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Seminars (ASC)

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Event When Speaker Title Presentation Material
ASC 10th January 2019
15:00 to 16:00
Peter Binev Greedy algorithms in reduced modeling
ASC 18th January 2019
11:00 to 12:00
Andreas Seeger A maximal function for families of Hilbert transforms along homogeneous curves
Let H(u) be the Hilbert transform along the parabola (t; ut2) where u 2 R. For a set U of positive numbers consider the maximal function HUf = supfH(u)f : u 2 Ug. We obtain (essentially) optimal results for the
Lp operator norm of HU when 2 < p < 1. The results are proved for families of Hilbert transforms along more general non- at homogeneous curves. Joint work with Shaoming Guo, Joris Roos and Po-Lam Yung.

ASC 24th January 2019
15:00 to 16:00
Elijah Liflyand The Fourier transform of a function of bounded variation: symmetry and asymmetry
New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. The main result is an asymptotic formula for the  cosine Fourier transform. Such relations have previously been known only for the sine Fourier transform. To prove the mentioned result, not only a different space is considered but also a new way of proving such theorems is applied. Interrelations of various function spaces are studied in this context. The obtained results are used for proving new estimates for the Fourier transform of a radial function and completely new results on the integrability  of trigonometric series.




ASC 29th January 2019
15:00 to 16:00
Mikhail Tyaglov Hurwitz stable and self-interlacing orthogonal polynomials.
In this survey talk, we consider a few examples of Hurwitz stable and their dual (the so-called) self-interlacing polynomials and discuss a number of interrelations between theory of orthogonal polynomials, moment theory, and the root distributions of polynomials. Some specific relations between the Jacobi polynomials and certain  Hurwitz stable polynomials will be presented and an answer to a question of Geno Nikolov on interlacing properties of the Jacobi polynomials will be provided. We will also  discuss the asymptotic behaviour of sequences of Hurwitz stable polynomials.




ASC 11th February 2019
15:00 to 16:30
Erich Novak Lecture 1: Some old and new results on Information-Based Complexity

We give a short introduction to IBC and present some basic definitionsand a few results. The general question is: How many function values (or values of other functionals) of $f$ do we need to compute $S(f)$ up to an error $\epsilon$? Here $S(f)$ could be the integral or the maximum of $f$.
In particular we study the question: Which problems are tractable? When do we have the curse of dimension?
 In the second talk we discuss complexity results for numerical integration. In particular we present results for the star discrepancy, the curse of dimension for $C^k$ functions, and results for randomized algorithms



 

ASC 12th February 2019
15:00 to 16:30
Ben Adcock Lecture 1: Overview and Theory
Lecture 1: Overview and Theory  In these lectures I will present an introduction to compressed sensing and sparse approximation.  The first lecture gives an overview of compressed sensing and its standard theory.  Next, I will focus on two major areas of application.  The second lecture considers image reconstruction, and its application to medical and scientific imaging.  The third lecture considers high-dimensional approximation via compressed sensing, with application to parametric PDEs in Uncertainty Quantification.




ASC 13th February 2019
15:00 to 16:30
Erich Novak Lecture 2: Complexity results for integration.
We give a short introduction to IBC and present some basic definitions and a few results. The general question is: How many function values (or values of other functionals) of $f$ do we need to compute $S(f)$
up to an error $\epsilon$? Here $S(f)$ could be the integral or the maximum of $f$.
In particular we study the question: Which problems are tractable? When do we have the curse of dimension?
 In this second talk we discuss complexity results for numerical integration. In particular we present results for the star discrepancy, the curse of dimension for $C^k$ functions, and results for randomized algorithms




ASC 14th February 2019
15:00 to 16:30
Ben Adcock Lecture 2: Compressive Imaging
In these lectures I will present an introduction to compressed sensing and sparse approximation.  The first lecture gives an overview of compressed sensing and its standard theory.  Next, I will focus on two major areas of application.  The second lecture considers image reconstruction, and its application to medical and scientific imaging.  The third lecture considers high-dimensional approximation via compressed sensing, with application to parametric PDEs in Uncertainty Quantification.




ASC 15th February 2019
15:00 to 16:30
Ben Adcock Lecture 3: High-Dimensional Polynomial Approximation
In these lectures I will present an introduction to compressed sensing and sparse approximation.  The first lecture gives an overview of compressed sensing and its standard theory.  Next, I will focus on two major areas of application.  The second lecture considers image reconstruction, and its application to medical and scientific imaging.  The third lecture considers high-dimensional approximation via compressed sensing, with application to parametric PDEs in Uncertainty Quantification.




ASCW01 18th February 2019
09:40 to 10:15
Henryk Wozniakowski Exponential tractability of weighted tensor product problems
ASCW01 18th February 2019
11:00 to 11:35
Aicke Hinrichs Random sections of ellipsoids and the power of random information
We study the circumradius of the intersection of an $m$-dimensional ellipsoid~$\mathcal E$ with half axes $\sigma_1\geq\dots\geq \sigma_m$ with random subspaces of codimension $n$. We find that, under certain assumptions on $\sigma$, this random radius $\mathcal{R}_n=\mathcal{R}_n(\sigma)$ is of the same order as the minimal such radius $\sigma_{n+1}$ with high probability. In other situations $\mathcal{R}_n$ is close to the maximum~$\sigma_1$. The random variable $\mathcal{R}_n$ naturally corresponds to the worst-case error of the best algorithm based on random information for $L_2$-approximation of functions from a compactly embedded Hilbert space $H$ with unit ball $\mathcal E$.

In particular, $\sigma_k$ is the $k$th largest singular value of the embedding $H\hookrightarrow L_2$. In this formulation, one can also consider the case $m=\infty$, and we prove that random information behaves very differently depending on whether $\sigma \in \ell_2$ or not. For $\sigma \notin \ell_2$ random information is completely useless. For $\sigma \in \ell_2$ the expected radius of random information tends to zero at least at rate $o(1/\sqrt{n})$ as $n\to\infty$.

In the proofs we use a comparison result for Gaussian processes a la Gordon, exponential estimates for sums of chi-squared random variables, and estimates for the extreme singular values of (structured) Gaussian random matrices.

