Zurich Colloquium in Applied and Computational Mathematics

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Archive 2019

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Speaker

Title

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27 February 2019
16:15-17:15

Prof. Dr. Benjamin Stamm
RWTH Aachen University

Event Details

Speaker invited by

Remi Abgrall

Abstract

In this talk we provide two examples of models and numerical methods involving N-body polarization effects. One characteristic feature of simulations involving molecular systems is that the scaling in the number N of atoms or particles is important and traditional computational methods, like domain decomposition methods for example, may behave differently than problems with a fixed computational domain.
We will first see an example in the context of the Poisson-Boltzmann continuum solvation model and present a numerical method that relies on an integral equation coupled with a domain decomposition strategy. Numerical examples illustrate the behaviour of the proposed method.
In a second case, we consider a N-body problem of interacting dielectric charged spheres whose solution satisfies an integral equation of the second kind. We present results from an a priori analysis with error bounds that are independent of the number particles N allowing for, in combination with the Fast Multipole Method (FMM), a linear scaling method. Towards the end, we finish the talk with applications to dynamic processes and enhanced stabilization of binary superlattices through polarization effects.

Efficient numerical methods for polarization effects in molecular systems

Y27 H25

6 March 2019
16:15-17:15

Dr. Felix Voigtlaender
Catholic University of Eichstätt-Ingolstadt

Event Details

Speaker invited by

Rima Alaifari

Abstract

We present a systematic approach towards understanding the sparsity properties of different frame constructions like Gabor systems, wavelets, shearlets, and curvelets.
We use the following terminology: Analysis sparsity means that the frame coefficients are sparse (in an \ell^p sense), while synthesis sparsity means that the function can be written as a linear combination of the frame elements using sparse coefficients. While these two notions are completely distinct for general frames, we show that if the frame in question is sufficiently nice, then both forms of sparsity of a function are equivalent to membership of the function in a certain decomposition space.
These decomposition spaces are a common generalization of Besov spaces and modulation spaces. While Besov spaces can be defined using a dyadic partition of unity on the Fourier domain, modulation spaces employ a uniform partition of unity, and general decomposition spaces use an (almost) arbitrary partition of unity on the Fourier domain.
To each decomposition space, there is an associated frame construction: Given a generator, the resulting frame consists of certain translated, modulated and dilated versions of the generator. These are chosen so that the frequency concentration of the frame is similar to the frequency partition of the decomposition space. For instance, Besov spaces yield wavelet systems, while modulation spaces yield Gabor systems.
We give conditions on the (possibly compactly supported!) generator of the frame which ensure that analysis sparsity and synthesis sparsity of a function are both equivalent to membership of the function in the decomposition space.

Understanding sparsity properties of frames using decomposition spaces

Y27 H25

13 March 2019
16:15-17:15

Prof. Dr. Ulrik Fjordholm
University of Oslo

Event Details

Speaker invited by

Siddhartha Mishra

Abstract

Hyperbolic conservation laws abound in the physical and engineering sciences, but their nonlinear and discontinuous nature necessitate numerical approximations. While the stability, compactness and convergence theory for these PDE is at a mature state, the available theory on convergence _rates_ is at best sub-optimal, and in many cases altogether lacking. In this talk I will describe some recent developments in this field. In particular, we argue that the Wasserstein distance is a natural metric in which to measure convergence. This is joint work with Susanne Solem.

