2,065 research outputs found
Gradient-Based Estimation of Uncertain Parameters for Elliptic Partial Differential Equations
This paper addresses the estimation of uncertain distributed diffusion
coefficients in elliptic systems based on noisy measurements of the model
output. We formulate the parameter identification problem as an infinite
dimensional constrained optimization problem for which we establish existence
of minimizers as well as first order necessary conditions. A spectral
approximation of the uncertain observations allows us to estimate the infinite
dimensional problem by a smooth, albeit high dimensional, deterministic
optimization problem, the so-called finite noise problem in the space of
functions with bounded mixed derivatives. We prove convergence of finite noise
minimizers to the appropriate infinite dimensional ones, and devise a
stochastic augmented Lagrangian method for locating these numerically. Lastly,
we illustrate our method with three numerical examples
Pulse shape optimization for electron-positron production in rotating fields
We optimize the pulse shape and polarization of time-dependent electric
fields to maximize the production of electron-positron pairs via strong field
quantum electrodynamics processes. The pulse is parametrized in Fourier space
by a B-spline polynomial basis, which results in a relatively low-dimensional
parameter space while still allowing for a large number of electric field
modes. The optimization is performed by using a parallel implementation of the
differential evolution, one of the most efficient metaheuristic algorithms. The
computational performance of the numerical method and the results on pair
production are compared with a local multistart optimization algorithm. These
techniques allow us to determine the pulse shape and field polarization that
maximize the number of produced pairs in computationally accessible regimes.Comment: 16 pages, 10 figure
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification
We present a new approach to convexification of the Tikhonov regularization
using a continuation method strategy. We embed the original minimization
problem into a one-parameter family of minimization problems. Both the penalty
term and the minimizer of the Tikhonov functional become dependent on a
continuation parameter.
In this way we can independently treat two main roles of the regularization
term, which are stabilization of the ill-posed problem and introduction of the
a priori knowledge. For zero continuation parameter we solve a relaxed
regularization problem, which stabilizes the ill-posed problem in a weaker
sense. The problem is recast to the original minimization by the continuation
method and so the a priori knowledge is enforced.
We apply this approach in the context of topology-to-shape geometry
identification, where it allows to avoid the convergence of gradient-based
methods to a local minima. We present illustrative results for magnetic
induction tomography which is an example of PDE constrained inverse problem
On the monotone and primal-dual active set schemes for -type problems,
Nonsmooth nonconvex optimization problems involving the quasi-norm,
, of a linear map are considered. A monotonically convergent
scheme for a regularized version of the original problem is developed and
necessary optimality conditions for the original problem in the form of a
complementary system amenable for computation are given. Then an algorithm for
solving the above mentioned necessary optimality conditions is proposed. It is
based on a combination of the monotone scheme and a primal-dual active set
strategy. The performance of the two algorithms is studied by means of a series
of numerical tests in different cases, including optimal control problems,
fracture mechanics and microscopy image reconstruction
An augmented lagrangian approach for Euler-Korteweg type equations
On présente un modèle hyperbolique quasi-linéaire de premier ordre approximant les équations d'Euler-Korteweg (E-K), qui décrivent des écoulements de fluides compressibles dont l'énergie dépend du gradient de la densité. Le système E-K peut être vu comme les équations d'Euler-Lagrange d'un Lagrangien soumis à la conservation de la masse. Vu la présence du gradient de la densité dans le Lagrangien, des dérivées d'ordre élevé de la densité apparaissent dans les équations du mouvement. L'approche présentée ici permet d'obtenir un système d'équations hyperboliques qui approxime le système E-K. L'idée est d'introduire un nouveau paramètre d'ordre qui approxime la densité via une méthode de pénalisation classique. Le gradient de cette nouvelle variable remplace alors le gradient de la densité dans le Lagrangien, ce qui permet de construire le Lagrangien augmenté. Les équations d'Euler-Lagrange associées à celui-ci, sont des équations hyperboliques avec des termes sources raides et des vitesses de caractéristiques rapides. Ce système est analysé puis résolu numériquement en utilisant des schémas de type IMEX. En particulier, cette approche a été appliquée à l'équation de Schrödinger non-linéaire défocalisante (qui peut