This is joint work with David Krieg, Erich Novak, Joscha Prochno and Mario Ullrich.
ASCW01 18th February 2019
11:40 to 12:15
Yuri Malykhin On some lower bounds for Kolmogorov widths
ASCW01 18th February 2019
13:40 to 14:15
Jan Vybiral Approximation of Ridge Functions and Sparse Additive Models
The approximation of smooth multivariate functions is known to suffer the curse of dimension. We discuss approximation of structured multivariate functions, which take the form of a ridge, their sum, or of the so-called sparse additive models. We give also results about optimality of such algorithms.
ASCW01 18th February 2019
14:20 to 14:55
Alexander Litvak Order statistics and Mallat--Zeitouni problem
Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not necessarily identically distributed coordinates and let $T$ be an orthogonal transformation of $\mathbb{R}^n$. We show that the random vector $Y=T(X)$ satisfies $$\mathbb{E} \sum \limits_{j=1}^k j\mbox{-}\min _{i\leq n}{X_{i}}^2 \leq C \mathbb{E} \sum\limits_{j=1}^k j\mbox{-}\min _{i\leq n}{Y_{i}}^2$$ for all $k\leq n$, where ``$j\mbox{-}\min$'' denotes the $j$-th smallest component of the corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regarding optimality of the Karhunen--Lo\`eve basis for the nonlinear reconstruction. We also show some relations for order statistics of random vectors (not only Gaussian), which are of independent interest. This is a joint work with Konstantin Tikhomirov.


ASCW01 18th February 2019
15:30 to 16:05
Mario Ullrich Construction of high-dimensional point sets with small dispersion
Based on deep results from coding theory, we present an deterministic algorithm that contructs a point set with dispersion at most $\eps$ in dimension $d$ of size $poly(1/\eps)*\log(d)$, which is optimal with respect to the dependence on $d$. The running time of the algorithms is, although super-exponential in $1/\eps$, only polynomial in $d$.
ASCW01 18th February 2019
16:10 to 16:45
Dicussion
ASCW01 19th February 2019
09:00 to 09:35
Winfried Sickel The Haar System and Smoothness Spaces built on Morrey Spaces
For some  Nikol'skij-Besov spaces $B^s_{p,q}$ the orthonormal Haar system can be used as an unconditional Schauder basis. Nowadays necessary and sufficient conditions with respect to $p,q$ and $s$ are known for this property. In recent years in a number of papers some modifications of Nikol'skij-Besov spaces based on Morrey spaces have been investigated. In my talk I will concentrate on a version called Besov-type spaces and denoted by $B^{s,\tau}_{p,q}$. It will be my aim to discuss some necessary and some sufficient conditions on the parameters $p,q,s,\tau$ such that one can characterize these classes by means of the Haar system. This is joined work with Dachun Yang and Wen Yuan (Beijing Normal University).
ASCW01 19th February 2019
09:40 to 10:15
Dachun Yang Ball Average Characterizations of Function Spaces

It is well known that function spaces play an important role in the study on various problems from analysis. In this talk, we present pointwise and ball average characterizations of function spaces including Sobolev spaces, Besov spaces and Triebel-Lizorkin spaces on the Euclidean spaces. These characterizations have the advantages so that they can be used as the definitions of these function spaces on metric measure spaces. Some open questions are also presented in this talk.

ASCW01 19th February 2019
11:00 to 11:35
Wen Yuan Embedding and continuity envelopes of Besov-type spaces

In this talk, we discuss about the sharp embedding properties between Besov-type spaces and Triebel-Lizorkin-type and present some related necessary and sufficient conditions for these embedding. The corresponding continuity envelopes are also worked out.

ASCW01 19th February 2019
11:40 to 12:15
Bin Han Directional Framelets with Low Redundancy and Directional Quasi-tight Framelets

Edge singularities are ubiquitous and hold key information for many high-dimensional problems. Consequently, directional representation systems are required to effectively capture edge singularities for high-dimensional problems. However, the increased angular resolution often significantly increases the redundancy rates of a directional system. High redundancy rates lead to expensive computational costs and large storage requirement, which hinder the usefulness of such directional systems for problems in moderately high dimensions such as video processing. In this talk, we attack this problem by using directional tensor product complex tight framelets with mixed sampling factors. Such introduced directional system has good directionality with a very low redundancy rate $\frac{3^d-1}{2^d-1}$, e.g., the redundancy rates are $2$, $2\frac{2}{3}$, $3\frac{5}{7}$, $5\frac{1}{3}$ and $7\frac{25}{31}$ for dimension $d=1,\ldots,5$. Our numerical experiments on image/video denoising and inpainting show that the performance of our proposed directional system with low redundancy rate is comparable or better than several state-of-the-art methods which have much higher redundancy rates. In the second part, we shall discuss our recent developments of directional quasi-tight framelets in high dimensions. This is a joint work with Chenzhe Diao, Zhenpeng Zhao and Xiaosheng Zhuang.

ASCW01 19th February 2019
13:40 to 14:15
Clayton Webster Polynomial approximation via compressed sensing of high-dimensional functions on lower sets
This talk will focus on compressed sensing approaches to sparse polynomial approximation of complex functions in high dimensions. Of particular interest is the parameterized PDE setting, where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we will present and analyze several procedures for exactly reconstructing a set of (jointly) sparse vectors, from incomplete measurements.  These include novel weighted $\ell_1$ minimization, improved iterative hard thresholding, mixed convex relaxations, as well as nonconvex penalties. Theoretical recovery guarantees will also be presented based on improved bounds for the restricted isometry property, as well as unified null space properties that encompass all currently proposed nonconvex minimizations.  Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the described compressed sensing methods. 
ASCW01 19th February 2019
14:20 to 14:55
Oscar Dominguez Characterizations of Besov spaces in terms of K-functionals
Besov spaces occur naturally in many fields of analysis. In this talk, we discuss various characterizations of Besov spaces in terms of different K-functionals. For instance, we present descriptions via oscillations, Bianchini-type norms and approximation methods. This is a joint work with S. Tikhonov (Barcelona).
ASCW01 19th February 2019
15:30 to 16:05
Wenrui Ye Local restriction theorem and maximal Bochner-Riesz operator for the Dunkl transforms

In this talk, I will mainly describe joint work with Dr. Feng Dai on the critical index for the almost everywhere convergence of the Bochner-Riesz means in weighted Lp-spaces with p=1 or p>2. Our results under the case p>2 are in full analogy with the classical result of M. Christ on estimates of the maximal Bochner-Riesz means of Fourier integrals and the classical result of A. Carbery, José L. Rubio De Francia and L. Vega on a.e. convergence of Fourier integrals. Besides, I will also introduce several new results that are related to our main results, including: (i) local restriction theorem for the Dunkl transform which is significantly stronger than the global one, but more difficult to prove; (ii) the weighted Littlewood Paley inequality with Ap-weights in the Dunkl non-commutative setting; (iii) sharp local point-wise estimates of several important kernel functions.

