54,867 research outputs found
On the optimal control of some nonsmooth distributed parameter systems arising in mechanics
Variational inequalities are an important mathematical tool for modelling
free boundary problems that arise in different application areas. Due to the
intricate nonsmooth structure of the resulting models, their analysis and
optimization is a difficult task that has drawn the attention of researchers
for several decades. In this paper we focus on a class of variational
inequalities, called of the second kind, with a twofold purpose. First, we aim
at giving a glance at some of the most prominent applications of these types of
variational inequalities in mechanics, and the related analytical and numerical
difficulties. Second, we consider optimal control problems constrained by these
variational inequalities and provide a thorough discussion on the existence of
Lagrange multipliers and the different types of optimality systems that can be
derived for the characterization of local minima. The article ends with a
discussion of the main challenges and future perspectives of this important
problem class
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 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
Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation
There has been a large increase in the amount of work on hierarchical
low-rank approximation methods, where the interest is shared by multiple
communities that previously did not intersect. This objective of this article
is two-fold; to provide a thorough review of the recent advancements in this
field from both analytical and algebraic perspectives, and to present a
comparative benchmark of two highly optimized implementations of contrasting
methods for some simple yet representative test cases. We categorize the recent
advances in this field from the perspective of compute-memory tradeoff, which
has not been considered in much detail in this area. Benchmark tests reveal
that there is a large difference in the memory consumption and performance
between the different methods.Comment: 19 pages, 6 figure
Suitable Spaces for Shape Optimization
The differential-geometric structure of certain shape spaces is investigated
and applied to the theory of shape optimization problems constrained by partial
differential equations and variational inequalities. Furthermore, we define a
diffeological structure on a new space of so-called -shapes. This can
be seen as a first step towards the formulation of optimization techniques on
diffeological spaces. The -shapes are a generalization of smooth
shapes and arise naturally in shape optimization problems
Derivative-free optimization methods
In many optimization problems arising from scientific, engineering and
artificial intelligence applications, objective and constraint functions are
available only as the output of a black-box or simulation oracle that does not
provide derivative information. Such settings necessitate the use of methods
for derivative-free, or zeroth-order, optimization. We provide a review and
perspectives on developments in these methods, with an emphasis on highlighting
recent developments and on unifying treatment of such problems in the
non-linear optimization and machine learning literature. We categorize methods
based on assumed properties of the black-box functions, as well as features of
the methods. We first overview the primary setting of deterministic methods
applied to unconstrained, non-convex optimization problems where the objective
function is defined by a deterministic black-box oracle. We then discuss
developments in randomized methods, methods that assume some additional
structure about the objective (including convexity, separability and general
non-smooth compositions), methods for problems where the output of the
black-box oracle is stochastic, and methods for handling different types of
constraints
Discontinuous Galerkin methods for fractional elliptic problems
We provide a mathematical framework for studying different versions of
discontinuous Galerkin (DG) approaches for solving 2D Riemann-Liouville
fractional elliptic problems on a finite domain. The boundedness and stability
analysis of the primal bilinear form are provided. A priori error estimate
under energy norm and optimal error estimate under norm are obtained
for DG methods of the different formulations. Finally, the performed numerical
examples confirm the optimal convergence order of the different formulations
Photoacoustic imaging taking into account thermodynamic attenuation
In this paper we consider a mathematical model for photoacoustic imaging
which takes into account attenuation due to thermodynamic dissipation. The
propagation of acoustic (compressional) waves is governed by a scalar wave
equation coupled to the heat equation for the excess temperature. We seek to
recover the initial acoustic profile from knowledge of acoustic measurements at
the boundary.
We recognize that this inverse problem is a special case of boundary
observability for a thermoelastic system. This leads to the use of
control/observability tools to prove the unique and stable recovery of the
initial acoustic profile in the weak thermoelastic coupling regime. This
approach is constructive, yielding a solvable equation for the unknown acoustic
profile. Moreover, the solution to this reconstruction equation can be
approximated numerically using the conjugate gradient method. If certain
geometrical conditions for the wave speed are satisfied, this approach is
well--suited for variable media and for measurements on a subset of the
boundary. We also present a numerical implementation of the proposed
reconstruction algorithm
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
A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes
In this paper we propose a novel arbitrary high order accurate semi-implicit
space-time DG method for the solution of the three-dimensional incompressible
Navier-Stokes equations on staggered unstructured curved tetrahedral meshes. As
typical for space-time DG schemes, the discrete solution is represented in
terms of space-time basis functions. This allows to achieve very high order of
accuracy also in time, which is not easy to obtain for the incompressible
Navier-Stokes equations. Similar to staggered finite difference schemes, in our
approach the discrete pressure is defined on the primary tetrahedral grid,
while the discrete velocity is defined on a face-based staggered dual grid. A
very simple and efficient Picard iteration is used in order to derive a
space-time pressure correction algorithm that achieves also high order of
accuracy in time and that avoids the direct solution of global nonlinear
systems. Formal substitution of the discrete momentum equation on the dual grid
into the discrete continuity equation on the primary grid yields a very sparse
five-point block system for the scalar pressure, which is conveniently solved
with a matrix-free GMRES algorithm. From numerical experiments we find that the
linear system seems to be reasonably well conditioned, since all simulations
shown in this paper could be run without the use of any preconditioner. For a
piecewise constant polynomial approximation in time and proper boundary
conditions, the resulting system is symmetric and positive definite. This
allows us to use even faster iterative solvers, like the conjugate gradient
method. The proposed method is verified for approximation polynomials of degree
up to four in space and time by solving a series of typical 3D test problems
and by comparing the obtained numerical results with available exact analytical
solutions, or with other numerical or experimental reference data
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