62,474 research outputs found
On Adaptive Eulerian-Lagrangian Method for Linear Convection-Diffusion Problems
In this paper, we consider the adaptive Eulerian--Lagrangian method (ELM) for
linear convection-diffusion problems. Unlike the classical a posteriori error
estimations, we estimate the temporal error along the characteristics and
derive a new a posteriori error bound for ELM semi-discretization. With the
help of this proposed error bound, we are able to show the optimal convergence
rate of ELM for solutions with minimal regularity. Furthermore, by combining
this error bound with a standard residual-type estimator for the spatial error,
we obtain a posteriori error estimators for a fully discrete scheme. We present
numerical tests to demonstrate the efficiency and robustness of our adaptive
algorithm.Comment: 30 page
A Multiscale Method for Model Order Reduction in PDE Parameter Estimation
Estimating parameters of Partial Differential Equations (PDEs) is of interest
in a number of applications such as geophysical and medical imaging. Parameter
estimation is commonly phrased as a PDE-constrained optimization problem that
can be solved iteratively using gradient-based optimization. A computational
bottleneck in such approaches is that the underlying PDEs needs to be solved
numerous times before the model is reconstructed with sufficient accuracy. One
way to reduce this computational burden is by using Model Order Reduction (MOR)
techniques such as the Multiscale Finite Volume Method (MSFV).
In this paper, we apply MSFV for solving high-dimensional parameter
estimation problems. Given a finite volume discretization of the PDE on a fine
mesh, the MSFV method reduces the problem size by computing a
parameter-dependent projection onto a nested coarse mesh. A novelty in our work
is the integration of MSFV into a PDE-constrained optimization framework, which
updates the reduced space in each iteration. We also present a computationally
tractable way of differentiating the MOR solution that acknowledges the change
of basis. As we demonstrate in our numerical experiments, our method leads to
computational savings particularly for large-scale parameter estimation
problems and can benefit from parallelization.Comment: 22 pages, 4 figures, 3 table
A Lagrangian Gauss-Newton-Krylov Solver for Mass- and Intensity-Preserving Diffeomorphic Image Registration
We present an efficient solver for diffeomorphic image registration problems
in the framework of Large Deformations Diffeomorphic Metric Mappings (LDDMM).
We use an optimal control formulation, in which the velocity field of a
hyperbolic PDE needs to be found such that the distance between the final state
of the system (the transformed/transported template image) and the observation
(the reference image) is minimized. Our solver supports both stationary and
non-stationary (i.e., transient or time-dependent) velocity fields. As
transformation models, we consider both the transport equation (assuming
intensities are preserved during the deformation) and the continuity equation
(assuming mass-preservation).
We consider the reduced form of the optimal control problem and solve the
resulting unconstrained optimization problem using a discretize-then-optimize
approach. A key contribution is the elimination of the PDE constraint using a
Lagrangian hyperbolic PDE solver. Lagrangian methods rely on the concept of
characteristic curves that we approximate here using a fourth-order Runge-Kutta
method. We also present an efficient algorithm for computing the derivatives of
final state of the system with respect to the velocity field. This allows us to
use fast Gauss-Newton based methods. We present quickly converging iterative
linear solvers using spectral preconditioners that render the overall
optimization efficient and scalable. Our method is embedded into the image
registration framework FAIR and, thus, supports the most commonly used
similarity measures and regularization functionals. We demonstrate the
potential of our new approach using several synthetic and real world test
problems with up to 14.7 million degrees of freedom.Comment: code available at:
https://github.com/C4IR/FAIR.m/tree/master/add-ons/LagLDDM
A discrete Liouville identity for numerical reconstruction of Schr\"odinger potentials
We propose a discrete approach for solving an inverse problem for
Schr\"odinger's equation in two dimensions, where the unknown potential is to
be determined from boundary measurements of the Dirichlet to Neumann map. For
absorptive potentials, and in the continuum, it is known that by using the
Liouville identity we obtain an inverse conductivity problem. Its discrete
analogue is to find a resistor network that matches the measurements, and is
well understood. Here we show how to use a discrete Liouville identity to
transform its solution to that of Schr\"odinger's problem. The discrete
Schr\"odinger potential given by the discrete Liouville identity can be used to
reconstruct the potential in the continuum in two ways. First, we can obtain a
direct but coarse reconstruction by interpreting the values of the discrete
Schr\"odinger potential as averages of the continuum Schr\"odinger potential on
a special sensitivity grid. Second, the discrete Schr\"odinger potential may be
used to reformulate the conventional nonlinear output least squares
optimization formulation of the inverse Schr\"odinger problem. Instead of
minimizing the boundary measurement misfit, we minimize the misfit between the
discrete Schr\"odinger potentials. This results in a better behaved
optimization problem that converges in a single Gauss-Newton iteration, and
gives good quality reconstructions of the potential, as illustrated by the
numerical results.Comment: 20 pages, 8 figure
A Unified Study of Continuous and Discontinuous Galerkin Methods
A unified study is presented in this paper for the design and analysis of
different finite element methods (FEMs), including conforming and nonconforming
FEMs, mixed FEMs, hybrid FEMs,discontinuous Galerkin (DG) methods, hybrid
discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG
and WG are shown to admit inf-sup conditions that hold uniformly with respect
to both mesh and penalization parameters. In addition, by taking the limit of
the stabilization parameters, a WG method is shown to converge to a mixed
method whereas an HDG method is shown to converge to a primal method.