ASCW01 19th February 2019
16:10 to 16:45
Discussion
ASCW01 20th February 2019
09:00 to 09:35
Holger Rauhut Recovery of functions of many variables via compressive sensing
The talk will report on the use of compressive sensing for the recovery of functions of many variables from sample values. We will cover trigonometric expansions as well as expansions in tensorized orthogonal polynomial systems and provide convergence rates in terms of the number of samples which avoid the curse of dimensionality. The technique can be used for the numerical solution of parametric operator equations.
ASCW01 20th February 2019
09:40 to 10:15
Dũng Dinh Dimension-dependence error estimates for sampling recovery on Smolyak grids

We investigate dimension-dependence estimates of the approximation error for linear algorithms of sampling recovery on Smolyak grids parametrized by $m$, of periodic $d$-variate functions from the space with Lipschitz-H\"older mixed smoothness $\alpha > 0$. For the subsets of the unit ball in this space of functions with homogeneous condition and of functions depending on $\nu$ active variables ($1 \le \nu \le d$), respectively, we prove some upper bounds and lower bounds (for $\alpha \le 2$) of the error of the optimal sampling recovery on Smolyak grids, explicit in $d$, $\nu$, $m$ when $d$ and $m$ may be large. This is a joint work with Mai Xuan Thao, Hong Duc University, Thanh Hoa, Vietnam.

ASCW01 20th February 2019
11:00 to 11:35
Martin Buhmann Recent Results on Rational Approximation and Interpolation with Completely and Multiply Monotone Radial Basis Functions
We will report on new results about approximations to continuous functions of multiple variables. We shall use either approximation with interpolation or approximation by rational functions. For these kinds of approximations, radial basis functions are particularly attractive, as they provide regular, positive definite or conditionally positive definite approximations, independent of the spatial dimension and independent the distribution of the data points we wish to work with. These interpolants have very many applications for example in solving nonlinear partial differential equations by collocation. In this talk, we classify radial basis and other functions that are useful for such scattered data interpolation or for rational approximations from vector spaces spanned by translates of those basis functions (kernels); for this we study in particular multiply and/or completely monotone functions. We collect special properties of such monotone functions, generalise them and find larger classes than the well known monotone functions for multivariate interpolation. Furthermore, we discuss efficient ways to compute rational approximations using the same type of kernels.
ASCW01 20th February 2019
11:40 to 12:15
Lutz Kaemmerer Multiple Rank-1 Lattices as Sampling Schemes for Approximation
The approximation of functions using sampling values along single rank-1 lattices leads to convergence rates of the approximation errors that are far away from optimal ones in spaces of dominating mixed smoothness. A recently published idea that uses sampling values along several rank-1 lattices in order to reconstruct multivariate trigonometric polynomials accompanied by fast methods for the construction of these sampling schemes as well as available fast Fourier transform algorithms motivates investigations on the approximation properties of the arising sampling operators applied on functions of specific smoothness, in particular functions of dominating mixed smoothness which naturally leads to hyperbolic cross approximations.
ASCW01 21st February 2019
09:00 to 09:35
Thomas Kuehn Preasymptotic estimates for approximation of multivariate periodic Sobolev functions
Approximation of Sobolev functions is a topic with a long history and many applications in different branches of mathematics. The asymptotic order as $n\to\infty$ of the approximation numbers $a_n$ is well-known for embeddings of isotropic Sobolev spaces and also for Sobolev spaces of dominating mixed smoothness. However, if the dimension $d$ of the underlying domain is very high, one has to wait exponentially long until the asymptotic rate becomes visible. Hence, for computational issues this rate is useless, what really matters is the preasymptotic range, say $n\le 2^d$.  
In the talk I will first give a short overview over this relatively new field. Then I will present some new preasymptotic estimates for $L_2$-approximation of periodic Sobolev functions, which improve the previously known results. I will discuss the cases of isotropic and dominating mixed smoothness, and also $C^\infty$-functions of Gevrey type. Clearly, on all these spaces there are many equivalent norms. It is an interesting effect that - in contrast to the asymptotic rates - the preasymptotic behaviour strongly depends on the chosen norm.
ASCW01 21st February 2019
09:40 to 10:15
Konstantin Ryutin Best m-term approximation of the "step-function" and related problems
The main point of the talk is  the problem of approximation    of the step-function by $m$-term trigonometric polynomials  and some closely related problems: the approximate rank of a specific triangular matrix,  the Kolmogorov width of BV functions. This problem has its origins  in approximation theory (best sparse approximation and Kolmogorov widths) as well as in computer science (approximate rank of a matrix). There are different approaches and techniques: $\gamma_2$--norm, random approximations, orthomassivity of a set....  I plan to show what can be achieved by these techniques.
ASCW01 21st February 2019
11:00 to 11:35
Michael Gnewuch Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration
Smolyak's method, also known as hyperbolic cross approximation or sparse grid method, is a powerful %black box tool to tackle multivariate tensor product problems just with the help of efficient algorithms for the corresponding univariate problem. We provide upper and lower error bounds for randomized Smolyak algorithms with fully explicit dependence on the number of variables and the number of information evaluations used. The error criteria we consider are the worst-case root mean square error (the typical error criterion for randomized algorithms, often referred to as ``randomized error'') and the root mean square worst-case error (often referred to as ``worst-case error''). Randomized Smolyak algorithms can be used as building blocks for efficient methods, such as multilevel algorithms, multivariate decomposition methods or dimension-wise quadrature methods, to tackle successfully high-dimensional or even infinite-dimensional problems. As an example, we provide a very general and sharp result on infinite-dimensional integration on weighted reproducing kernel Hilbert spaces and illustrate it for the special case of weighted Korobov spaces. We explain how this result can be extended, e.g., to spaces of functions whose smooth dependence on successive variables increases (``spaces of increasing smoothness'') and to the problem of L_2-approximation (function recovery).
ASCW01 21st February 2019
11:40 to 12:15
Heping Wang Monte Carlo methods for $L_q$ approximation on periodic Sobolev spaces with mixed smoothness
In this talk we consider multivariate approximation of compact embeddings of periodic Sobolev spaces of dominating mixed smoothness into the $L_q,\ 2< q\leq \infty$ space by linear Monte Carlo methods that use arbitrary linear information. We construct linear Monte Carlo methods and obtain explicit-in-dimension upper estimates. These estimates catch up with the rate of convergence.
ASCW01 21st February 2019
13:40 to 14:15
Robert J. Kunsch Optimal Confidence for Monte Carlo Integration of Smooth Functions
We study the complexity $n(\varepsilon,\delta)$ of approximating the integral of smooth functions at absolute precision $\varepsilon > 0$ with confidence level $1 - \delta \in (0,1)$ using function evaluations as information within randomized algorithms. Methods that achieve optimal rates in terms of the root mean square error (RMSE) are not always optimal in terms of error at confidence, usually we need some non-linearity in order to suppress outliers. Besides, there are numerical problems which can be solved in terms of error at confidence but no algorithm can guarantee a finite RMSE, see [1]. Hence, the new error criterion seems to be more general than the classical RMSE. The sharp order for multivariate functions from classical isotropic Sobolev spaces $W_p^r([0,1]^d)$ can be achieved via control variates, as long as the space is embedded in the space of continuous functions $C([0,1]^d)$. It turns out that the integrability index $p$ has an effect on the influence of the uncertainty $\delta$ to the complexity, with the limiting case $p = 1$ where deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the effort we need to take in order to increase the confidence level. Determining the complexity $n(\varepsilon,\delta)$ is much more challenging for mixed smoothness spaces $\mathbf{W}_p^r([0,1]^d)$. While optimal rates are known for the classical RMSE (as long as $\mathbf{W}_p^r([0,1]^d)$ is embedded in $L_2([0,1]^d)$), see [2], basic modifications of the corresponding algorithms fail to match the theoretical lower bounds for approximating the integral with prescribed confidence.

