Detection of Singularities by Discrete Multiscale Directional Representations

One of the most remarkable properties of the continuous curvelet and shearlet transforms is their sensitivity to the directional regularity of functions and distributions. As a consequence of this property, these transforms can be used to characterize the geometry of edge singularities of functions and distributions by their asymptotic decay at fine scales. This ability is a major extension of the conventional continuous wavelet transform which can only describe pointwise regularity properties. However, while in the case of wavelets it is relatively easy to relate the asymptotic properties of the continuous transform to properties of discrete wavelet coefficients, this problem is surprisingly challenging in the case of discrete curvelets and shearlets where one wants to handle also the geometry of the singularity. No result for the discrete case was known so far. In this paper, we derive non-asymptotic estimates showing that discrete shearlet coefficients can detect, in a precise sense, the location and orientation of curvilinear edges. We discuss connections and implications of this result to sparse approximations and other applications.     

Sparse regularization in limited angle tomography

We investigate the reconstruction problem of limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, electron microscopy, etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, e.g. filtered backprojection (FBP), do not perform well in such situations.To stabilize the reconstruction procedure additional prior knowledge about the unknown object has to be integrated into the reconstruction process. In this work, we propose the use of the sparse regularization technique in combination with curvelets. We argue that this technique gives rise to an edge-preserving reconstruction. Moreover, we show that the dimension of the problem can be significantly reduced in the curvelet domain. To this end, we give a characterization of the kernel of the limited angle Radon transform in terms of curvelets and derive a characterization of solutions obtained through curvelet sparse regularization. In numerical experiments, we will show that the theoretical results directly translate into practice and that the proposed method outperforms classical reconstructions.     

New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities

This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C² edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle‐shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2^?j, each element has an envelope that is aligned along a “ridge” of length 2^?j/2 and width 2^?j. We prove that curvelets provide an essentially optimal representation of typical objects f that are C² except for discontinuities along piecewise C² curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n‐term partial reconstruction f obtained by selecting the n largest terms in the curvelet series obeys This rate of convergence holds uniformly over a class of functions that are C² except for discontinuities along piecewise C² curves and is essentially optimal. In comparison, the squared error of n‐term wavelet approximations only converges as n^?1 as n → ∞, which is considerably worse than the optimal behavior. © 2003 Wiley Periodicals, Inc.     

Short communication: Dimensionality reduction of curvelet sparse regularizations in limited angle tomography

We investigate the reconstruction problem for limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, such as filtered backprojection (FBP), do not perform well in such situations. To stabilize the inversion we propose the use of a sparse regularization technique in combination with curvelets. We argue that this technique has the ability to preserve edges. As our main result, we present a characterization of the kernel of the limited angle Radon transform in terms of curvelets. Moreover, we characterize reconstructions which are obtained via curvelet sparse regularizations at a limited angular range. As a result, we show that the dimension of the limited angle problem can be significantly reduced in the curvelet domain. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parabolic Molecules

Anisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significant attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities. In this paper, with the introduction of the notion of parabolic molecules, we aim to provide a comprehensive framework which includes customarily employed representation systems based on parabolic scaling such as curvelets and shearlets. It is shown that pairs of parabolic molecules have the fundamental property to be almost orthogonal in a particular sense. This result is then applied to analyze parabolic molecules with respect to their ability to sparsely approximate data governed by anisotropic features. For this, the concept of sparsity equivalence is introduced which is shown to allow the identification of a large class of parabolic molecules providing the same sparse approximation results as curvelets and shearlets. Finally, as another application, smoothness spaces associated with parabolic molecules are introduced providing a general theoretical approach which even leads to novel results for, for instance, compactly supported shearlets.     

Representation of Fourier Integral Operators Using Shearlets

Traditional methods of time-frequency and multiscale analysis, such as wavelets and Gabor frames, have been successfully employed for representing most classes of pseudodifferential operators. However, these methods are not equally effective in dealing with Fourier Integral Operators in general. In this article, we show that the shearlets, recently introduced by the authors and their collaborators, provide very efficient representations for a large class of Fourier Integral Operators. The shearlets are an affine-like system of well-localized waveforms at various scales, locations and orientations, which are particularly efficient in representing anisotropic functions. Using this approach, we prove that the matrix representation of a Fourier Integral Operator with respect to a Parseval frame of shearlets is sparse and well-organized. This fact recovers a similar result recently obtained by Candès and Demanet using curvelets, which illustrates the benefits of directional multiscale representations (such as curvelets and shearlets) in the study of those functions and operators where traditional multiscale methods are unable to provide the appropriate geometric analysis in the phase space. The second author was supported in part by a National Science Foundation grant DMS 0604561.     

Directional Haar wavelet frames on triangles

Traditional wavelets are not very effective in dealing with images that contain orientated discontinuities (edges). To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C²-curves. While curvelets and shearlets have compact support in frequency domain, we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets, being based on a generalized multiresolution analysis (MRA) associated with a dilation matrix and a finite collection of ‘shear’ matrices. The complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. The freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising.     

A relationship between λ‐symmetries and first integrals for ordinary differential equations

In this paper, we provide some geometric properties of λ‐symmetries of ordinary differential equations using vector fields and differential forms. According to the corresponding geometric representation of λ‐symmetries, we conclude that first integrals can also be derived if the equations do not possess enough symmetries. We also investigate the properties of λ‐symmetries in the sense of the deformed Lie derivative and differential operator. We show that λ‐symmetries have the exact analogous properties as standard symmetries if we take into consideration the deformed cases.     

