Loop space blow-up and scale calculus

In order to provide a local description of a regular function in a small neighbourhood of a point $x$, it is sufficient by Taylor's theorem to know the value of the function as well as all of its derivatives up to the required order at the point $x$ itself. In other words, one could say that a regular function is locally modelled by the set of polynomials. The theory of regularity structures due to Hairer generalizes this observation and provides an abstract setup, which in the application of singular SPDE extends the set of polynomials by functionals constructed from, e.g., white noise. In this context, the notion of Taylor polynomials is lifted to the notion of so-called modelled distributions. The celebrated reconstruction theorem, which in turn was inspired by Gubinelli's sewing lemma, is of paramount importance for the theory. It enables to reconstruct a modelled distribution as a true distribution on $R^d$ which is locally approximated by this extended set of models or "monomials". In the original work of Hairer, the error is measured based on Hölder norms. This was then generalized to the whole scale of Besov spaces by Hairer and Labbé in subsequent papers. It is the aim of this work to adapt the analytic part of the theory of regularity structures to the scale of Triebel-Lizorkin spaces.

This paper discusses a generalization of spectral representations related to convex one-homogeneous regularization functionals, e.g. total variation or \(\ell ^1\)-norms. Those functionals serve as a substitute for a Hilbert space structure (and the related norm) in classical linear spectral transforms, e.g. Fourier and wavelet analysis. We discuss three meaningful definitions of spectral representations by scale space and variational methods and prove that (nonlinear) eigenfunctions of the regularization functionals are indeed atoms in the spectral representation. Moreover, we verify further useful properties related to orthogonality of the decomposition and the Parseval identity.

This paper discusses the use of absolutely one-homogeneous regularization functionals in a variational, scale space, and inverse scale space setting to define a nonlinear spectral decomposition of input data. We present several theoretical results that explain the relation between the different definitions. Additionally, results on the orthogonality of the decomposition, a Parseval-type identity, and the notion of generalized (nonlinear) eigenvectors closely link our nonlinear multiscale decompositions to the well-known linear filtering theory. Numerical results are used to illustrate our findings.

For image filtering applications, it has been observed recently that both diffusion filtering and associated regularization models provide similar filtering properties. The comparison has been performed for regularization functionals with convex penalization functional. In this paper we discuss the relation between non-convex regularization functionals and associated time dependent diffusion filtering techniques (in particular the Mean Curvature Flow equation). Here, the general idea is to approximate an evolution process by a sequence of minimizers of iteratively convexified energy (regularization) functionals.

Adaptive global thresholding on the sphere

This paper is concerned with the estimation of the partial derivatives of a probability density function of directional data on the d-dimensional torus within the local thresholding framework. The estimators here introduced are built by means of the toroidal needlets, a class of wavelets characterised by excellent concentration properties in both the real and the harmonic domains. In particular, we discuss the convergence rates of the -risks for these estimators, investigating their minimax properties and proving their optimality over a scale of Besov spaces, here taken as nonparametric regularity function spaces.

Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain rule, and Jensen’s inequality on time scales, we can establish some comprehensive and generalize a number of classical reverse Hilbert-type inequalities to a general time scale space. In time scale calculus, results are unified and extended. At the same time, the theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on symmetrical properties which play an essential role in determining the correct methods to solve inequalities. As a special case of our results when the supermultiplicative function represents the identity map, we obtain some results that have been recently published.

Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods in Hilbert spaces. Besides simplifying computations it allows to provide a good understanding of properties of the forward problem compared to the prior information introduced by the regularization methods. In the last decade nonlinear variational approaches such as ` or total variation regularizations became quite prominent regularization techniques with certain properties being superior to standard methods. In the analysis of those, singular values and vectors did not play any role so far, for the obvious reason that these problems are nonlinear, together with the issue of defining singular values and singular vectors in the first place. In this paper however we want to start a study of singular values and vectors for nonlinear variational regularization of linear inverse problems, with particular focus on singular onehomogeneous regularization functionals. A major role is played by the smallest singular value, which we define as the ground state of an appropriate functional combining the (semi)norm introduced by the forward operator and the regularization functional. The optimality condition for the ground state further yields a natural generalization to higher singular values and vectors involving the subdifferential of the regularization functional, although we shall see that the Rayleigh principle may fail for higher singular values. Using those definitions of singular values and vectors, we shall carry over two main properties from the world of linear regularization. The first one is gaining information about scale, respectively the behavior of regularization techniques at different scales. This also leads to novel estimates at different scales, generalizing the estimates for the coefficients in the linear singular value expansion. The second one is to provide classes of exact solutions for variational regularization methods. We will show that all singular vectors can be reconstructed up to a scalar factor by the standard Tikhonov-type regularization approach even in the presence of (small) noise. Moreover, we will show that they can even be reconstructed without any bias by the recently popularized inverse scale space method.