Joint work with Daniel Rudolf 

[1]  R.J. Kunsch, E. Novak, D. Rudolf. Solvable integration problems and optimal sample size selection. To appear in Journal of Complexity.
[2]  M. Ullrich. A Monte Carlo method for integration of multivariate smooth functions. SIAM Journal on Numerical Analysis, 55(3):1188-1200, 2017.


ASCW01 21st February 2019
14:20 to 14:55
Markus Weimar Optimal recovery using wavelet trees
This talk is concerned with the approximation of embeddings between Besov-type spaces defined on bounded multidimensional domains or (patchwise smooth) manifolds. We compare the quality of approximations of three different strategies based on wavelet expansions. For this purpose, sharp rates of convergence corresponding to classical uniform refinement, best $N$-term, and best $N$-term tree approximation will be presented. In particular, we will see that whenever the embedding of interest is compact, greedy tree approximation schemes are as powerful as abstract best $N$-term approximation and that (for a large range of parameters) they can outperform uniform schemes based on a priori fixed (hence non-adaptively chosen) subspaces. This observation justifies the usage of adaptive non-linear algorithms in computational practice, e.g., for the approximate solution of boundary integral equations arising from physical applications. If time permits, implications for the related concept of approximation spaces associated to the three approximation strategies will be discussed.
ASCW01 21st February 2019
15:30 to 16:05
Discussion
ASCW01 21st February 2019
16:10 to 16:45
Discussion
ASCW01 22nd February 2019
09:00 to 09:35
Jürgen Prestin Shift-invariant Spaces of Multivariate Periodic Functions
One of the underlying ideas of multiresolution and wavelet analysis consists in the investigation of shift-invariant function spaces. In this talk one-dimensional shift-invariant spaces of periodic functions are generalized to multivariate shift-invariant spaces on non-tensor product patterns. These patterns are generated from a regular integer matrix. The decomposition of these spaces into shift-invariant subspaces can be discussed by the properties of these matrices. For these spaces we study different bases and their time-frequency localization. Of particular interest are multivariate orthogonal Dirichlet and de la Valle\'e Poussin kernels and the respective wavelets. This approach also leads to an adaptive multiresolution. Finally, with these methods we construct shearlets and show how we can detect jump discontinuities of given cartoon-like functions.
ASCW01 22nd February 2019
09:40 to 10:15
Bastian Bohn Least squares regression on sparse grids
In this talk, we first recapitulate the framework of least squares regression on certain sparse grid and hyperbolic cross spaces. The underlying numerical problem can be solved quite efficiently with state-of-the-art algorithms. Analyzing its stability and convergence properties, we can derive the optimal coupling between the number of necessary data samples and the degrees of freedom in the ansatz space.Our analysis is based on the assumption that the least-squares solution employs some kind of Sobolev regularity of dominating mixed smoothness, which is seldomly encountered for real-world applications. Therefore, we present possible extensions of the basic sparse grid least squares algorithm by introducing suitable a-priori data transformations in the second part of the talk. These are tailored such that the resulting transformed problem suits the sparse grid structure.

Co-authors: Michael Griebel (University of Bonn), Jens Oettershagen (University of Bonn), Christian Rieger (University of Bonn)
ASCW01 22nd February 2019
11:00 to 11:35
Song Li Some Sparse Recovery Methods in Compressed Sensing
In this talk, I shall investigate some sparse recovery methods in Compressed Sensing. In particular, I will focus on RIP approach and D-RIP approach.  As a result, we confirmed a conjecture on RIP, which is related to Terence. Tao and Jean. Bourgain's works in this fields.  Then, I will also investigate the relations between our works and statistics.
ASCW01 22nd February 2019
11:40 to 12:15
tba
ASCW01 22nd February 2019
13:40 to 16:45
Discussion
ASC 25th February 2019
15:00 to 16:00
Michael Lacey Discrete Spherical Averages
The strongest inequalities concerning continuous spherical averages are phrased in the language of $L^p$ improving inequalities.  Replace the  continuous averages by discrete averages, that is average over lattice points on a sphere. These inequalities then engage the continuous versions, the Hardy-Littlewood circle method, and Kloosterman sums. We will report on progress understanding these inequalities. Joint work with Robert Kesler, and Dario Mena.  



