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Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs
Get PDFThis paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on with , and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters . We
establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence of the random inputs, and prove convergence rates of best -term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best -term truncations can practically be computed, by greedy-type algorithms
as in [SG, Gi1], or by multilevel Monte-Carlo methods as in
[KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]
Uniform estimates for positive solutions of a class of semilinear elliptic equations and related Liouville and one-dimensional symmetry results
Get PDFWe consider the semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain . Under suitable assumptions on the potential , including the double well potential that gives rise to the Allen-Cahn equation, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that
the estimate is independent of . The main advantage of our approach is that it allows us to remove a restrictive monotonicity assumption on that was imposed in the recent paper by G. Fusco, F. Leonetti and C. Pignotti. In addition, we can remove a non-degeneracy condition on the global minimum of that
was assumed in the latter reference. Furthermore, we can generalize an old result of P. Hess and D.G. De Figueiredo, concerning semilinear elliptic nonlinear eigenvalue problems. Moreover, we study the boundary layer of global minimizers of the corresponding singular perturbation problem. For the above applications, our approach is based on a refinement of a useful result that dates back to P. Clement and G. Sweers, concerning the behavior of global
minimizers of the associated energy over large balls, subject to Dirichlet conditions. Combining this refinement with global bifurcation theory and the celebrated sliding method, we can prove uniform estimates for solutions away from their nodal set, refining a lemma from a well known paper of H. Berestycki, L. A. Caffarelli and L. Nirenberg. In particular, combining our approach with a-priori estimates that we obtain by blow-up, the doubling lemma of P. Polacik, P. Quittner, and P. Souplet and known Liouville type theorems, we can give a new proof of a Liouville type theorem of Y. Du and L. Ma, without using boundary blow-up solutions. We can also provide an alternative proof, and a useful extension, of a Liouville theorem of
H. Berestycki, F. Hamel, and H. Matano, involving the presence of an obstacle. Making use of the latter
extension, we consider the singular perturbation problem with mixed boundary conditions. Moreover, we prove some new one-dimensional symmetry properties of certain entire solutions to Allen-Cahn type equations, by exploiting for the first time an old result of Caffarelli, Garofalo, and Segala, and we suggest a connection with the theory of minimal surfaces. Using this approach, we also provide new proofs of well known symmetry results in half-spaces with Dirichlet boundary conditions. Lastly, we study
the one-dimensional symmetry of solutions in convex cylindrical domains with Neumann boundary conditions, and in convex epigraphs with partially over-determined boundary conditions
Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants
Get PDFWe revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points.
Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring).
All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface.
For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system.
This yields an efficient, output-sensitive algorithm for
computing the discriminant polynomial
Multispecies Virial Expansions
Get PDFWe study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange-Good inversion formula, which has other applications such as counting coloured trees or studying probability generating functions in multi-type branching processes. We prove that the virial expansion converges absolutely in a domain of small densities. In addition, we establish that the virial coefficients can be expressed in terms of two-connected graphs
Consistent Discretizations for Vanishing Regularization Solutions to Image Processing Problems
Get PDFA model problem is used to represent a typical image processing problem of reconstructing an unknown in the face of incomplete data. A consistent discretization for a vanishing regularization solution is defined so that, in the absence of noise, limits first with respect to regularization and then with respect to grid refinement agree with a continuum counterpart defined in terms of a saddle point formulation. It is proved and demonstrated computationally for an artificial example and for a realistic example with magnetic resonance images that a mixed finite element discretization is consistent in the sense defined here. On the other hand, it is demonstrated computationally that a standard finite element discretization is not consistent, and the reason for the inconsistency is suggested in terms of theoretical and computational evidence
The problem of dynamic cavitation in nonlinear elasticity
Get PDFThe notion of singular limiting induced from continuum solutions (slic-solutions) is applied to
the problem of cavitation in nonlinear elasticity, in order to re-assess an example of non-uniqueness of
entropic weak solutions (with polyconvex energy) due to a forming cavity
Singular limiting induced from continuum solutions and the problem of dynamic cavitation
Get PDFIn the works of
K.A. Pericak-Spector and S. Spector [Pericak-Spector, Spector 1988, 1997] a class of self-similar
solutions are constructed for the equations of radial isotropic elastodynamics
that describe cavitating solutions. Cavitating solutions decrease the total
mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions
(for polyconvex energies) due to point-singularities at the cavity. To resolve
this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution),
according to which a discontinuous motion is a slic-solution if its averages
form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for
creating the cavity, which is captured by the notion of slic-solution but neglected by the
usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the
cavitating solution is in fact larger than that of the homogeneously deformed state.
We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture,
and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan
Advanced trajectory engineering of diffraction-resisting laser beams
Get PDFWe introduce an analytical technique for engineering the trajectory of diffraction-resisting laser beams. The generated beams have a Bessel-like transverse field distribution and can be navigated along rather arbitrary curved paths in free space, thus being an advanced hybrid between accelerating and non-accelerating diffraction-free optical waves. The method involves phase-modulating the wavefront of a Gaussian laser beam to create a continuum of conical ray bundles whose apexes define a prespecified focal curve, along which a nearly perfect circular intensity lobe propagates without diffracting. Through extensive numerical simulations, we demonstrate the great flexibility in the design of a gamut of different beam trajectories. Propagation around obstructions and self-healing scenarios are also investigated. The proposed wave entities can be used extensively for light trajectory control in applications such as laser microfabrication, optical tweezers and curved plasma filamentation spectroscopy
Signal to Noise Ratio estimation in passive correlation-based imaging
Get PDFWe consider imaging with passive arrays of sensors using as illumination
ambient noise sources. The first step for imaging under such circumstances is
the computation of the cross correlations of the recorded signals, which have
attracted a lot of attention recently because of their numerous applications
in seismic imaging, volcano monitoring, and petroleum prospecting. Here, we
use these cross correlations for imaging reflectors with travel-time
migration. While the resolution of the image obtained this way has been
studied in detail, an analysis of the signal-to-noise ratio (SNR) is
presented in this paper along with numerical simulations that support the
theoretical results. It is shown that the SNR of the image inherits the SNR
of the computed cross correlations and therefore it is proportional to the
square root of the bandwidth of the noise sources times the recording
time. Moreover, the SNR of the image is proportional to the array size. This
means that the image can be stabilized by increasing the size of the array
when the recorded signals are not of long duration, which is important in
applications such as non-destructive testing