Zurich Colloquium in Applied and Computational Mathematics

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Archive 2016

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Speaker

Title

Location

2 March 2016
16:15-17:15

Prof. Dr. Alexandre Ern
Université Paris-Est, CERMICS

Event Details

Speaker invited by

Remi Abgrall

Abstract

We introduce a quasi-interpolation operator for scalar- and vector-valued finite element spaces with some continuity across mesh interfaces. This operator is stable in L1, leaves the corresponding finite element space point-wise invariant, whether homogeneous boundary conditions are imposed or not, and, assuming regularity in the fractional Sobolev spaces W^{s,p} where p\in [1,\infty] and s can be arbitrarily close to zero, gives optimal local approximation estimates in any Lp-norm. The theory is illustrated on H1-, H(curl)- and H(div)-conforming spaces.

Finite element quasi-interpolation and best approximation

9 March 2016
16:15-17:15

Prof. Dr. John Ball
University of Oxford

Event Details

Speaker invited by

Habib Ammari

Abstract

When a new phase is nucleated in a martensitic solid phase transformation, it has to fit geometrically onto the parent phase, forming interfaces between the phases accompanied by possibly complex microstructure. The talk will describe some mathematical issues involved in understanding such questions of compatibility and their influence on metastability, as illustrated by recent experimental discoveries. For liquid crystals planar (as opposed to point and line) defects are not usually considered, but there are some situations in which they seem to be relevant, such as for smectic A thin films where compatibility issues not unlike those for martensitic materials arise.

Interfaces and metastability in solid and liquid crystals

16 March 2016
16:15-17:15

Dr. Holger Heumann
INRIA Sophia-Antipolis

Event Details

Speaker invited by

Ralf Hiptmair

Abstract

The Grad/Hogan model for plasma equilibrium evolution at the resistive diffusion timescale separates into two low-dimensional subproblems: The axisymmetric free-boundary plasma equilibrium problem and the one-dimensional system of transport and diffusion equations. The equilibrium problem, also known as the Grad-Shafranov equation, is a non-linear elliptic problem for the poloidal flux function. The transport and diffusion equations, basically hydrodynamic equations and resistive diffusion formulated in the curvilinear coordinate system induced by the level lines of the poloidal flux function, are non-linear advection-diffusion equations. The unknowns of the system of transport and diffusion equations determine in some non-linear fashion the profile of the toroidal current density, the non-linear righthand side of the Grad-Shafranov equation and the poloidal flux on the other hand determines coefficients in the system of transport and diffusion equations. Devising stable numerical methods for a self-consistent simulation of equilibrium and transport and diffusion is an active area of research with many open problems, but highly important for scenario development and realtime control in experimental tokamak devices such as ITER. We will present the details of the Grad/Hogan model with a focus on numerical solutions methods towards automated scenario development.

Numerical methods for tokamak plasma equilibrium evolution at the resistive diffusion timescale

17 March 2016
15:15-16:15

Prof. Dr. Philippe Ciarlet
City University of Hong Kong

Event Details

Speaker invited by

Stefan Sauter

Abstract

A nonlinear Korn inequality on a surface asserts that the distance, measured by means of an appropriate norm, between a surface and a deformed surface is, up to rigid body motions, "controlled" by the distances between their fundamental forms, likewise measured by means of appropriate norms. In this talk, we review various recently established nonlinear Korn inequalities on a surface, either in spaces of continuously differentiable functions or in Sobolev spaces. We also briefly discuss some of their potential applications beyond differential geometry per se, such as the intrinsic approach in nonlinear shell theory, or the modelling of the Earth surface. References: P.G. Ciarlet: the continuity of a surface as a function of its two fundamental forms, J. Math. Pures Appl. 82 (2003), 253-274. P.G. Ciarlet, C. Mardare: Recovery of a surface with boundary and its continuity as a function of its two fundamental forms, Anal. Appl. 3 (2005), 99-1117. P.G. Ciarlet, L. Gratie, C. Mardare: A nonlinear Korn inequality on a surface, J. Math. Pures Appl. 85 (2006), 2-16. P.G. Ciarlet, M. Malin, C. Mardare: In preparation.