ASC 12th March 2019
15:00 to 16:30
Elena Berdysheva Metric Approximation of Set-Valued Functions
We study approximation of set-valued functions (SVFs) | functions mapping a real interval to compact sets in Rd. In addition to the theoretical interest in this subject, it is relevant to various applications in elds where SVFs are used, such as economy, optimization, dynamical systems, control theory, game theory, dierential inclusions, geometric modeling. In particular, SVFs are relevant to the problem of the reconstruction of 3D objects from their parallel cross-sections. The images (values) of the related SVF are the cross-sections of the 3D object, and the graph of this SVF is the 3D object. Adaptations of classical sample-based approximation operators, in particular, of positive operators for approximation of SVFs with convex images were intensively studied by a number of authors. For example, R.A Vitale studied an adaptation of the classical Bernstein polynomial operator based on Minkowski linear combination of sets which converges to the convex hull of the image. Thus, the limit SVF is always a function with
convex images, even if the original function is not. This eect is called convexication and is observed in various adaptations based on Minkowski linear combinations. Clearly such adaptations work for set-valued functions with convex images, but are useless for the approximation of SFVs with non-convex images. Also the standard construction of an integral of set-valued functions | the Aumann integral | possesses the property of convexication. Dyn, Farkhi and Mokhov developed in a series of work a new approach that is free of convexication | the so-called metric linear combinations and the metric integral.
Adaptations of classical approximation operators to continuous SFVs were studied by Dyn, Farkhi and Mokhov. Here, we develop methods for approximation of SFVs that are not necessarily contin- uous. As the rst step, we consider SVFs of bounded variation in the Hausdor metric.
In particular, we adapt to SVFs local operators such as the symmetric Schoenberg spline operator, the Bernstein polynomial operator and the Steklov function. Error bounds, obtained in the averaged Hausdor metric, provide rates of approximation similar to those for real-valued functions of bounded variation.
Joint work with Nira Dyn, Elza Farkhi and Alona Mokhov (Tel Aviv University, Israel).
ASC 25th March 2019
11:00 to 12:00
Dorothee Haroske Morrey sequence spaces
Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type were studied in recent years quite intensively and systematically. Decomposition methods like atomic or wavelet characterisations require suitably adapted sequence spaces. This has been done to some extent already. However, based on some discussion at a conference in Poznan in 2017 we found that Morrey sequence spaces $m_{u,p}=m_{u,p}(\mathbb{Z}^d)$, $0 We consider some basic features, embedding properties, a pre-dual, a corresponding version of Pitt's compactness theorem, and can further characterise the compactness of embeddings of related finite dimensional spaces in terms of their entropy numbers.
This is joint work with Leszek Skrzypczak (Poznan).
ASC 8th April 2019
15:00 to 16:00
Luz Roncal Hardy-type inequalities for fractional powers of the Dunkl--Hermite operator
We prove Hardy-type inequalities for the conformally invariant fractional powers of the Dunkl--Hermite operator. Consequently, we also obtain Hardy inequalities for the fractional harmonic oscillator as well.
The strategy is as follows: first, by introducing suitable polar coordinates, we reduce the problem to the Laguerre setting. Then, we push forward an argument developed by R. L. Frank, E. H. Lieb and R. Seiringer, initially developed in the Euclidean setting, to get a Hardy inequality for the fractional-type Laguerre operator. Such argument is based on two facts: first, to get an integral representation for the corresponding fractional operator, and second, to write a proper ground state representation.
This is joint work with \'O. Ciaurri (Universidad de La Rioja, Spain) and S. Thangavelu (Indian Institute of Science of Bangalore, India).
ASC 29th April 2019
14:00 to 16:00
Yuan Xu Simultaneous approximation by polynomials
Least square polynomials in an $L^2$ space are partial sums of the Fourier orthogonal expansions. If we were to approximate functions and their derivatives simultaneously on a domain in $R^d$ (as desired in spectral method), we would need to consider orthogonal expansions in a Sobolev space, for which the orthogonality is defined with respect to an inner product that contains derivatives. Since multiplication operators are no longer self-adjoint under such an inner product, the orthogonality is hard to understand and analyze. In the talk we will explain what is known.



ASC 1st May 2019
13:00 to 14:45
Marta Betcke Photoacoustic tomography with incomplete data
In photoacoustic tomography, the acoustic propagation time across the specimen constitutes the ultimate limit on sequential sampling frequency. Furthermore, the state-of-the art PAT systems are still remote from realising this limit. Hence, for high resolution imaging problems, the acquisition of a complete set of data can be impractical or even not possible e.g. the underlying dynamics causes the object to evolve faster than measurements can be acquired. To mitigate this problem we revert to parallel data acquisition along with subsampling/compressed sensing techniques. We consider different regularisation assumptions such as edge sparsity, sparsity of image representation and wave field propagation in Curvelet frame as well as learnt regularisation. We discuss the benefits and limitations of the proposed approaches in PAT context



ASC 9th May 2019
14:00 to 15:00
Nira Dyn Reconstruction of a 3D object from a finite number of its 1D parallel cross-sections
The problem of reconstruction of a 3D object from its parallel 2D cross sections has been
considered by many researchers. In some previous works we suggested to regard the problem as an approximation of a set-valued function from a finite number of its samples, which are 2D sets. We used approximation methods for single-valued functions by applying operations between sets instead of operations between numbers.
Since 2D sets are much more complicated than 1D sets, we suggest here to regard 3D objects as bivariate  functions  with 1D sets as samples, and to use the analogue of piecewise linear interpolation on a triangulation as the approximation method.
In this talk we present our method, and discuss the properties of the resulting interpolants, including continuity and approximation rates. Few examples will be presented.