On a wave map equation arising in general relativity

We consider a class of space‐times for which the essential part of Einstein's equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1 + 1 nonlinear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations as t → ∞. For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to 0 as t^?1/2 as t → ∞. In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behavior need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the nonhomogeneous case, one gets the following asymptotic behavior in the upper half‐plane (after applying an isometry of hyperbolic space if necessary):

• 1 The solution converges to a point.
• 2 The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary).
• 3 The solution goes to infinity along a curve y = const.
• 4 The solution oscillates around a circle inside the upper half‐plane.

Thus, even though the solutions become spatially homogeneous in the sense that the spatial variations die out, the asymptotic behavior may be radically different from anything observed for spatially homogeneous solutions of the equations. This analysis can then be applied to draw conclusions concerning the associated class of space‐times. For instance, one obtains the leading‐order behavior of the functions appearing in the metric, and one can conclude future causal geodesic completeness. © 2004 Wiley Periodicals, Inc.     

A Feynman–Kac formula for stochastic Dirichlet problems

A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that appear in the usual formulation are not even defined in the Itô sense.     

Compactly supported shearlets are optimally sparse

Cartoon-like images, i.e., C² functions which are smooth apart from a C² discontinuity curve, have by now become a standard model for measuring sparse (nonlinear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit sparse approximations within this model, which are optimal up to a log-factor. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications.In this paper, we present the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of (non-tight) shearlet frames with shearlet generators having compact support and satisfying some weak directional vanishing moment conditions.     

Microlocal Analysis of the Geometric Separation Problem

Image data are often composed of two or more geometrically distinct constituents; in galaxy catalogs, for instance, one sees a mixture of pointlike structures (galaxy superclusters) and curvelike structures (filaments). It would be ideal to process a single image and extract two geometrically “pure” images, each one containing features from only one of the two geometric constituents. This seems to be a seriously underdetermined problem but recent empirical work achieved highly persuasive separations. We present a theoretical analysis showing that accurate geometric separation of point and curve singularities can be achieved by minimizing the ?₁ norm of the representing coefficients in two geometrically complementary frames: wavelets and curvelets. Driving our analysis is a specific property of the ideal (but unachievable) representation where each content type is expanded in the frame best adapted to it. This ideal representation has the property that important coefficients are clustered geometrically in phase space, and that at fine scales, there is very little coherence between a cluster of elements in one frame expansion and individual elements in the complementary frame. We formally introduce notions of cluster coherence and clustered sparsity and use this machinery to show that the underdetermined systems of linear equations can be stably solved by ?₁ minimization; microlocal phase space helps organize the calculations that cluster coherence requires. © 2012 Wiley Periodicals, Inc.     

Asymptotic Behavior of Solutions of Dynamic Equations

We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.     

Global solution to the Cauchy problem in non-linear hyperbolic thermoelasticity

We prove the existence of global solutions for small data to the initial value problem for the non-linear hyperbolic system of partial differential equations describing a thermoelastic medium in a three-dimensional space under the assumption that the coefficients in the non-linear terms are smooth functions of their arguments and behave like 0(∣η∣) for k₀ ≥ 2 near the origin. The asymptotic behaviour of the solution as t → ∞ is also described.     

On construction of weak solutions with higher integrable gradients to linear hyperbolic partial differential systems

Taking linear hyperbolic partial differential equations as an illustration, we attempt to construct weak solutions with higher integrable gradients, in the sense of Gehring, to hyperbolic diffeential equations with initial and boundary conditions. We adopt Rothe's method and follow the calculation which has been expanded by Giaquinta and Struwe in dealing with parabolic equations. To establish the scheme, we evaluate some local estimates for solutions to Rothe's approximations to hyperbolic differential equations. Bibliography: 6 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 30–52.     

Weak Solutions of Semilinear PDEs in Sobolev Spaces and Their Probabilistic Interpretation via the FBSDEs

Abstract

The article is devoted to representation of weak solutions (in Sobolev sense) of degenerate parabolic partial differential equations through forward-backward stochastic differential equations. Before, we prove a weak version of a norm equivalence result.     

Differentialoperatoren bei partiellen Differentialgleichungen

Summary In the present paper those formally hyperbolic differential equations are characterized for which solutions can be represented by means of differential operators acting on holomorphic functions. This is done by a necessary and sufficient condition on the coefficients of the differential equation. These operators are determined simultaneously. By it a general procedure is presented to construct differential equations and corresponding differential operators which map holomorphic functions onto solutions of the differential equations. We also discuss the question under which circumstances all the solutions of a differential equation can be represented by differential operators. For the equations characterized previously we determine the Riemann function. Some special classes of differential equations are investigated in detail. Furthermore the possibility of a representation of pseudoanalytic functions and the corresponding Vekua resolvents by differential operators is discussed.

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Herrn Prof. Dr. K. W. Bauer zum 60. Geburtstag gewidmet     

Riemann integral operator for nonstationary processes in the axisymmetric case

A generalization of the Riemann operator method is proposed, which can be used to analyze in a unified framework the linear equations of nonstationary processes in the axisymmetric case. An integral representation of the solutions of a hyperbolic and a parabolic equation is constructed. The use of the apparatus of special functions produces a simple representation of solutions of partial differential equations, which is convenient for analysis.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 16–22, 1990.     

Mean-field stochastic control with elephant memory in finite and infinite time horizon

ABSTRACT

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.

• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.

• For infinite horizon, we derive sufficient and necessary maximum principles.

  As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.