dependent wave equation. Two different approaches are considered here based either on the first or second-order formulation of the wave equation. Both are extended to general boundary-value problems governed by the Helmholtz equation and lead to robust and inherently parallel algorithms. Numerical results illustrate the accuracy, convergence and strong scalability of controllability methods for the solution of high frequency Helmholtz equations with up to a billion unknowns on massively parallel architectures
An adaptive finite element method for high-frequency scattering problems with variable coefficients
We introduce a new method for the numerical approximation of time-harmonic
acoustic scattering problems stemming from material inhomogeneities. The method
works for any frequency , but is especially efficient for
high-frequency problems. It is based on a time-domain approach and consists of
three steps: \emph{i)} computation of a suitable incoming plane wavelet with
compact support in the propagation direction; \emph{ii)} solving a scattering
problem in the time domain for the incoming plane wavelet; \emph{iii)}
reconstruction of the time-harmonic solution from the time-domain solution via
a Fourier transform in time. An essential ingredient of the new method is a
front-tracking mesh adaptation algorithm for solving the problem in \emph{ii)}.
By exploiting the limited support of the wave front, this allows us to make the
number of the required degrees of freedom to reach a given accuracy
significantly less dependent on the frequency , as shown in the
numerical experiments. We also present a new algorithm for computing the
Fourier transform in \emph{iii)} that exploits the reduced number of degrees of
freedom corresponding to the adapted meshes
- …