ASC 21st May 2019
16:00 to 17:00
Ronald DeVore Rothschild Distinguished Visiting Fellow Lecture: Optimality of Algorithms for Approximation/Computation
OFBW46 23rd May 2019
10:00 to 10:10
Jane Leeks, David Abrahams Welcome and Introduction
OFBW46 23rd May 2019
10:10 to 10:20
Anders Hansen Introduction
OFBW46 23rd May 2019
10:20 to 11:00
Alhussein Fawzi Robustness and Geometry of Deep Neural Networks
OFBW46 23rd May 2019
11:20 to 12:00
Hamza Fawzi Fundamental Limitations on Adversarial Robustness
OFBW46 23rd May 2019
12:00 to 12:40
Jennifer Boon The Ethics of Algorithmic Decision Making
OFBW46 23rd May 2019
13:40 to 14:00
Vegard Antun On Instabilities of Deep Learning in Image Reconstruction - Part I
OFBW46 23rd May 2019
14:00 to 14:20
Matthew Colbrook On Instabilities of Deep Learning in Image Reconstruction - Part II
OFBW46 23rd May 2019
14:20 to 14:40
Laura Thesing A Stable Learning Framework
OFBW46 23rd May 2019
14:40 to 15:20
Thomas Strohmer Privacy Preserving Machine Learning: A Human Imperative?
OFBW46 23rd May 2019
15:40 to 16:20
Neil Lawrence Meta Modelling and Deploying Machine Learning Software
OFBW46 23rd May 2019
16:20 to 17:00
Pearse Keane The Moorfields - DeepMind Collaboration - Reinventing the Eye Exam using Deep Learning
ASC 7th June 2019
14:00 to 15:00
Guergana Petrova Approximation via Deep Neural Networks
We will discuss the approximation power of deep neural networks.
In particular, we will present classes of functions which can be
efficiently captured by neural networks where classical nonlinear methods
fall short of the task.





ASC 12th June 2019
16:00 to 17:00
Svitlana Mayboroda Kirk Distinguished Visiting Fellow Lecture: The hidden landscape of localization
Complexity of the geometry, randomness of the potential, and many other irregularities of the system can cause powerful, albeit quite different, manifestations of localization: a phenomenon of confinement of waves, or eigenfunctions, to a small portion of the original domain. In the present talk we show that behind a possibly disordered system there exists a clear structure, referred to as a landscape function, which predicts the location and shape of the localized eigenfunctions, a pattern of their exponential decay, and delivers accurate bounds for the corresponding eigenvalues in the range where, for instance, the classical Weyl law notoriously fails. We will discuss main features of this structure universally relevant for all elliptic operators, as well as specific applications to the Schrodinger operator with random potential and to the Poisson-Schrodinger drift-diffusion system governing carrier distribution and transport in semiconductor alloys.




ASC 13th June 2019
15:00 to 16:00
Maria del Carmen Reguera Rodriguez Sparse forms for Bochner-Riesz operators
Sparse operators are positive dyadic operators that have very nice boundedness properties. The L^p bounds and weighted L^p bounds with sharp constant are easy to obtain for these operators. In the recent years, it has been proven that singular integrals (cancellative operators) can be pointwise controlled by sparse operators. This has made the sharp weighted theory of singular integrals quite straightforward. The current efforts focus in understanding the use of sparse operators to bound rougher operators, such a oscillatory integrals. Following this direction, our goal in this talk is to describe the control of Bochner-Riesz operators by sparse operators.




ASCW03 17th June 2019
09:50 to 10:40
Akram Aldroubi Dynamical sampling and frames generated from powers of exponential operators
In this talk, I will give a brief review of the problem of frame generation from operator powers of exponentials acting on a set of vectors. I will discuss its relation to dynamical sampling, review some of the previous results and present several new ones.
ASCW03 17th June 2019
11:10 to 12:00
Denka Kutzarova Transportation cost spaces on finite metric spaces
Transportation cost spaces are studied by several groups of researchers, for different reasons and under different names. The term Lipschitz-free spaces is commonly used in Banach space theory.
We prove that the transportation cost space on any finite metric space contains a large well-complemented subspace which is close to $\ell_1^n$.
We show that transportation cost spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.
In the particular case of diamond graphs we prove that their cycle space is spanned by even levels of Haar functions. It is curious that the subspaces generated by all the even/odd levels of the Haar functions also appear in the study of quasi-greedy basic sequences in $L_1[0,1]$.
This research is joint with Stephen Dilworth and Mikhail Ostrovskii.
ASCW03 17th June 2019
13:30 to 14:20
Albert Cohen Optimal sampling for approximation on general domains
We consider the approximation of an arbirary function in any dimension from point samples. Approximants are picked from given or adaptively chosen finite dimensional spaces. Various recent works reveal that optimal approximations can be constructed at minimal sampling budget by least-squares methods with particular sampling measures. In this talk, we discuss strategies to construct these measures and their samples in the adaptive context and in general non-tensor-product multivariate domains.

ASCW03 17th June 2019
14:20 to 15:10
Peter Binev High Dimensional Approximation via Sparse Occupancy Trees
Adaptive domain decomposition is often used in finite elements methods for solving partial differential equations in low space dimensions. The adaptive decisions are usually described by a tree. Assuming that can find the (approximate) error for approximating a function on each element of the partition, we have shown that a particular coarse-to-fine method provides a near-best approximation. This result can be extended to approximating point clouds any space dimension provided that we have relevant information about the errors and can organize properly the data. Of course, this is subject to the curse of dimensionality and nothing can be done in the general case. In case the intrinsic dimensionality of the data is much smaller than the space dimension, one can define algorithms that defy the curse. This is usually done by assuming that the data domain is close to a low dimensional manifold and first approximating this manifold and then the function defined by it. A few years ago, together with Philipp Lamby, Wolfgang Dahmen, and Ron DeVore, we proposed a direct method (without specifically identifying any low dimensional set) that we called "sparse occupancy trees". The method defines a piecewise constant or linear approximation on general simplicial partitions. This talk considers an extension of this method to find a similar approximation on conforming simplicial partitions following an idea from a recent result together with Francesca Fierro and Andreas Veeser about near-best approximation on conforming triangulations.
ASCW03 17th June 2019
15:40 to 16:30
Claire Boyer Representer theorems and convex optimization
We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and elements of the extreme rays of the regularizer level sets. As a side result, we characterize the minimizers of the total gradient variation. As an ongoing work, we will also study the geometry of the total gradient variation ball.
This is a joint work with Antonin Chambolle, Yohann De Castro, Vincent Duval, Frédéric de Gournay, and Pierre Weiss.
ASCW03 18th June 2019
09:00 to 09:50
Simon Foucart  Functions of Few Coordinate Variables: Sampling Schemes and Recovery Algorithms
I will revisit in this talk the task of approximating multivariate functions that depend on only a few of their variables. The number of samples required to achieve this task to a given accuracy has been determined for Lipschitz functions several years ago. However, two questions of practical interest remain: can we provide an explicit sampling strategy and can we efficiently produce approximants? I will (attempt to) answer these questions under some additional assumptions on the target function. Firstly, if it is known to be linear, then the problem is exactly similar to the standard compressive sensing problem, and I will review some of recent contributions there. Secondly, if the target function is quadratic, then the problem connects to sparse phaseless recovery and to jointly low-rank and bisparse recovery, for which some results and open questions will be presented. Finally, if the target function is known to increase coordinatewise, then the problem reduces to group testing, from which I will draw the sought-after sampling schemes and recovery algorithms.
ASCW03 18th June 2019
09:50 to 10:40
Alexander Olevskii Discrete translates in function spaces
Given a Banach function space on R^n,  does there exist a uniformly discrete set of translates of a single function, which spans the space? I'll present a survey on the problem and discuss recent results, joint with A.Ulanovskii.
ASCW03 18th June 2019
11:10 to 12:00
Olga Mula Optimal algorithms for state estimation using reduced models
ASCW03 18th June 2019
13:30 to 14:20
Joaquim Ortega-Cerdà A sequence of well-conditioned polynomials