être réduite au système E-K via la transformée de Madelung), pour laquelle des comparaisons avec des solutions exactes et asymptotiques ont été faites, notamment pour des solitons gris et des ondes de choc dispersives. La même approche a été également appliquée aux équations de filmes minces avec capillarité, pour lesquelles une comparaison avec des résultats numériques de référence et des résultats expérimentaux a été faite. Il a été démontré que le modèle augmenté peut aussi bien s'appliquer pour des modèles dont le terme de capillarité est non-linéaire. Dans ce même cadre, une étude de gouttes stationnaires sur un substrat solide horizontal a été établie afin de classifier les profils possibles de gouttes selon leur énergie. Ceci a permis également de faire des comparaisons du modèle augmenté sur des solutions stationnaires. Enfin, une partie indépendante de ce travail est consacrée à l'étude des équations équivalentes associées aux schémas numériques, où l'on démontre que les conditions de stabilité qui dérivent d'une troncature de l'équation équivalente, n'a du sens que si la série correspondante dans l'espace de Fourier est convergente, sur les longueurs d'onde admissibles dans la pratique.An approximate first order quasilinear hyperbolic model for Euler-Korteweg (E-K) equations, describing compressible fluid flows whose energy depend on the gradient of density, is derived. E-K system can be seen as the Euler-Lagrange equations to a Lagrangian submitted to the mass conservation constraint. Due to the presence of the density gradient in the Lagrangian, one recovers high-order derivatives of density in the motion equations. The approach presented here permits us to obtain a system of hyperbolic equations that approximate E-K system. The idea is to introduce a new order parameter which approximates the density via a carefully chosen penalty method. The gradient of this new independent variable will then replace the original gradient of density in the Lagrangian, resulting in the so-called augmented Lagrangian. The Euler-Lagrange equations of the augmented Lagrangian result in a first order hyperbolic system with stiff source terms and fast characteristic speeds. Such a system is then analyzed and solved numerically by using IMEX schemes. In particular, this approach was applied to the defocusing nonlinear Schrödinger equation (which can be reduced to the E-K equations via the Madelung transform), for which a comparison with exact and asymptotic solutions, namely gray solitons and dispersive shock waves was performed. Then, the same approach was extended to thin film flows with capillarity, for which comparison of the numerical results with both reference numerical solutions and experimental results was performed. It was shown that the augmented model is also extendable to models with full nonlinear surface tension. In the same setting, a study of stationary droplets on a horizontal solid substrate was conducted in an attempt to classify droplet profiles depending on their energy forms. This also allowed to compare the augmented Lagrangian approach in the case of stationary solutions, and which showed excellent agreement with the reference solutions. Lastly, an independent part of this work is devoted to the study of modified equations associated to numerical schemes for stability purposes. It is shown that for a linear scheme, stability conditions which are obtained from a truncation of the associated modified equation, are only relevant if the corresponding series in Fourier space is convergent for the admissible wavenumbers
The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach
We study the combination of the alternating direction method of multipliers
(ADMM) with physics-informed neural networks (PINNs) for a general class of
nonsmooth partial differential equation (PDE)-constrained optimization
problems, where additional regularization can be employed for constraints on
the control or design variables. The resulting ADMM-PINNs algorithmic framework
substantially enlarges the applicable range of PINNs to nonsmooth cases of
PDE-constrained optimization problems. The application of the ADMM makes it
possible to untie the PDE constraints and the nonsmooth regularization terms
for iterations. Accordingly, at each iteration, one of the resulting
subproblems is a smooth PDE-constrained optimization which can be efficiently
solved by PINNs, and the other is a simple nonsmooth optimization problem which
usually has a closed-form solution or can be efficiently solved by various
standard optimization algorithms or pre-trained neural networks. The ADMM-PINNs
algorithmic framework does not require to solve PDEs repeatedly, and it is
mesh-free, easy to implement, and scalable to different PDE settings. We
validate the efficiency of the ADMM-PINNs algorithmic framework by different
prototype applications, including inverse potential problems, source
identification in elliptic equations, control constrained optimal control of
the Burgers equation, and sparse optimal control of parabolic equations
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