Nonlinear Korn inequalities on a surface

Y27 H 12

6 April 2016
16:15-17:15

Prof. Dr. Frédéric Coquel
CMAP - Ecole polytechnique

Event Details

Speaker invited by	Remi Abgrall
Abstract	+++Cancelled+++!

+++Cancelled+++!

11 April 2016
16:15-17:15

Prof. Dr. Ian Sloan
Department of Mathematics, University of New South Wales, Sydney, Australia

Event Details

Speaker invited by

Christoph Schwab

Abstract

This talk describes recent computational developments in partial differential equations with random coefficients treated as a high-dimensional problem. The prototype of such problems is the underground flow of water or oil through a porous medium, with the permeability of the material treated as a random field. (The stochastic dimension of the problem is high if the random field needs a large number of random variables for its effective description.). There are many approaches to the problem, ranging from the polynomial chaos method initiated by Norbert Wiener to the Monte Carlo and (of particular interest to the UNSW group) Quasi-Monte Carlo methods. In recent years there have been significant progress in the development and analysis of algorithms in these areas.

PDE with random coefficients as a high-dimensional problem

13 April 2016
17:30-18:30

Prof. Dr. Youssef Marzouk
MIT

Event Details

Speaker invited by

Christoph Schwab

Abstract

Bayesian inference provides a natural framework for quantifying uncertainty in model parameters and predictions, and for combining heterogeneous sources of information. But the computational demands of the Bayesian framework constitute a major bottleneck in large-scale applications. We will discuss how transport maps, i.e., deterministic couplings between probability measures, can enable useful new approaches to Bayesian computation. A first use involves a combination of measure transport and Metropolis correction; here, we use continuous transportation to transform typical MCMC proposals into adapted non-Gaussian proposals, both local and global. Second, we discuss a variational approach to Bayesian inference that constructs a deterministic transport from a reference distribution to the posterior, without resorting to MCMC. Independent and unweighted posterior samples can then be obtained by pushing forward reference samples through the map. Making either approach efficient in high dimensions, however, requires identifying and exploiting low-dimensional structure. We present new results relating sparsity of transport maps to the conditional independence structure of the target distribution, and discuss how this structure can be revealed through the analysis of certain average derivative functionals. A connection between transport maps and graphical models yields many useful algorithms for efficient ordering and decomposition---here, generalized to the continuous and non-Gaussian setting. The resulting inference algorithms involve either the direct identification of sparse maps or the composition of low-dimensional maps and rotations. We demonstrate our approaches on Bayesian inference problems arising in spatial statistics and in partial differential equations. This is joint work with Matthew Parno and Alessio Spantini. Please note that this is a Colloquium of the Computational Science Zurich Distinguished Lecture.

Bayesian inference and the low-dimensional structure of measure transport

KOL G201 UZH

20 April 2016
16:15-17:15

Prof. Dr. Daniele A. Di Pietro
University of Montpellier

Event Details

Speaker invited by

Remi Abgrall

Abstract

We develop a family of arbitrary-order primal methods for (possibly degenerate) diffusion problems on general polygonal/polyhedral meshes. The degrees of freedom are scalar-valued polynomials of the same order at mesh elements and faces. The cornerstones of the method are local reconstruction operators defined at the element level. The design of the method additionally hinges on a least-squares penalty term on faces weakly enforcing the matching between local element- and face-based degrees of freedom. We first address the pure diffusion case with regular to lie the cornerstones of the method. In the lowest-order case, equivalence with the Hybrid Finite Volume method is shown. A first extension is to variable diffusion problems, where the diffusion coefficient is embedded in the construction of the discrete gradient. A second, more challenging, extension is to possibly degenerate advection-diffusion problems, where the advection term is discretized by means of a discrete reconstruction of the advective derivative and upwind stabilization. Here, the main difficulty is to ensure consistency of the method when the exact solution jumps at the diffusive/advective interface. The theoretical results are confirmed by numerical experiments on both standard and polygonal meshes.

