59,807 research outputs found
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
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
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
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
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
Exact determination of the volume of an inclusion in a body having constant shear modulus
We derive an exact formula for the volume fraction of an inclusion in a body
when the inclusion and the body are linearly elastic materials with the same
shear modulus. Our formula depends on an appropriate measurement of the
displacement and traction around the boundary of the body. In particular, the
boundary conditions around the boundary of the body must be such that they
mimic the body being placed in an infinite medium with an appropriate
displacement applied at infinity
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
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
Backstepping Control of the One-Phase Stefan Problem
In this paper, a backstepping control of the one-phase Stefan Problem, which
is a 1-D diffusion Partial Differential Equation (PDE) defined on a time
varying spatial domain described by an ordinary differential equation (ODE), is
studied. A new nonlinear backstepping transformation for moving boundary
problem is utilized to transform the original coupled PDE-ODE system into a
target system whose exponential stability is proved. The full-state boundary
feedback controller ensures the exponential stability of the moving interface
to a reference setpoint and the -norm of the distributed
temperature by a choice of the setpint satisfying given explicit inequality
between initial states that guarantees the physical constraints imposed by the
melting process.Comment: 6 pages, 4 figures, The 2016 American Control Conferenc
Preconditioning Parametrized Linear Systems
Preconditioners are generally essential for fast convergence in the iterative
solution of linear systems of equations. However, the computation of a good
preconditioner can be expensive. So, while solving a sequence of many linear
systems, it is advantageous to recycle preconditioners, that is, update a
previous preconditioner and reuse the updated version. In this paper, we
introduce a simple and effective method for doing this. Although our approach
can be used for matrices changing slowly in any way, we focus on the important
case of sequences of the type , where the right hand side
may or may not change. More general changes in matrices will be discussed in a
future paper. We update preconditioners by defining a map from a new matrix to
a previous matrix, for example the first matrix in the sequence, and combine
the preconditioner for this previous matrix with the map to define the new
preconditioner. This approach has several advantages. The update is entirely
independent from the original preconditioner, so it can be applied to any
preconditioner. The possibly high cost of an initial preconditioner can be
amortized over many linear solves. The cost of updating the preconditioner is
more or less constant and independent of the original preconditioner. There is
flexibility in balancing the quality of the map with the computational cost. In
the numerical experiments section we demonstrate good results for several
applications.Comment: V2 Model Reduction discussion. V3 Theoretical section replaced with
experimental analysis of sparsity patterns. Top opt application added. Rail
replaced with Flow. Only early shifts for THT. ILUTP, SAM implementations
more efficient; results updated. ILUTP m-file included. New citations added.
V4 New pattern added to top opt sparsity pattern analysis/results. ILUTP
m-file minor update
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