Furthermore, a special class of DG methods, known as the mixed DG methods, is
presented to fill a gap revealed in the unified framework.Comment: 39 page
Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids
In this paper two new families of arbitrary high order accurate spectral DG
finite element methods are derived on staggered Cartesian grids for the
solution of the inc.NS equations in two and three space dimensions. Pressure
and velocity are expressed in the form of piecewise polynomials along different
meshes. While the pressure is defined on the control volumes of the main grid,
the velocity components are defined on a spatially staggered mesh. In the first
family, h.o. of accuracy is achieved only in space, while a simple
semi-implicit time discretization is derived for the pressure gradient in the
momentum equation. The resulting linear system for the pressure is symmetric
and positive definite and either block 5-diagonal (2D) or block 7-diagonal (3D)
and can be solved very efficiently by means of a classical matrix-free
conjugate gradient method. The use of a preconditioner was not necessary. This
is a rather unique feature among existing implicit DG schemes for the NS
equations. In order to avoid a stability restriction due to the viscous terms,
the latter are discretized implicitly. The second family of staggered DG
schemes achieves h.o. of accuracy also in time by expressing the numerical
solution in terms of piecewise space-time polynomials. In order to circumvent
the low order of accuracy of the adopted fractional stepping, a simple
iterative Picard procedure is introduced. In this manner, the symmetry and
positive definiteness of the pressure system are not compromised. The resulting
algorithm is stable, computationally very efficient, and at the same time
arbitrary h.o. accurate in both space and time. The new numerical method has
been thoroughly validated for approximation polynomials of degree up to N=11,
using a large set of non-trivial test problems in two and three space
dimensions, for which either analytical, numerical or experimental reference
solutions exist.Comment: 46 pages, 15 figures, 4 table
The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
We list the so-called Askey-scheme of hypergeometric orthogonal polynomials.
In chapter 1 we give the definition, the orthogonality relation, the three term
recurrence relation and generating functions of all classes of orthogonal
polynomials in this scheme. In chapeter 2 we give all limit relation between
different classes of orthogonal polynomials listed in the Askey-scheme.
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme.
We give their definition, orthogonality relation, three term recurrence
relation and generating functions. In chapter 4 we give the limit relations
between those basic hypergeometric orthogonal polynomials. Finally in chapter 5
we point out how the `classical` hypergeometric orthogonal polynomials of the
Askey-scheme can be obtained from their q-analogues
A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System based on Fourier, Legendre and Hermite Polynomials
In this work, we apply a semi-Lagrangian spectral method for the
Vlasov-Poisson system, previously designed for periodic Fourier
discretizations, by implementing Legendre polynomials and Hermite functions in
the approximation of the distribution function with respect to the velocity
variable. We discuss second-order accurate-in-time schemes, obtained by
coupling spectral techniques in the space-velocity domain with a BDF
time-stepping scheme. The resulting method possesses good conservation
properties, which have been assessed by a series of numerical tests conducted
on the standard two-stream instability benchmark problem. In the Hermite case,
we also investigate the numerical behavior in dependence of a scaling parameter
in the Gaussian weight. Confirming previous results from the literature, our
experiments for different representative values of this parameter, indicate
that a proper choice may significantly impact on accuracy, thus suggesting that
suitable strategies should be developed to automatically update the parameter
during the time-advancing procedure.Comment: arXiv admin note: text overlap with arXiv:1803.0930
First and Second Order Shape Optimization based on Restricted Mesh Deformations
We consider shape optimization problems subject to elliptic partial
differential equations. In the context of the finite element method, the
geometry to be optimized is represented by the computational mesh, and the
optimization proceeds by repeatedly updating the mesh node positions. It is
well known that such a procedure eventually may lead to a deterioration of mesh
quality, or even an invalidation of the mesh, when interior nodes penetrate
neighboring cells. We examine this phenomenon, which can be traced back to the
ineptness of the discretized objective when considered over the space of mesh
node positions. As a remedy, we propose a restriction in the admissible mesh
deformations, inspired by the Hadamard structure theorem. First and second
order methods are considered in this setting. Numerical results show that mesh
degeneracy can be overcome, avoiding the need for remeshing or other
strategies. FEniCS code for the proposed methods is available on GitHub
Decoupled, Energy Stable Scheme for Hydrodynamic Allen-Cahn Phase Field Moving Contact Line Model
In this paper, we present an efficient energy stable scheme to solve a phase
field model incorporating contact line condition. Instead of the usually used
Cahn-Hilliard type phase equation, we adopt the Allen-Cahn type phase field
model with the static contact line boundary condition that coupled with
incompressible Navier-Stokes equations with Navier boundary condition. The
projection method is used to deal with the Navier-Stokes equa- tions and an
auxiliary function is introduced for the non-convex Ginzburg-Landau bulk
potential. We show that the scheme is linear, decoupled and energy stable.
Moreover, we prove that fully discrete scheme is also energy stable. An
efficient finite element spatial discretization method is implemented to verify
the accuracy and efficiency of proposed schemes. Numerical results show that
the proposed scheme is very efficient and accurat
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