Hybrid High-Order methods on general meshes

27 April 2016
16:15-17:15

Prof. Dr. Barbara Kaltenbacher
University of Klagenfurt

Event Details

Speaker invited by

Habib Ammari

Abstract

Parameter identification problems typically consist of a model equation, e.g. a (system of) ordinary or partial differential equation(s), and the observation equation. In the conventional reduced setting, the model equation is eliminated via the parameter-to-state map. Alternatively, one might consider both sets of equations (model and observations) as one large system, to which some regularization method is applied. The choice of the formulation (reduced or all-at-once) can make a large difference computationally, depending on which regularization method is used: Whereas almost the same optimality system arises for the reduced and the all-at-once Tikhonov method, the situation is different for iterative methods, especially in the context of nonlinear models. In this talk we will exemplarily provide some convergence results for all-at-once versions of variational, Newton type and gradient based regularization methods. Moreover we will compare the implementation requirements for the respective all-at-one and reduced versions, as well as their computational performance by means of some numerical tests.

Regularization based on all-at-once formulations for inverse problems

4 May 2016
16:15-17:15

Prof. Dr. Gabriel Lord
Heriot-Watt University

Event Details

Speaker invited by

Remi Abgrall

Abstract

We examine new numerical methods to approximate SPDEs and discuss both convergence and efficiency. We are particularly interested in the time discretisation of multiplicative noise. Our techniques are primarily based on approximating the mild solution of the SPDE where we can try and exploit exact solutions in the numerics. Proofs of convergence are for globally Lipschitz nonlinearities. In the case where this condition does not hold, rather than tamed methods, we examine instead using an adaptive timestep for the SPDE. We take as applications SPDEs aising from models of neural and also from models of reactive single phase flow in a porous media.

Efficient numerical simulation of SPDEs

11 May 2016
16:15-17:15

Prof. Dr. Bruno Despres
Laboratoire Jacques Louis Lions, Univ. Paris VI

Event Details

Speaker invited by

Siddhartha Mishra

Abstract

Resonant Maxwell's equations are relevant for the heating of a plasma in a Tokamak. An important difficulty is the mathematical theory of these equations, which shows the possibility of a physical solution in the form of a Dirac mass inside the domain. I will draw a parallel between the search of good entropies for hyperbolic systems and the limit absorption principle. In this setting good entropies become manufactured solutions. It yields original weak formulations of the problem, which are well posed (existence and uniqueness of the solution) in convenient functional spaces.

New weak formulations for resonant Maxwell's equations

18 May 2016
16:15-17:15

Prof. Dr. Ilaria Perugia
Universitaet Wien

Event Details

Speaker invited by

Ralf Hiptmair

Abstract

The virtual element method (VEM) is a generalisation of the finite element method recently introduced by Beirao da Veiga, Brezzi, Cangiani, Manzini, Marini and Russo in 2013, which takes inspiration from modern mimetic finite difference schemes, and allows one to use very general polygonal/polyhedral meshes. This talk is concerned with a new method based on inserting plane wave basis functions within the VEM framework in order to construct an H1-conforming, high-order method for the discretisation of the Helmholtz problem, in the spirit of the partition of unity method. The main ingredients of this plane wave VEM scheme (PW-VEM) are: i) a low order VEM space whose basis functions, which form a partition of unity and are associated to the mesh vertices, are not explicitly computed in the element interiors; ii) a local projection operator onto the plane wave space; iii) an approximate stabilization term. The PW-VEM will be derived, and an outline of its convergence analysis will be presented, as well as some numerical tests. These results have been obtained in collaboration with Paola Pietra (IMATI-CNR "E. Magenes'', Pavia, Italy) and Alessandro Russo (Università di Milano Bicocca, Milano, Italy).

A plane wave virtual element method for the Helmholtz problem

25 May 2016
16:15-17:15

Dr. Xavier Claeys
LJLL, UPMC, Paris

Event Details

Speaker invited by

Ralf Hiptmair

Abstract

Local Multi-Trace Formulations (local MTF) are block-sparse boundary integral equations adapted to elliptic PDEs with piece-wise constant coefficients (typically multi-subdomain scattering problems) only recently introduced in [Hiptmair & Jerez-Hanckes, 2012]. In these formulations, transmission conditions are enforced by means of local operators, so that only adjacent subdomains communicate. Although they provide an appealing framework for domain decomposition, present literature only offers two contributions in this direction. In [Hiptmair, Jerez-Hanckes, Lee, Peng, 2013] a new version of local MTF is proposed that involves a relaxation parameter in the enforcement of transmission conditions. In [Dolean & Gander, 2014] the authors conduct a basic explicit study of this modified local MTF in a 1-D setting with 2 subdomains and determine a critical value for the relaxation parameter that minimises the spectral radius of block-Jacobi iteration operators. In the present talk, we describe new contributions extending these results to arbitrary geometrical settings in 2-D and 3-D, assuming that the subdomain partition does not involve any junction point.

