Sergio Conti, Janusz Ginster and Martin Rumpf A $BV$ Functional and its Relaxation for Joint Motion Estimation and Image Sequence Recovery ESAIM: Mathematical Modelling and Numerical Analysis, 49(5): 1463-1487 2015 http://dx.doi.org/10.1051/m2an/2015036
Abstract: The estimation of motion in an image sequence is a fundamental task in image processing. Frequently, the image sequence is corrupted by noise and one simultaneously asks for the underlying motion field and a restored sequence. In smoothly shaded regions of the restored image sequence the brightness constancy assumption along motion paths leads to a pointwise differential condition on the motion field. At object boundaries which are edge discontinuities both for the image intensity and for the motion field this condition is no longer well defined. In this paper a total-variation type functional is discussed for joint image restoration and motion estimation. This functional turns out not to be lower semicontinuous, and in particular fine-scale oscillations may appear around edges. By the general theory of vector valued $BV$ functionals its relaxation leads to the appearance of a singular part of the energy density, which can be determined by the solution of a local minimization problem at edges. Based on bounds for the singular part of the energy and under appropriate assumptions on the local intensity variation one can exclude the existence of microstructures and obtain a model well-suited for simultaneous image restoration and motion estimation. Indeed, the relaxed model incorporates a generalized variational formulation of the brightness constancy assumption. The analytical findings are related to ambiguity problems in motion estimation such as the proper distinction between foreground and background motion at object edges.
Sergio Conti, Adriana Garroni and Michael Ortiz The line-tension approximation as the dilute limit of linear-elastic dislocations Arch. Ration. Mech. Anal., 218(2): 699-755 2015 http://dx.doi.org/10.1007/s00205-015-0869-7
Abstract: We prove that the classical line-tension approximation for dislocations in crystals, i.e., the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the $\Gamma$-limit of regularized linear-elasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length $\psi(b,t)$, which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length $\psi_0(b,t)$ obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation the classical energy down to its $\calH^1$-elliptic envelope.
Abstract: Dislocations in crystals can be studied by a Peierls-Nabarro type model, which couples linear elasticity with a nonconvex term modeling plastic slip. In the limit of small lattice spacing, and for dislocations restricted to planes, we show that it reduces to a line-tension model, with an energy depending on the local orientation and Burgers vector of the dislocation. This model predicts, for specific geometries, spontaneous formation of microstructure, in the sense that straight dislocations are unstable towards a zig-zag pattern. Coupling between dislocations in different planes can lead to microstructures over several length scales.
Sergio Conti, Adriana Garroni and Annalisa Massaccesi Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity Calc. Var. PDE, 54(2): 1847-1874 2015 http://dx.doi.org/10.1007/s00526-015-0846-x
Abstract: In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may be identified with divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In this paper we develop the theory of relaxation for these energies and provide one physically motivated example in which the relaxation for some Burgers vectors is nontrivial and can be determined explicitly. From a technical viewpoint the key ingredients are an approximation and a structure theorem for 1-currents with multiplicity in a lattice.
Dinh Dũng and Michael Griebel Hyperbolic cross approximation in infinite dimensions Journal of Complexity 2015 http://arxiv.org/pdf/1501.01119v1
Abstract: We give tight upper and lower bounds of the cardinality of
the index sets of certain hyperbolic crosses which reflect mixed
Sobolev–Korobov-type smoothness and mixed Sobolev-analytictype
smoothness in the infinite-dimensional case where specific
summability properties of the smoothness indices are fulfilled.
These estimates are then applied to the linear approximation of
functions from the associated spaces in terms of the ε-dimension
of their unit balls. Here, the approximation is based on linear
information. Such function spaces appear for example for the
solution of parametric and stochastic PDEs. The obtained upper
and lower bounds of the approximation error as well as of the
associated ε-complexities are completely independent of any parametric
or stochastic dimension. Moreover, the rates are independent
of the parameters which define the smoothness properties
of the infinite-variate parametric or stochastic part of the solution.
These parameters are only contained in the order constants.
This way, linear approximation theory becomes possible in the
infinite-dimensional case and corresponding infinite-dimensional
problems get tractable.
Benedict Geihe and Martin Rumpf A posteriori error estimates for sequential laminates in shape optimization In DCDS-S Special issue on Homogenization-Based Numerical Methods 2015 http://arxiv.org/abs/1501.07461
Abstract: A posteriori error estimates are derived in the context of two-dimensional structural elastic shape optimization under the compliance objective. It is known that the optimal shape features are microstructures that can be constructed using sequential lamination. The descriptive parameters explicitly depend on the stress. To derive error estimates the dual weighted residual approach for control problems in PDE constrained optimization is employed, involving the elastic solution and the microstructure parameters. Rigorous estimation of interpolation errors ensures robustness of the estimates while local approximations are used to obtain fully practical error indicators. Numerical results show sharply resolved interfaces between regions of full and intermediate material density.