Convergence rates of numerical approximations of hyperbolic conservation laws

Y27 H25

27 March 2019
16:15-17:15

Prof. Dr. Robert Scheichl
IWR, Uni Heidelberg

Event Details

Speaker invited by

Christoph Schwab

Abstract

Sample-based multilevel uncertainty quantification tools, such as multilevel Monte Carlo, multilevel quasi-Monte Carlo or multilevel stochastic collocation, have recently gained huge popularity due to their potential to efficiently compute robust estimates of quantities of interest (QoI) derived from PDE models that are subject to uncertainties in the input data (coefficients, boundary conditions, geometry, etc). Especially for problems with low regularity, they are asymptotically optimal in that they can provide statistics about such QoIs at (asymptotically) the same cost as it takes to compute one sample to the target accuracy. However, when the data uncertainty is localised at random locations, such as for manufacturing defects in composite materials, the cost per sample can be reduced significantly by adapting the spatial discretisation individually for each sample. Moreover, the adaptive process typically produces coarser approximations that can be used directly for the multilevel uncertainty quantification. In this talk, I will present two novel developments that aim to exploit these ideas. In the first part I will present Continuous Level Monte Carlo (CLMC), a generalisation of multilevel Monte Carlo (MLMC) to a continuous framework where the level parameter is a continuous variable. This provides a natural framework to use sample-wise adaptive refinement strategy, with a goal-oriented error estimator as our new level parameter. We introduce a practical CLMC estimator (and algorithm) and prove a complexity theorem showing the same rate of complexity as for MLMC. Also, we show that it is possible to make the CLMC estimator unbiased with respect to the true quantity of interest. Finally, we provide two numerical experiments which test the CLMC framework alongside a sample-wise adaptive refinement strategy, showing clear gains over a standard MLMC approach with uniform grid hierarchies. In the second part, I will show how to extend the sample-adaptive strategy to multilevel stochastic collocation (MLSC) methods providing a complexity estimate and numerical experiments for a MLSC method that is fully adaptive in the dimension, in the polynomial degrees and in the spatial discretisation.
This is joint work with Gianluca Detommaso (Bath), Tim Dodwell (Exeter) and Jens Lang (Darmstadt).

Multilevel Uncertainty Quantification with Sample-Adaptive Model Hierarchies

Y27 H 25

3 April 2019
16:15-17:15

Dr. Victorita Dolean Maini
University of Strathclyde Glasgow

Event Details

Speaker invited by

Stefan Sauter

Abstract

Solving wave propagation problems in harmonic regime is a very challenging task because of their indefinite nature and highly oscillatory solution when the wavenumber k is high. Although there have been different attempts to solve them efficiently, we believe that there is no established and robust preconditioner, whose behaviour is independent of k, for general decompositions into subdomains. Several attempts have been made in the literature, e.g. by Conen et al (2014) for Helmholtz problems with heterogeneous coefficients or Graham et al. (2017) for Helmholtz equations and later on extended to Maxwell’s equations with specific boundary conditions but the mechanism and the limits of applicability of such methods are far from being fully understood. In this talk we will present a few recent methods and illustrate their application on several difficult examples.

Robust preconditioners for time harmonic wave propagation problems

Y27 H 25

10 April 2019
16:15-17:15

Prof. Dr. Praveen Chandrasekhar
TIFR Bangalore, India

Event Details

Speaker invited by

Siddhartha Mishra

Abstract

Some PDE models like MHD and Maxwell's equations contain magnetic field as a dependent variable which must be divergence-free due to the non-existence of magnetic monopoles. This is an inherent constraint satisfied by the induction equation due to its curl structure. Numerical schemes may not preserve this structure unless they are specifically designed for this purpose. A staggered storage of variables is useful to satisfy such constraints by a numerical scheme. In this talk, I will describe two approaches to construct high order numerical approximations based on discontinuous Galerkin method that are constraint preserving. In the first approach, we perform a divergence-free reconstruction of the magnetic field while in the second approach, the divergence constraint is automatically satisfied by the numerical scheme due to the use of H(div) elements. The numerical flux used in such DG methods must satisfy a consistency condition between the 1-D and multi-D Riemann solvers, and we construct HLL-type schemes for MHD that exhibit such consistency. These methods are useful in applications where explicit time stepping schemes can be used and I will show some results for MHD and Maxwell's equations.