Analysis of Block-Jacobi Preconditioners for Local Multi-Trace Formulations

28 September 2016
16:15-17:15

Prof. Dr. John Butcher
University of Auckland, New Zealand

Event Details

Speaker invited by

Siddhartha Mishra

Abstract

A general linear method is "G-symplectic" if a certain algebraic condition is satisfied. Many examples of these methods are known and their behaviour is now well understood, both theoretically and experimentally. The focus is now on deriving high order methods in the anticipation that they will provide accurate and efficient integration schemes for mechanical and other problems. One of the starting points is an analysis of the order conditions (Butcher, J., Imran, G., Order conditions for G-symplectic methods, BIT, 55 (2015), 927-948. It was shown that the order conditions are related to unrooted trees in a similar way to what is known for symplectic Runge-Kutta methods (Sanz-Serna J. M., Abia L., Order conditions for canonical Runge-Kutta schemes, SIAM J. Numer. Anal. 28, 1081-1096 (1991)). Starting from the order conditions, simplifications can be made by assuming time-reversal symmetry and enhanced stage order. Even after high order methods have been found, the construction of suitable starting schemes is usually an essential step before working algorithms can be built. But sometimes this can be avoided.

The construction of high order G-symplectic methods

5 October 2016
16:15-17:15

Prof. Dr. Martin Burger
Universität Münster

Event Details

Speaker invited by

Habib Ammari

Abstract

Variational techniques in the regularization of inverse problems have evolved to become a standard tool in the field. In particular in image reconstruction, the use of nonquadratic regularization functionals (and possibly data fidelities) such as sparsity-promoting l1-minimization or edge-enhancing techniques based on total variation made enormous impact. Consequently novel questions with respect to the quantification of uncertainties were raised, questions of particular relevance being the relation to appropriate Bayesian prior and posterior models on the one hand and the estimation of errors caused by random noise in the data. In this talk we will discuss several recent developments in these problems in important case of convex regularization functionals (log-concave priors in the Bayesian setup), based on the use of Bregman distances and other dual error measures. We discuss a quite general approach to estimate errors in the solutions of the variational regularization methods, which is based on duality techniques for convex optimization and can treat large (unbounded) noise. As a direct consequence we obtain estimates on the expected error for a setup with white noise in the data. Moreover, we discuss the characterization of the minimizer as a maximum a-posteriori probability (MAP) estimate for an appropriate model. For this sake we introduce the novel concept of weak MAP estimates and relate those to minimizers of a natural Bayes cost. This talk is based on joint work with Tapio Helin (Helsinki), Felix Lucka (UCL) and Hanne Kekkonen (Warwick)

Uncertainty Quantification in the Variational Regularization of Inverse Problems

12 October 2016
16:15-17:15

Prof. Dr. Gianluigi Rozza
SISSA, Triest

Event Details

Speaker invited by

Christoph Schwab

Abstract

In this talk we deal with the state of the art of Reduced Order Methods (ROM) for parametric Partial Differential Equations (PDEs) and we provide some perspectives in their current trends and developments, with a special interest in Computational Fluid Dynamics (CFD) parametric problems. Systems modelled by PDEs are depending by several complex parameters in need of being reduced, even before the computational phase in a pre-processing step. Efficient parametrizations (random inputs, geometry, physics) are very important to be able to properly address an offline-online decoupling of the computational procedures and to allow competitive computational performances. Current ROM developments include: a better use of high fidelity methods, also spectral element method, enhancing the quality of the reduced model too; more efficient sampling techniques to reduce the number of the basis functions, retained as snapshots, and the dimension of online systems; the improvements of the certification of accuracy based on residual based error bounds and stability factors; for nonlinear system also investigations on bifurcations of parametric solutions is crucial and it may be obtained thanks to a reduced eigenvalue analysis. All the previous aspects are very important in CFD problems in order to be able to study complex industrial and biomedical flow problems in real time, and to couple viscous flows -velocity, pressure, and also thermal field - with a structural field or a porous medium. This last task requires also an efficient reduced parametric treatment of interfaces.