We investigate the inverse scale space flow as a decomposition method for decomposing data into generalised singular vectors. We show that the inverse scale space flow, based on convex and even and positively one-homogeneous regularisation functionals, can decompose data represented by the application of a forward operator to a linear combination of generalised singular vectors into its individual singular vectors. We verify that for this decomposition to hold true, two additional conditions on the singular vectors are sufficient: orthogonality in the data space and inclusion of partial sums of the subgradients of the singular vectors in the subdifferential of the regularisation functional at zero.We also address the converse question of when the inverse scale space flow returns a generalised singular vector given that the initial data is arbitrary (and therefore not necessarily in the range of the forward operator). We prove that the inverse scale space flow is guaranteed to return a singular vector if the data satisfies a novel dual singular vector condition.We conclude the paper with numerical results that validate the theoretical results and that demonstrate the importance of the additional conditions required to guarantee the decomposition result.

A regular (weakly) transitive translation function for a fibre bundle induces an H-space antihomomorphism from the loop space of the base into the group of the bundle. The question when such translation functions exist has had three answers to date. E. Brown [1, p. 226] states that every bundle with paracompact base admits weakly transitive translation functions. However, E. Fadell has pointed out an essential error in the proof. (Namely, [1, p. 227] v.(a), v.(f8) and v,,(a+o) are unrelated.) In Steenrod's book [4, p. 59] we find that if a bundle has totally disconnected group, then it admits transitive translation functions. J. Schlesinger [3] has proved a converse to this result for a restricted class of bundles. Schlesinger's theorem fails to contradict Brown's claim for the following reasons: (1) Brown employs Moore paths while Schlesinger uses ordinary (unit domain) paths. (2) Schlesinger discusses transitivity while Brown claims only weak transitivity. This paper removes the first of these distinctions; namely we prove:

Inverse scale space methods are derived as asymptotic limits of iterative regularization methods. They have proven to be efficient methods for denoising of gray valued images and for the evaluation of unbounded operators. In the beginning, inverse scale space methods have been derived from iterative regularization methods with squared Hilbert norm regularization terms, and later this concept was generalized to Bregman distance regularization (replacing the squared regularization norms); therefore allowing for instance to consider iterative total variation regularization. We have proven recently existence of a solution of the associated inverse total variation flow equation. In this paper we generalize these results and prove existence of solutions of inverse flow equations derived from iterative regularization with general convex regularization functionals. We present some applications to filtering of color data and for the stable evaluation of the diZenzo edge detector.

In this chapter we investigate one procedure for describing the dynamics of complex webs when the differential equations of ordinary dynamics are no longer adequate, that is, the webs are fractal. We described some of the essential features of fractal functions earlier, starting from the simple dynamical processes described by functions that are fractal, such as the Weierstrass function, which are continuous everywhere but are nowhere differentiable. This idea of non-differentiability suggests introducing elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. The relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, changes in time of phenomena that are described by fractal functions are probably best described by fractional equations of motion. In any event, this perspective is the one we developed elsewhere [31] and we find it useful here for discussing some properties of complex webs. The separation of time scales in complex physical phenomena allows smoothing over the microscopic fluctuations and the construction of differentiable representations of the dynamics on large space scales and long time scales. However, such smoothing is not always possible.

We show that the higher Grothendieck–Witt groups, a.k.a. algebraic hermitian $$K$$ -groups, are represented by an infinite orthogonal Grassmannian in the $$\mathbb {A}^1$$ -homotopy category of smooth schemes over a regular base for which $$2$$ is a unit in the ring of regular functions. We also give geometric models for various $$\mathbb {P}^1$$ - and $$S^1$$ -loop spaces of hermitian $$K$$ -theory.