Divergence-free discontinuous Galerkin method for MHD and Maxwell's equations

Y27 H25

17 April 2019
16:15-17:15

Prof. Dr. Paul Ledger
College of Engineering, Swansea University

Event Details

Speaker invited by

Ralf Hiptmair

Abstract

Locating and identifying hidden conducting objects has a range of important applications in metal detection including searching for buried treasure, identifying land mines and in the early detection of concealed terrorist threats. There is a need to distinguish between multiple objects, for example, in benign situations, such as coins and keys accidentally left in a pocket during a security search or a treasure hunter becoming lucky and discovering a hoard of Roman coins, as well as in threat situations, where the risks need to be clearly identified from the background clutter. Furthermore, objects are also often inhomogeneous and made up of several different metals. For instance, the barrel of a gun is invariably steel while the frame could be a lighter alloy, jacketed bullets have a lead shot and a brass jacket and modern coins often consist of a cheaper metal encased in nickel or brass alloy. Thus, in practical metal detection applications, it is important to be able to characterise both multiple objects and inhomogeneous objects. Traditional approaches to the metal detection involve determining the conductivity and permeability distributions in the eddy current approximation of Maxwell's equations and lead to an ill-posed inverse problem. On the other hand, practical engineering solutions in hand held metal detectors use simple thresholding and are not able to discriminate between small objects close to the surface and larger objects buried deeper underground. In this talk, an alternative approach in which prior information about the form of the conducting permeable object has been introduced will be discussed.
Ammari, Chen, Chen, Garnier and Volkov [1] have obtained the leading order term in an asymptotic expansion of the perturbed magnetic field, due to the presence of a homogeneous conducting permeable object, as the object size tends to zero. This expansion separates the object's position from its shape and material description, offering considerable advantages in case of isolated objects. We have shown that this leading order term simplifies for orthonormal coordinates and results in a characterisation of a conducting permeable object by a complex symmetric rank 2 magnetic polarizability tensor (MPT) for the eddy current case [2]. Interestingly, the MPT is different to the symmetric rank 2 Poyla-Szegö polarizability tensor (also known as the Poyla-Szegö polarisation tensor) that is known to characterise small permeable objects in magnetostatics and small conducting objects in electrical impedance tomography [3]. For instance, computing the coefficients of the MPT rely on the solution of vectorial curl-curl transmission problems while the latter on simpler scalar transmission problems. The topology ofan object plays an interesting role in the MPT coefficients. Including more terms in the asymptotic expansion increases the accuracy of the representation of the perturbed field. The higher order terms contain higher order tensors, which provide more information about the shape and material parameters of an object, and can help to improve object identification. Complete asymptotic expansions of the perturbed field for the electrical impedance tomography problem have been obtained by Ammari and Kang [3] and provide acomplete characterisation of an object by generalised polarisation tensors. For the eddy current case, we have extended the leading order term obtained in [1] to a complete expansion of the perturbed magnetic field. The higher order terms in our expansion characterise a conducting permeable object in terms of a new class of generalised magnetic polarizability tensors [4], of which the rank 2 MPT is the simplest case. Practical metal detection problems contain multiple and inhomogeneous objects. We have also provided an extension of [1, 2] to characterise multiple inhomogeneous objects, including objectsthat are closely spaced, in terms of MPTs [5].
The talk will review recent work on MPTs and explore the interesting properties exhibited by these tensors for shapes and topologies of objects. The role that a dictionary of MPTs for different shaped objects can play in object classification will also be described.
References:
[1] H. Ammari, J. Chen, Z. Chen, J. Garnier and D. Volkov, Target detection and characterizationfrom electromagnetic induction data, J. Math. Pure Appl. 2014: 101: 54-75.
[2] P.D. Ledger and W.R.B. Lionheart, Characterising the shape and material properties of hidden targets from magnetic induction data, IMA J. Appl. Math. 2015: 80: 1776-1798.
[3] H. Ammari and H. Kang, Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory, New-York, Springer, 2007.
[4] P.D. Ledger and W.R.B. Lionheart, Generalised magnetic polarizability tensors, Math. Meth. Appl. Sci., 2018: 41: 3175-3196.
[5] P.D. Ledger, W.R.B. Lionheart and A.A.S. Amad, Characterisation of multiple conducting permeable objects in metal detection by polarizability tensors, Math. Meth. Appl. Sci., 2019:42: 830-860.