Reduced Order Methods: state of the art and perspectives: focus on Computational Fluid Dynamics

19 October 2016
16:15-17:15

Dr. Mario Hefter
University of Kaiserslautern, Germany

Event Details

Speaker invited by

Arnulf Jentzen

Abstract

In recent years, strong (pathwise) approximation of stochastic differential equations (SDEs) has intensively been studied for SDEs of the form \begin{align*} \mathrm{d} X_t = (a-bX_t)\mathrm{d} t + \sigma\sqrt{X_t}\mathrm{d} W_t, \end{align*} with a scalar Brownian motion $W$ and parameters $a,\sigma>0$, $b\in\mathbb{R}$. These SDEs are, e.g., used to describe the volatility in the Heston model and the interest rate in the Cox-Ingersoll-Ross model. In the particular case of $b=0$ and $\sigma=2$ the solution is a squared Bessel process of dimension $a$. We propose a tamed Milstein scheme $Y^N$, which uses $N\in\mathbb N$ values of the driving Brownian motion $W$, and prove positive polynomial convergence rates for all parameter constellations. More precisely, we show that for every $1\leq p<\infty$ and every $\varepsilon>0$ there exists a constant $C>0$ such that \begin{align*} \sup_{0\leq t\leq 1}\left( E{ |X_t-Y_t^N|^p } \right)^{1/p} \leq C\cdot \frac{1}{N^{\min(1,\delta)/(2p)-\varepsilon}} \end{align*} for all $N\in\mathbb N$, where $\delta = {4a}/{\sigma^2}$. This is joint work with Andrï¿½ Herzwurm.

Strong Convergence Rates for Cox-Ingersoll-Ross Processes: Full Parameter Range

2 November 2016
16:15-17:15

Dr. Andrea Moiola
University of Reading

Event Details

Speaker invited by

Christoph Schwab

Abstract

The scattering of a time-harmonic acoustic wave by a planar screen with Lipschitz boundary is classically modelled by boundary integral equations (BIEs). If the screen is not Lipschitz, e.g. has fractal boundary, the correct Sobolev space setting to pose the problem is not obvious, because many of the relations between the standard definitions of Sobolev spaces on subsets of Euclidean space (e.g. restriction, completion of spaces of smooth functions, interpolation...) that hold in the Lipschitz case, fail to hold in general. To extend the BIE framework to general screens, we study properties of the classical fractional Sobolev spaces (or Bessel potential spaces)on general non-Lipschitz subsets of Rn. In particular, we extend results about duality, s-nullity (whether a set with empty interior can support distributions with given Sobolev regularity), and about the equivalence or not between alternative space definitions, providing several examples. An interesting application is the approximation of variational problems posed on fractal sets by problems posed on prefractal approximations. This is a joint work with S.N. Chandler-Wilde (Reading) and D.P. Hewett (UCL).

Sobolev spaces on non-Lipschitz sets with application to boundary integral equations on fractal screens

9 November 2016
16:15-17:15

Prof. Dr. Philippe Ciarlet
City University Hong Kong

Event Details

Speaker invited by

Habib Ammari

Abstract

The fundamental theorem of surface theory asserts that a surface can be recovered from the knowledge of its two fundamental forms if these two forms satisfy the Gauss and Codazzi-Mainardi equations on a simply-connected open subset of the plane, in which case it is uniquely defined up to translations and rotations. While this well-known existence and uniqueness result is classically established in spaces of continuously differentiable functions, it has been recently extended to other function spaces, in particular to Sobolev spaces. A related question is to examine whether such a recovered surface is a continuous function of its fundamental forms. A first positive answer was given by the author when the spaces are those of continuously differentiable functions equipped with their Frï¿½chet topologies. More recently, it was shown in various works, by Liliana Gratie, Maria Malin, Cristinel Mardare, and the author, that such a continuity result holds as well when the immersions defining the surfaces belong to ad hoc Sobolev spaces. In this talk, we will briefly review these recent advances and also consider some of their potential applications, for instance to the intrinsic approach to nonlinearly elastic shell theory, where the fundamental forms of the unknown deformed middle surface of a shell are taken as the new unknowns.