Nicola Gigli, Tapio Rajala and Karl-Theodor Sturm Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below J. Geom. Anal. 2015 http://arxiv.org/abs/1305.4849
Abstract: We prove existence and uniqueness of optimal maps on RCD∗(K,N) spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation and to the local-to-global property of RCD∗(K,N) bounds.
Michael Griebel, Christian Rieger and Barbara Zwicknagl Multiscale approximation and reproducing kernel Hilbert space methods SIAM Journal on Numerical Analysis, 53(2): 852-873 2015 http://dx.doi.org/10.1137/130932144
Michael Griebel, Alexander Hullmann and Oeter Oswald Optimal scaling parameters for sparse grid discretizations Numerical Linear Algebra with Applications, 22(1): 76-100 2015 http://dx.doi.org/10.1002/nla.1939
Abstract: We apply iterative subspace correction methods to elliptic PDE problems discretized by generalized sparse grid systems. The involved subspace solvers are based on the combination of all anisotropic full grid spaces that are contained in the sparse grid space. Their relative scaling is at our disposal and has significant influence on the performance of the iterative solver. In this paper, we follow three approaches to obtain close-to-optimal or even optimal scaling parameters of the subspace solvers and thus of the overall subspace correction method. We employ a Linear Program that we derive from the theory of additive subspace splittings, an algebraic transformation that produces partially negative scaling parameters which result in improved asymptotic convergence properties, and finally we use the OptiCom method as a variable non-linear preconditioner.
2014
Véronique Gayrard and Adéla Švejda Convergence of clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model on $\mathbbZ^d$ Lat. Am. J. Probab. Math. Stat., 11(2): 78-822 2014 http://alea.impa.br/articles/v11/11-35.pdf
Abstract: Using a method developed by Durrett and Resnick, [23], we establish general criteria for the convergence of properly rescaled clock processes of random dynamics in random environments on infinite graphs. This extends the results of Gayrard, [27], Bovier and Gayrard, [20], and Bovier, Gayrard, and Svejda, [21], and gives a unified framework for proving convergence of clock processes. As a first application we prove that Bouchaud's asymmetric trap model on \(\mathbb{Z}^d\) exhibits a normal aging behavior for all \(d \geq 2\). Namely, we show that certain two-time correlation functions, among which the classical probability to find the process at the same site at two time points, converge, as the age of the process diverges, to the distribution function of the arcsine law. As a byproduct we prove that the fractional kinetics process ages.
Michael Griebel and Alexander Hullmann Dimensionality Reduction of High-Dimensional Data with a NonLinear Principal Component Aligned Generative Topographic Mapping SIAM J. Sci. Comput., 36(3): A1027-A1047 2014 http://dx.doi.org/10.1137/130931382
Michael Griebel and Helmut Harbrecht On the convergence of the combination technique In Sparse grids and Applications, Volume 97 of Lecture Notes in Computational Science and Engineering
page 55-74.
2014 http://dx.doi.org/10.1007/978-3-319-04537-5_3
Michael Griebel, Jan Hamaekers and Frederik Heber A bond order dissection ANOVA approach for efficient electronic structure calculations In Extraction of Quantifiable Information from Complex Systems, Volume 102 of Lecture Notes in Computational Science and Engineering
Chapter 11, page 211-235.
Publisher: Springer International
2014 http://dx.doi.org/10.1007/978-3-319-08159-5
Michael Griebel and Alexander Hullmann A Sparse Grid Based Generative Topographic Mapping for the Dimensionality Reduction of High-Dimensional Data In Modeling, Simulation and Optimization of Complex Processes - HPSC 2012
page 51-62.
2014 http://dx.doi.org/10.1007/978-3-319-09063-4_5
Michael Griebel and Jens Oettershagen Dimension-adaptive sparse grid quadrature for integrals with boundary singularities In Sparse grids and Applications, Volume 97 of Lecture Notes in Computational Science and Engineering
page 109-136.
2014 http://dx.doi.org/10.1007/978-3-319-04537-5_5
Michael Griebel and Jan Hamaekers Fast Discrete Fourier Transform on Generalized Sparse Grids In Sparse grids and Applications, Lecture Notes in Computational Science and Engineering Vol. 97, Springer, Volume 97 of Lecture Notes in Computational Science and Engineering
page 75-108.
2014 http://dx.doi.org/10.1007/978-3-319-04537-5_4
2013
Michael Griebel and Helmut Harbrecht A note on the construction of L-fold sparse tensor product spaces Constructive Approximation, 38(2): 235-251 2013 http://dx.doi.org/10.1007/s00365-012-9178-7
Abstract: In the present paper, we consider the construction of general sparse tensor product spaces in arbitrary space dimensions when the single subdomains are of different dimensionality and the associated ansatz spaces possess different approximation properties. Our theory extends the results from Griebel and Harbrecht (Math. Comput., 2013) for the construction of two-fold sparse tensor product space to arbitrary L-fold sparse tensor product spaces.