Characterisation of multiple inhomogeneous conducting objects in metal detection using magnetic polarizability tensors

Y27 H 25

24 April 2019
16:15-17:15

Prof. Dr. Giovanni S. Alberti
University of Genova

Event Details

Speaker invited by

Rima Alaifari

Abstract

In this talk I will discuss how ideas from applied harmonic analysis, in particular sampling theory and compressed sensing (CS), may be applied to inverse problems in PDEs. The focus will be on inverse boundary value problems for the conductivity and the Schrodinger equations, and I will give uniqueness and stability results, both in the linearized and in the nonlinear case. These results make use of a recent general theory of infinite-dimensional CS for deterministic and non-isometric operators, which will be briefly surveyed. This is joint work with Matteo Santacesaria.

Infinite-dimensional inverse problems with finite measurements

Y27 H 25

15 May 2019
16:15-17:15

Prof. Dr. Arnulf Jentzen
ETH Zurich

Event Details

Speaker invited by

Siddhartha Mishra

Abstract

Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. For example, stochastic PDEs are a fundamental ingredient in models for nonlinear filtering problems in chemical engineering and weather forecasting, deterministic Schroedinger PDEs describe the wave function in a quantum physical system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research to describe optimal control problems where companys aim to minimise their costs, and deterministic Black-Scholes-type PDEs are highly employed in portfolio optimization models as well as in state-of-the-art pricing and hedging models for financial derivatives. The PDEs appearing in such models are often high-dimensional as the number of dimensions, roughly speaking, corresponds to the number of all involved interacting substances, particles, resources, agents, or assets in the model. For instance, in the case of the above mentioned financial engineering models the dimensionality of the PDE often corresponds to the number of financial assets in the involved hedging portfolio. Such PDEs can typically not be solved explicitly and it is one of the most challenging tasks in applied mathematics to develop approximation algorithms which are able to approximatively compute solutions of high-dimensional PDEs. Nearly all approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula. In this talk we introduce a new nonlinear Monte Carlo algorithm for high-dimensional nonlinear PDEs. We prove that this algorithm does indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic PDE with a nonlinearity depending on the PDE solution can be solved approximatively without the curse of dimensionality.

Overcoming the curse of dimensionality: from nonlinear Monte Carlo to deep artificial neural networks

Y27 H 25

22 May 2019
16:15-17:15

Prof. Dr. Stefan Steinerberger
Yale University

Event Details

Speaker invited by

Rima Alaifari

Abstract

I will discuss the classical problem of finding good sampling points (say, for the purpose of numerical integration, approximation or interpolation) on Riemannian manifolds (with special focus on the sphere and the torus) and finite graphs. This has beautiful connections to classical Harmonic Analysis, Analytic Number Theory and Combinatorics. On finite Graphs we use an abstract spectral definition, one that can be used to construct the Platonic solids in IR^3, to recover special subsets of vertices ("the Platonic Bodies inside a Graph"). These objects are quite mysterious but very beautiful (many pictures provided) and I will discuss some open problems associated with them.