Continuity of a surface as a function of its fundamental forms: recent advances and applications

16 November 2016
16:15-17:15

Prof. Dr. Istvan Gyongy
University of Edinburgh

Event Details

Speaker invited by

Arnulf Jentzen

Abstract

Stochastic PDEs of nonlinear filtering are considered. These equations are given on the whole Euclidean space in the spatial variable. To solve them numerically we localise the equations onto large balls. This localisation reduces various kinds of spatial approximation schemes to finite dimensional problems. The error is shown to be exponentially small. As an application, a numerical scheme is presented which combines the localisation and the space and time discretisation, and thus is fully implementable. The talk is based on a joint work with Mate Gerencser.

On numerical solutions of parabolic stochastic PDEs given on the whole space

23 November 2016
16:15-17:15

Prof. Dr. Christiane Tretter
Universität Bern

Event Details

Speaker invited by	Habib Ammari
Abstract	In this talk variational principles for eigenvalues in gaps of the essential spectrum are presented which are used to derive two-sided eigenvalue bounds. Applications of the results include the Klein-Gordon equation, even when complex eigenvalues occur, and spectral problems arising in the analysis of 2D photonic crystals.

Variational principles for eigenvalues in spectral gaps and applications

30 November 2016
16:15-17:15

Prof. Dr. Jinchao Xu
PennState University

Event Details

Speaker invited by

Ralf Hiptmair

Abstract

In this talk, I will present a general framework for the design and analysis of Algebraic or Abstract Multi-Grid (AMG) methods. Given a smoother, such as Gauss-Seidel or Jacobi, we provide a general approach to the construction of a quasi-optimal coarse space and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation, and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, the classic AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG, and spectral AMGe.

A unified approach to the design and analysis of AMG

7 December 2016
16:15-17:15

Prof. Dr. Jens Markus Melenk
University of Technology Vienna

Event Details

Speaker invited by

Christoph Schwab

Abstract

Boundary Element Methods (BEM) are an important tool for the numerical solution of acoustic and electromagnetic scattering problems. These BEM matrices are fully populated so that data-sparse approximations are required to reduce the complexity from quadratic to log-linear. For the high-frequency case of large wavenumber, standard blockwise low-rank approaches are insufficient. One possible data-sparse matrix format for this problem class that can lead to log-linear complexity are directional H^2-matrices. We present a full analysis of a specific incarnation of this approach. Directional H^2-matrices are blockwise low rank matrices, where the block structure is determined by the so-called parabolic admissibility condition. In order to achieve log-linear complexity with this admissibility condition, a nested multilevel structure is essential that provides a data-sparse connection between clusters of source and target points on different levels. We present a particular variant of directional H^2-matrices in which all pertinent objects are obtained by polynomial interpolation. This allows us to rigorously establish exponential convergence in the block rank in conjunction with log-linear complexity. The work presented here is joint with S.~B\"orm (Kiel).

directional H^2-matrices for Helmholtz integral operators

14 December 2016
16:15-17:15

Prof. Dr. Stefan Volkwein
Universität Konstanz

Event Details

Speaker invited by

Ralf Hiptmair

Abstract

In the talk bicriterial optimal control problem governed by a parabolic partial differential equation (PDE) and bilateral control constraints is considered. For the numerical optimization the reference point method is utilized. The PDE is discretized by a Galerkin approximation utilizing the method of proper orthogonal decomposition (POD). POD is a powerful approach to derive reduced-order approximations for evolution problems. Numerical examples illustrate the efficiency of the proposed strategy.

POD-Based Multicriterial Optimal Control by the Reference Point Method (joint work with S. Banholzer and D. Beermann)

15 December 2016
17:15-18:15

Dr. Konstantinos Dareiotis
Uppsala University, Sweden

Event Details

Speaker invited by

Arnulf Jentzen

Abstract

We will present L^{\infty}-estimates for solutions of Stochastic PDEs (SPDEs) of parabolic type obtained by techniques motivated by the works of De Giorgi and Moser in the deterministic setting. The global estimates will then be applied in order to prove solvability for a class of semilinear SPDEs, while the local estimates will be used in order to obtain a weak Harnack-type inequality for solutions of linear equations, which is in turn used to deduce information about the oscillation of the solutions. The results are from joint work with Mate Gerencser (IST, Austria).

L^{infty}-estimates for Stochastic PDEs of parabolic type and applications

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