We find an explicit sequence of polynomials of arbitrary degree with
small condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993.
This is joint work together with Carlos Beltran, Ujué Etayo and Jordi Marzo.

ASCW03 18th June 2019
14:20 to 15:10
Holger Rauhut Linear and one-bit compressive sensing with subsampled random convolutions
Compressive sensing predicts that sparse vectors can recovered from incomplete linear measurements with efficient algorithms in a stable way. While many theoretical results work with Gaussian random measurement matrices, practical applications usually demand for structure. The talk covers the particular case of structured random measurements defined via convolution with a random vector and subsampling (deterministic or random as well). We will give an overview on the corresponding theory and will cover also recent results concerning recovery from one-bit measurements arising in quantized compressive sensing.
Based on joint works with Felix Krahmer, Shahar Mendelson, Sjoerd Dirksen and Hans-Christian Jung.
ASCW03 18th June 2019
15:40 to 16:30
Lukas Herrmann Quasi-Monte Carlo integration in uncertainty quantification of elliptic PDEs with log-Gaussian coefficients

Quasi-Monte Carlo (QMC) rules are suitable to overcome the curse of dimension in the numerical integration of high-dimensional integrands.
Also the convergence rate of essentially first order is superior to Monte Carlo sampling.
We study a class of integrands that arise as solutions of elliptic PDEs with log-Gaussian coefficients.
In particular, we focus on the overall computational cost of the algorithm.
We prove that certain multilevel QMC rules have a consistent accuracy and computational cost that is essentially of optimal order in terms of the degrees of freedom of the spatial Finite Element
discretization for a range of infinite-dimensional priors.
This is joint work with Christoph Schwab.

References:
[L. Herrmann, Ch. Schwab: QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights, Numer. Math. 141(1) pp. 63--102, 2019],
[L. Herrmann, Ch. Schwab: Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients, to appear in ESAIM:M2AN],
[L. Herrmann: Strong convergence analysis of iterative solvers for random operator equations, SAM report, 2017-35, in review]

ASCW03 19th June 2019
09:00 to 09:50
Andreas Seeger Basis properties of the Haar system in various function spaces

We present recent results on the Haar system in Besov and Triebel-Lizorkin spaces, with an emphasis on endpoint results.
Joint work with Gustavo Garrigós and Tino Ullrich.

ASCW03 19th June 2019
09:50 to 10:40
Christoph Thiele On singular Brascamp Lieb integrals
We will give an overview over singular
Brascamp Lieb integrals and discuss some
recent results based on recent arxiv
postings with Polona Durcik.
ASCW03 19th June 2019
11:10 to 12:00
Hans Feichtinger Approximation of continuous problems in Fourier Analysis by finite dimensional ones: The setting of the Banach Gelfand Triple
When it comes to the constructive realization of operators arising in Fourier Analysis, be it the Fourier transform itself, or some convolution operator, or more generally an (underspread) pseudo-diferential operator it is natural to make use of sampled version of the ingredients. The theory around the Banach Gelfand Triple (S0,L2,SO') which is based on methods from Gabor and time-frequency analysis, combined with the
relevant function spaces (Wiener amalgams and modulation spaces) allows to provide what we consider the appropriate setting and possibly the starting point for qualitative as well as later on more quantitative
error estimates.

ASCW03 20th June 2019
09:00 to 09:50
Sinan Güntürk Extracting bits from analog samples: in pursuit of optimality
Sampling theorems, linear or non-linear, provide the basis for obtaining digital representations of analog signals, but often it is not obvious how to quantize these samples in order to achieve the best rate-distortion trade off possible, especially in the presence of redundancy. We will present a general approach called "distributed beta encoding" which can achieve superior (and often near-optimal) rate-distortion performance in a wide variety of sampling scenarios. These will include some classical problems in the linear setting such as Fourier and Gabor sampling, and some others in the nonlinear setting, such as compressive sampling, spectral super-resolution, and phase retrieval.
ASCW03 20th June 2019
09:50 to 10:40
Peter Richtarik Stochastic Quasi-Gradient Methods: Variance Reduction via Jacobian Sketching
We develop a new family of variance reduced stochastic gradient descent methods for minimizing the average of a very large number of smooth functions. Our method—JacSketch—is motivated by novel de- velopments in randomized numerical linear algebra, and operates by maintaining a stochastic estimate of a Jacobian matrix composed of the gradients of individual functions. In each iteration, JacSketch efficiently updates the Jacobian matrix by first obtaining a random linear measurement of the true Jacobian through (cheap) sketching, and then projecting the previous estimate onto the solution space of a linear matrix equa- tion whose solutions are consistent with the measurement. The Jacobian estimate is then used to compute a variance-reduced unbiased estimator of the gradient, followed by a stochastic gradient descent step. Our strategy is analogous to the way quasi-Newton methods maintain an estimate of the Hessian, and hence our method can be seen as a stochastic q uasi-gradient method. Indeed, quasi-Newton methods project the current Hessian estimate onto a solution space of a linear equation consistent with a certain linear (but non-random) measurement of the true Hessian. Our method can also be seen as stochastic gradient descent applied to a controlled stochastic optimization reformulation of the original problem, where the control comes from the Jacobian estimates. We prove that for smooth and strongly convex functions, JacSketch converges linearly with a meaningful rate dictated by a single convergence theorem which applies to general sketches. We also provide a refined convergence theorem which applies to a smaller class of sketches, featuring a novel proof technique based on a stochastic Lyapunov function. This enables us to obtain sharper complexity results for variants of JacSketch with importance sampling. By specializing our general approach to specific sketching strategies, JacSketch reduces to the celebrated stochastic average gradient (SAGA) method, and