Sampling on Manifolds and Finite Graphs

Y27 H25

18 September 2019
16:15-17:15

Prof. Dr. Erik Burman
University College London

Event Details

Speaker invited by

Christoph Schwab

Abstract

In this talk we will consider some ill-posed elliptic equations and their discretization using Finite Element Methods. The standard approach to ill-posed problems is to regularize the continuous problem so that existence and uniqueness is guaranteed. The regularized problem can then be solved using standard Finite Element Methods. When using this strategy, in order to optimize accuracy, the regularization parameter must be chosen as a function both of the stability properties of the ill-posed problem, the mesh parameter and perturbations in data. Here we will propose a different approach [1], where the ill-posed pde is discretized in an optimization framework, prior to regularization. To ensure discrete well-posedness we add stabilizing terms to the formulation, drawing on experience from stabilized FEM and discontinuous Galerkin methods. The error in the resulting Finite Element reconstructions is then analyzed using Carleman estimates on the continuous problem. This results in approximations that are optimal with respect to the approximation order of the Finite Element space and the stability of the computed quantity. The mesh parameter here plays the role of the regularization parameter. Mesh resolution can be chosen independently of the stability properties of the physical problem, but must match perturbations in data, in a way made explicit in the estimates. Some examples of problems analyzed in this framework will be presented, selectedfrom recent work on the Helmholtz equation [4], the convection diffusion equation [5], Stokes'equations [2] and Darcy's equation [3].
Keywords: ill-posed problems, data assimilation, stabilized Finite Element Methods
Mathematics Subject Classifications (2010): 65N30, 35R25, 65N20.
[1] E. Burman. Stabilised Finite Element Methods for ill-posed problems with conditional stability. Building bridges: connections and challenges in modern approaches to numerical partial differential equations, 93127,Lect. Notes Comput. Sci. Eng., 114, Springer, [Cham], 2016., Dec. 2015.
[2] E. Burman, P. Hansbo, Stabilized nonconforming Finite Element Methods for data assimilation in incompressible flows. Math. Comp. 87, no. 311, 2018.
[3] E. Burman, M. G. Larson, L. Oksanen. Primal dual mixed Finite Element Methods for the elliptic Cauchy problem, arXiv:1712.10172, Siam J. Num. Anal., to appear, 2018.
[4] E. Burman, M. Nechita, L. Oksanen. Unique continuation for the Helmholtz equation using stabilized Finite Element Methods. Journal de Mathematiques Pures et Appliquees,https://doi.org/10.1016/j.matpur.2018.10.003.
[5] E. Burman, M. Nechita, L. Oksanen. A stabilized Finite Element Method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime. arXiv:1811.00431, 2018.

Ill-posed problems and stabilized Finite Element Methods

HG E 1.2

25 September 2019
16:15-17:15

Prof. Dr. Wolfgang Hackbusch
Max-Planck-Institut für Mathematik

Event Details

Speaker invited by

Stefan Sauter

Abstract

Exponential sums consist of the terms a_i*exp(-b_ix). Such approximations for the functions 1/x and 1/sqrt(x) are of particular interest. We give examples of their use in quantum chemistry and tensor calculus. The best approximation of 1/x with respect to the maximum norm is theoretically well understood, and rather sharp error estimates are known. The error decays exponentially in the number of terms. The optimal exponential sum is characterised by the equioscillation property. The Remez algorithm is the standard method in the case of polynomial approximation. The existing literature about the numerical computation of the best approximation by exponential sums shows that all authors faced severe numerical difficulties. This may lead to the wrong impression that the problem is illposed. In the lecture a stable method is described which is used to compute best approximations for various parameters up to 57 terms. Literature: W. Hackbusch: Computation of best L^{\infty} exponential sums for 1/x by Remez' algorithm. Comput. Vis. in Sci. (2019) 20:1-11

Computation of Best Exponential Sums Approximating 1/x in the Maximum Norm

HG E 1.2

2 October 2019
16:15-17:15

Dr. Carolina Urzua-Torres
University of Oxford

Event Details

Speaker invited by

Ralf Hiptmair

Abstract

Space-time discretization methods are becoming increasingly popular, since they allow adaptivity in space and time simultaneously, and can use parallel iterative solution strategies for time-dependent problems. However, in order to exploit these advantages, one needs to have a complete numerical analysis of the corresponding Galerkin methods. Different strategies have been used to derive variational methods for the time domain boundary integral equations for the wave equation. The more established and succesful ones include weak formulations based on the Laplace transform, and also time-space energetic variational formulations. However, their corresponding numerical analyses are still incomplete and present difficulties that are hard to overcome, if possible at all. As an alternative, we pursue a new approach to formulate the boundary integral equations for the wave equation, which aims to provide the missing mathematical analysis for space-time boundary element methods. In this talk, I will give a short introduction to boundary element methods; briefly explain the current formulations for the wave equation; and discuss the new approach and our preliminary results.

A new approach to Space-Time Boundary Integral Equations for the Wave Equation

HG E 1.2

7 October 2019
16:15-17:15

Prof. Dr. Gitta Kutyniok
TU Berlin

Event Details

Speaker invited by

Christoph Schwab

Abstract

Inverse problems in imaging such as denoising, recovery of missing data, or the inverse scattering problem appear in numerous applications. However, due to their increasing complexity, model-based methods are often today not sufficient anymore. At the same time, we witness the tremendous success of data-based methodologies, in particular, deep neural networks for such problems. However, pure deep learning approaches often neglect known and valuable information from the modeling world and also currently still lack a profound theoretical understanding. In this talk, we will provide an introduction to this problem complex and then focus on the inverse problem of computed tomography, where one of the key issues is the limited angle problem. For this problem, we will demonstrate the success of hybrid approaches. We will develop a solver for this severely ill-posed inverse problem by combining the model-based method of sparse regularization by shearlets with the data-driven method of deep learning. Our approach is faithful in the sense that we only learn the part which cannot be handled by model-based methods, while applying the theoretically controllable sparse regularization technique to all other parts. We further show that our algorithm significantly outperforms previous methodologies, including methods entirely based on deep learning. Finally, we will discuss how similar ideas can also be used to solve related problems such as detection of wavefront sets.

Deep Learning meets Modeling: Taking the Best out of Both Worlds

HG G 19.2

30 October 2019
16:15-17:15

Prof. Dr. Ian Hawke
University of Southhampton

Event Details

Speaker invited by

Siddhartha Mishra

Abstract

In late 2017 the merger of two neutron stars was detected both in gravitational and electromagnetic waves. This gave unprecedented information on the behaviour of objects more massive than the sun, squashed into the size of a city, with magnetic fields many orders of magnitude larger than Earth's. As more detections come in, we will need to significantly improve the models we use to describe these events, and the numerical methods used to simulate them. Here I will discuss the difficulties in coupling Einstein's theory of General Relativity to ever more complex matter models, with a focus on recent multi-phase descriptions of charged species needed to describe non-ideal elastic and magnetic behaviour. This necessarily requires non-conservative numerical methods for the matter, numerically enforced constraints for the magnetic fields, and adaptive techniques to cover the large range of scales involved.

Numerical simulations of Gravitational Waves from Neutron Stars

HG E 1.2

4 December 2019
16:15-17:15

Gregor Gassner
Universität zu Köln

Event Details

Speaker invited by

Remi Abgrall

Abstract

In this talk, we show how to construct a discontinuous Galerkin type discretisation of the ideal MHD equations based on first principles. By carefully choosing the form of the PDEs (divergence, advective, splitform, etc) it is possible to design a compatible discretisation where e.g. kinetic energy is preserved, with the right Lorentz force behavior, where we recover a discrete entropy evolution and where zero divergence of the B field is satisfied discretely. We will demonstrate these properties for a 3D ideal MHD test case simulated with the open source framework FLUXO (github.com/project-fluxo).

Splitform Discontinuous Galerkin for the ideal MHD equations: energy, Lorentz force, entropy and divergence B

HG E 1.2

11 December 2019
16:15-17:15

Prof. Dr. Steffen Börm
Christian-Albrechts-Universität zu Kiel

Event Details

Speaker invited by

Ralf Hiptmair

Abstract

We consider the computation of electrostatic potentials by the boundary element method. In order to obtain O(h²) convergence of discrete solutions, we have to employ piecewise linear basis functions and piecewise quadratic parametrizations of the surface. Constructing the data-sparse approximations of the integral operators required for high accuracies poses several challenges: methods like ACA or GCA require the computation of individual matrix entries, and since the supports of basis functions are spread across multiple triangles, this computation is far more computationally expensive than for simple discontinuous basis functions. Alternative techniques like HCA allow us to significantly reduce the computational work. Another challenge is the parametrization of the curved triangles: the Gramian and the normal vector are no longer constant on each triangle, but have to be computed in each quadrature point, and this increases the necessary work even further. Combining efficient quadrature techniques with HCA matrix compression, algebraic coarsening and recompression, and Krylov solvers allows us to handle surface meshes with up to 18 million triangles on relatively affordable servers while preserving the theoretically predicted convergence rate of the underlying discretization scheme.

Fast Boundary Element Methods for Electrostatic Field Computations

HG E 1.2

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