Co-authors: Robert Mansel Gower (Telecom ParisTech), Francis Bach (INRIA - ENS - PSL Research University)
ASCW03 20th June 2019
11:10 to 12:00
Boris Kashin On some theorems on the restriction of operator to coordinate subspace
ASCW03 20th June 2019
13:30 to 14:20
Yuan Xu Orthogonal structure in and on quadratic surfaces
Orthogonal structure in and on quadratic surfaces Text of abstract: Spherical harmonics are orthogonal polynomials on the unit sphere. They are eigenfunctions of the Laplace-Beltrami operator on the sphere and they satisfy an addition formula (a closed formula for their reproducing kernel). In this talk, we consider orthogonal polynomials on quadratic surfaces of revolution and inside the domain bounded by quadratic surfaces.  We will define orthogonal polynomials on the surface of a cone that possess both characteristics of spherical harmonics. In particular, the addition formula on the cone has a one-dimensional structure, which leads to a convolution structure on the cone useful for studying Fourier orthogonal series. Furthermore, the same narrative holds for orthogonal polynomials defined on the solid cones.
ASCW03 20th June 2019
14:20 to 15:10
Gitta Kutyniok Beating the Curse of Dimensionality: A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
High-dimensional parametric partial differential equations (PDEs) appear in various contexts including control and optimization problems, inverse problems, risk assessment, and uncertainty quantification. In most such scenarios the set of all admissible solutions associated with the parameter space is inherently low dimensional. This fact forms the foundation for the so-called reduced basis method.

Recently, numerical experiments demonstrated the remarkable efficiency of using deep neural networks to solve parametric problems. In this talk, we will present a theoretical justification for this class of approaches. More precisely, we will derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric PDEs. In fact, without any knowledge of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical approximation results. We use this low-dimensionality to guarantee the existence of a reduced basis. Then, for a large variety of parametric PDEs, we construct neural networks that yield approximations of the parametric maps not suffering from a curse of dimensionality and essentially only depending on the size of the reduced basis.

This is joint work with Philipp Petersen (Oxford), Mones Raslan, and Reinhold Schneider.
ASCW03 21st June 2019
09:00 to 09:50
Aicke Hinrichs How good is random information compared to optimal information?
We study approximation and integration problems and compare the quality of optimal information with the quality of random information. For some problems random information is almost optimal and for other problems random information is much worse than optimal information. We give a survey about known and new results. Parts of the talk are based on joint work with D. Krieg, E. Novak, J. Prochno and M. Ullrich.
ASCW03 21st June 2019
09:50 to 10:40
Karlheinz Groechenig Totally positive functions in sampling theory and time-frequency analysis
Totally positive functions play an important role in approximation theory and statistics. In this talk I will present recent new applications of totally positive functions (TPFs) in sampling theory and time-frequency analysis.   (i) We study the sampling problem for shift-invariant spaces generated by a TPF. These spaces arise the span of the integer shifts of a TPF and are often used as a substitute for bandlimited functions.   We give a complete characterization of sampling sets for a shift-invariant space with a TPF generator of Gaussian type in the style of Beurling.   (ii) A related problem is the question of Gabor frames, i.e., the spanning properties of time-frequency shifts of a given function. It is conjectured that the lattice shifts of a TPF generate a frame, if and only if the density of the lattice  exceeds 1. At this time this conjecture has been proved  for two important subclasses of TPFs. For  rational lattices it is true for arbitrary TPFs.  So far, TPFs seem to be the only window functions for which the fine structure of the associated Gabor  frames is tractable.   (iii) Yet another question in time-frequency analysis is the existence of zeros of the Wigner distribution (or the radar ambiguity function). So far all examples of zero-free ambiguity functions are related to TPFs, e.g., the ambiguity function of the Gaussian is zero free.
ASCW03 21st June 2019
11:10 to 12:00
Feng Dai Integral norm discretization and related problems
In this talk, we will discuss the problem of replacing an integral norm with respect to a given probability measure by the corresponding integral
norm with respect to a discrete measure. We study the problem for elements of finite dimensional spaces in a general setting, paying a
special attention to the case of the multivariate trigonometric polynomials with frequencies from a finite set with fixed cardinality.
Both new results and a survey of known results will be presented.

This is a joint work with A. Prymak, V.N. Temlyakov and S. Tikhonov.
ASCW03 21st June 2019
13:30 to 14:20
Milana Gataric Imaging through optical fibres
In this talk, I'll present some recent results on reconstruction of optical vector-fields using measurements acquired by an optical fibre characterised by a non-unitary integral transform with an unknown spatially-variant kernel. A new imaging framework will be introduced, which through regularisation is able to recover an optical vector-field with respect to an arbitrary representation system potentially different from the one used for fibre calibration. In particular, this enables the recovery of an optical vector-field with respect to a Fourier basis, which is shown to yield indicative features of increased scattering associated with tissue abnormalities. The effectiveness of this framework is demonstrated using biological tissue samples in an experimental setting where measurements are acquired by a fibre endoscope, and it is observed that indeed the recovered Fourier coefficients are useful in distinguishing healthy tissues from lesions in early stages of cancer. If time permits, I'll also briefly present a new method that enables recovery of non-unitary fibre transmission matrices necessary for minimally invasive optical imaging in inaccessible areas of the body.
ASCW03 21st June 2019
14:20 to 15:10
Geno Nikolov Markov-type inequalities and extreme zeros of orthogonal polynomials
The talk is centered around the problem of finding (obtaining  tight two-sided bounds for)  the sharp constants in certain Markov-Bernstein type inequalities in weighted $L_2$ norms. It turns out that, under certain assumptions, this problem is equivalent to the estimation of the extreme zeros of orthogonal polynomials with respect to a measure supported on $R_{+}$. It will be shown how classical tools like the Euler-Rayleigh method and Gershgorin circle theorem produce surprisingly good bounds for the extreme zeros of the Jacobi, Gegenbauer and Laguerre polynomials. The sharp constants in the $L_2$  Markov inequalities with the Laguerre and Gegenbauer weight functions and in a discrete $\ell_2$ Markov-Bernstein inequality are investigated using the same tool.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons