Joint NA Seminars 2022 (Spring)
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Jan 27, Feb 3, Feb 10, Feb 17, Feb 24, Mar 3, Mar 10, Mar 17, Mar 24, Mar 31, Apr 7, Apr 14, Apr 21, Apr 28, May 5, May 12, May 19, May 23, Jun 2, Jun 9, Jun 16.
Week 4
Title: An asymptotic preserving numerical scheme and a reduced order model for the radiative transfer equation
Speaker: Zhichao Peng (Michigan State University)
Location: Zoom (https://kth-se.zoom.us/j/63299293349)
Time: Thursday, Jan 27, 2022, 14.00-15.00
Abstract: As the Knudsen number goes to zero, the radiative transfer equation (RTE) asymptotically converges to its diffusion limit. On one hand, it is a challenge to design efficient numerical schemes preserving the underlying physical limit. On the other hand, this limit also suggests the existence of a low-rank structure in the angular space which can be utilized to design reduced order models (ROMs). In the first part of this talk, we present an asymptotic preserving method solving time-dependent RTE based on the micro-macro decomposition and Schur complement. The proposed method is unconditionally stable in the diffusive regime and has standard CFL conditions in the transport regime. In the second part of the talk, we present a reduced basis method to build an angular-space ROM for the steady state RTE.
Week 5
Title: Numerical homogenization of geometric network models
Time: Thu 2022-02-03 14.00-15.00
Location: Zoom https://stockholmuniversity.zoom.us/j/62889720565
Speaker: Fredrik Hellman (Gothenburg University)
Abstract: A sheet of paper can structural mechanically be modeled as an elasticity equation on a graph with edges for the fibers and vertices for the fiber contact points. To solve such a fine model, however, the vast number of degrees of freedom poses a computational challenge. In this talk, we apply a homogenization technique to the fine model and obtain an upscaled problem expressed on a coarser finite element mesh and that is easier to solve. The introduction and analysis of a Scott–Zhang type interpolation operator and the necessary assumptions on the network are the main theoretical contributions of this work.
This is a joint work with Axel Målqvist, Morgan Görtz and Gustav Kettil.
Week 6
Title: High-Order CutFEM with BDF on Time-Dependent Domains
Time: Thu 2022-02-10 14.00-15.00
Location: Zoom https://kth-se.zoom.us/j/63299293349
Speaker: Simon Sticko (Uppsala University)
Abstract: We consider fully discrete methods for solving the advection-diffusion equation on time-dependent domains. These methods use the high-order stabilized cut finite element method as spatial discretization and backward difference formulas (BDF) for time-stepping. We consider both the case when the domain is a d-dimensional bounded subset of R^d and when it is a (d-1)-dimensional manifold. We focus on combining Lagrange elements of order p with BDF of order p+1, with p<5. We show numerical experiments to investigate the order of convergence. For some test cases, we observe optimal order: p+1, while for others suboptimal.
Week 7
Title: Finite difference methods for marine seismic exploration and earthquake sequence simulations
Time: Thu 2022-02-17 14.00-15.00
Location: Zoom https://stockholmuniversity.zoom.us/j/62889720565
Lecturer: Martin Almquist (Uppsala University)
Abstract: Marine seismic exploration uses pressure waves generated by an array of airguns, dragged behind a boat, to determine material properties of the subsurface. Mathematically, the problem may be formulated as the inverse problem of estimating parameters of the coupled acoustic-elastic wave equation. Given seismograms at point locations, we set up a misfit functional (MF) that measures the difference between simulated and recorded data. The gradient of the MF can be computed efficiently by solving the adjoint equations. We discuss the adjoint of the discrete equations and how seismogram data in the MF give rise to singular source terms. Earthquake sequence simulations, which simulate faults over multiple seismic events, are now widely used in the earthquake modeling community. Unlike a dynamic rupture simulation of a single event, earthquake sequence simulations can be used to estimate recurrence intervals and study how fault-slip history influences future events. We discuss results from a benchmark problem that involves an elastic solid and a nonlinear rate-and-state friction law on the fault.
Week 8
Title: Recent developments of a potential theory based Cartesian grid method for elliptic PDEs on irregular domains
Time: Thu 2022-02-24 14.00-15.00
Location: Zoom https://kth-se.zoom.us/j/63299293349
Speaker: Wenjun Ying (Shanghai Jiaotong University)
Abstract: This talk will be on a potential theory based Cartesian grid method for elliptic PDEs on irregular domains. The method solves a boundary value or interface problem of PDE in the framework of second-kind Fredholm boundary integral equations. It avoids some limitations of the traditional boundary integral method. It does not need to know or compute the fundamental solution or Green’s function of the PDE. Instead, it allows the solution of variable coefficients and nonlinear PDEs. The method evaluates boundary and volume integrals involved indirectly by solving equivalent but much simpler interface problems on Cartesian grids, based on properties of single, double layer boundary integrals and volume integrals in potential theory. In addition to its taking advantage of the well-conditioning property of the second-kind Fredholm boundary integral equations, the method makes full use of fast solvers on Cartesian grids. The Cartesian grid method can also accurately compute nearly singular and hypersingular boundary integrals. In this talk, I will present recent developments of the method.
Week 9
Cancelled
Week 10
Title: Spacetime finite element methods for control problems subject to the wave equation
Time: Thu 2022-03-10 14.00-15.00
Location: Zoom https://kth-se.zoom.us/j/63299293349
Speaker: Erik Burman (UCL)
Abstract: We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the control function given by the computational method. The proofs are based on the regularity properties of the control function given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. The order of the error estimates reflects the stability of the problem and the optimal approximation of the finite element spaces, but are slightly suboptimal compared to direct interpolation. We will also comment on the convergence of the scheme for solutions with the minimal regularity. Some numerical examples will be presented in one space dimension and time. The talk is based on joint work with Lauri Oksanen, Ali Feizmohammadi and Arnaud Münch.
Week 11
Time: Thu 2022-03-17 14.00-15.00 Cancelled
Speaker: Sven-Erik Ekström (Uppsala)
Title, abstract and location: TBA
Week 12
Title:Modern discontinuous Galerkin methods for computational fluid dynamics
Time: Thu 2022-03-24 14.00-15.00
Speaker: Andrew Winters (Linköping)
Location: KTH, 3721, Lindstedsvägen 25
Abstract: Discontinuous Galerkin (DG) methods have a long history in computational physics and engineering to approximate solutions of hyperbolic partial differential equations due to their high-order accuracy and geometric flexibility. However, DG is not perfect and there remain some issues. Over the past decade, DG has undergone an extensive transformation into its modern form; capable of robust, adaptive simulations for complex transient flows.
Solutions of non-linear conservation laws contain many complex phenomena such as discontinuities, singularities, and turbulence. These phenomena are all time dependent and feature multiple scales in space and time. A wide variety of low-order and high-order numerical methods have been developed over many decades. There is, however, a balancing act when applying either type of method to a given problem:
1) Low-order methods offer remarkable robustness but require a very large number of degrees of freedom (DOFs) to properly capture multi-scale non-linear phenomena.
2) High-order methods offer great capabilities to accurately capture non-linear phenomena while requiring a moderate number of DOFs. However, they often lack robustness.
One approach for a shock capturing framework is to blend a low-order finite volume method with a high-order nodal DG method. Through carful construction of the geometric terms, this strategy is generalisable to multi-dimensional curvilinear meshes.
The aim of this talk is to dissect and discuss a modern form of nodal DG spectral element methods. These DG methods combine favourable features of other methods, e.g., geometric flexibility of finite element methods, skew-symmetric formulations of finite difference methods, and entropy stable numerical fluxes from finite volume methods. Implementation aspects of this modern DG formulation will also be included.
Week 13
Title: Block preconditioning the p-Stokes equations in ice-sheet models
Time: Thu 2022-03-31 14.00-15.00
Speaker: Christian Helanow (SU)
Location: KTH, 3721, Lindstedsvägen 25
Abstract: The deformation of ice can accurately be modeled as non-linear Stokes flow. The constitutive equation for ice, relating strain rates to stresses, is that of a singular power law which when discretizing the system of equations gives rise to a poorly conditioned linear system. The system is of saddle-point nature and its character depends on e.g. the effective viscosity and chosen regularization parameter. This motivates the use of block preconditioners that explicitly take into account the properties of the underlying problem. In the context of ice-sheet modeling using the p-Stokes equations block preconditioners have been used and numerically investigated with a heuristic motivation. However, in more general non-Newtonian settings, with focus on Bingham fluids, bounds on the eigenvalues for the block-preconditioned Schur complement have been derived. We attempt to adapt the theory of these studies to include Schur-complement preconditioners for singular power-law fluids, and numerically investigate how the proposed block preconditioner is affected by the specifics of ice-sheet simulations.
Week 14
Title: Finite-difference methods for wave propagation problems including moving features
Time: Thu 2022-04-07 14.00-15.00
Speaker: Ylva Rydin (Uppsala)
Location: KTH, Seminar room 3418, Lindstedsvägen 25
Abstract: In this talk, high-order summation-by-parts (SBP) finite difference methods for time-dependent wave propagation problems will be discussed. It is well known that high-order finite difference methods have good properties for solving wave propagation problems efficiently. However, real-world problems often include complicated features such as complex geometries, non-linearities, or non-smooth data that may ruin the accuracy of SBP finite difference methods. In this talk, I will present methods for constructing accurate high-order finite difference schemes for problems including two such features: moving boundaries and moving point sources.
Week 15
Easter Break
Week 16
Title: Connecting random fields on manifolds and stochastic partial differential equations in simulations
Speaker: Annika Lang (Chalmers)
Location: Zoom https://kth-se.zoom.us/j/63299293349
Time: Thursday, Apr 21, 2022, 14.00-15.00
Abstract: Random fields on manifolds can be used as building blocks for solutions to stochastic partial differential equations or they can be described by stochastic partial differential equations. In this talk I present recent developments in numerical approximations of random fields and solutions to stochastic partial differential equations on manifolds and connect the two. More specifically, we look at the stochastic wave equation on the sphere and approximations of Gaussian random fields on manifolds using suitable finite element methods. Throughout the talk, theory and convergence analysis are combined with numerical examples and simulations.
This talk is based on joint work with David Cohen, Erik Jansson, Mihály Kovács, and Mike Pereira.
Week 17
Talk 1
Title: Parameter sensitivity study of dynamic ice sheet models
Time: Tue 2022-04-26 14.00-15.00
Location: KTH, 3721, Lindstedsvägen 25
Speaker: Cheng Gong (Dartmouth College)
Abstract: Predictions of future sea-level rise due to the mass loss from ice sheets are afflicted with uncertainty, caused mainly by insufficient understanding of spatiotemporally variable processes at the inaccessible base and the interior of ice sheets for which few direct observations exist and of which basal friction and ice rheology are the prime examples. Here, we present an inverse modeling framework for studying the relationship between bed and surface processes of ice sheets and glaciers. We derive time-dependent adjoint equations from a full Stokes model and a shallow-shelf/shelfy-stream approximation model, respectively, to determine the sensitivity of surface velocities to the perturbations in basal conditions. A closed form of the analytical solutions to the adjoint equations is given with a two-dimensional example for interpreting the physical meaning of the sensitivity analysis. Helheim Glacier, as an example, is then solved with Automatic Differentiation to quantify the transient sensitivity of the ice flux near the terminus to changes in basal frictions and ice rheology. These sensitivities highlight the regions where each parameter may contribute the most to changes in ice flux and which process should be properly captured by numerical models in order to accurately project the future response.
Talk 2
Title: Let’s use supercomputers to resolve ice flow
Time: Thu 2022-04-28 14.00-15.00
Location: Zoom https://stockholmuniversity.zoom.us/j/62889720565
Speaker: Ludovic Räss (ETH)
Abstract: Since over a decade, graphical processing units (GPUs) power supercomputers and are about to soon achieve exascale processing capabilities. However, the development of highly efficient, robust and scalable numerical algorithms lags behind this rapid increase in massive parallelism of modern hardware. This current situation challenges legacy solvers’ implementations and provides rooms for new development. In this presentation, I will discuss recent advances in the field of accelerated iterative solvers designed for massively parallel implementations. I will further discuss main performance limitations, scalability, and provide insights into some challenges related to resolving full-Stokes coupled to thermal and hydrological processes in 3D over complex topography.
Week 18
Title: Efficient Approaches for Time-Domain Wave Equations
Time: Thu 2022-05-05 14.00-15.00
Location: Zoom https://kth-se.zoom.us/j/63299293349
Speaker: Jeffrey Banks (RPI)
Abstract: For engineering or applied sciences, high-order accurate numerical methods are often desirable because they are potentially orders of magnitude more efficient than their low-order counterparts. However, realizing the potential payoff of high-order methods in complex domains, particularly for wave equations, has proven challenging. Here I highlight two aspects of our recent work on high-order accurate methods for the second-order formulation of the governing equations. Part I of the talk presents a numerical approach for dispersive Maxwell’s equations built around an efficient 3-level time stepping algorithm. Overlapping grids are used to address geometric complexity, and both second- and fourth-order accurate schemes are presented. Part II presents recent developments for Galerkin Differences (GD). Although GD is fundamentally a finite element approximation based on a Galerkin projection, the underlying GD space is nonstandard and is derived using profitable ideas from the finite difference literature. The resulting schemes possess remarkable properties including nodal superconvergence and the ability to use large CFL-one time steps.
Week 19
Moved to week 17, Apr. 26
Speaker: Chen Gong (Dartmouth)
Week 20
Title: Numerical Methods for Shallow Water Models
Time: Thu 2022-05-19 14.00-15.00
Location: Zoom https://kth-se.zoom.us/j/63299293349
Speaker: Yekaterina Epshteyn (Utah)
Abstract: In this talk, we will discuss design of structure-preserving central-upwind finite volume methods for shallow water models. Shallow water systems are widely used in many scientific and engineering applications related to modeling of water flows in rivers, lakes and coastal areas. Shallow water equations are examples of hyperbolic systems of balance laws and such mathematical models can present a significant challenge for the construction of accurate and efficient numerical algorithms. We will show that the developed structure-preserving numerical methods for shallow water equations deliver high-resolution, can handle complicated geometry, and satisfy necessary stability conditions. We will illustrate the performance of the designed algorithms on a number of challenging numerical tests. Current and future research will be discussed as well.
Week 21
Title: Side-effects of Learning from Low Dimensional Data Embedded in an Euclidean Space
Time: Mon 2022-05-23 14.00-15.00
Location: KTH, 3721, Lindstedsvägen 25
Speaker: Richard Tsai (UT Austin)
Abstract: The low dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low dimensional manifolds embedded in a high dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. We analyze the cases where the training data are distributed in a linear subspace of Rd. We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network’s depth and noise in the codimension of the data manifold.
Week 22
Title: Active thermal cloaking and mimicking
Time: Thu 2022-06-02 14.00-15.00
Location: Zoom https://kth-se.zoom.us/j/63299293349
Speaker: Maxence Cassier (CNRS, Institut Fresnel)
Abstract: In this talk, we present two active cloaking methods for the parabolic heat equation to hide objects or sources in a homogeneous isotropic medium. By active cloaking, we mean that it relies on designing a distribution of heat sources to make a source or an object invisible to an observer from the perspective of thermal measurements. The same techniques can be used for mimicking, in other words to make a source or an object look like a different one outside of the cloaked region. Our first method [1], based on Green identities, relies on a time-dependent potential formula that requires the knowledge of the temperature field and heat flux of the incident field (used to detect the object) on an active surface (the cloak). More precisely, using these data, it consists on designing distributions of monopole and dipole heat sources on the boundary of the region to be cloaked. This can be physically realised with thermal heat pumps such as Peltier devices. We show here “some stability” of this approach and analyse the numerical error in our simulations via the maximum principle of parabolic equations. However, one caveat of this approach is that the cloak has to surround the object. In our second method [2], we use a frequency-to-time approach, via a Fourier-Laplace transform in time, to address this drawback and allow “exterior cloaking” where the cloak does not surround the object. Furthermore, this second method based on Graf’s addition formulas on the Helmholtz equation with complex wavenumbers can be applied to a large class of linear partial differential equations that arise in physics. In this context, we provide quantitative bounds that evaluate the quality effect in the frequency domain and we analyse the cloaking error by using the maximum principle applied to Helmholtz equations whose complex-wave number lies in the conical domain {k ∈ C | | Im(k)| > | Re(k)|}.
In collaboration with Trent DeGiovanni (University of Utah), Sébastien Guenneau (UMI Abraham De Moivre, CNRS, Imperial College of London) and Fernando Guevara Vasquez (University of Utah).
References related to the talk:
- M. Cassier, T. DeGiovanni, S. Guenneau and F. Guevara Vasquez, Active Thermal Cloaking and Mimicking, Proc. R. Soc. A vol. 477, p. 20200941, 2021.
- M. Cassier, T. DeGiovanni, S. Guenneau and F. Guevara Vasquez, Active exterior cloaking for the 2D Helmholtz equation with complex wavenumbers and application to thermal cloaking, to appear in Philosophical Transactions of the Royal Society A, preprint available on Arxiv at https://arxiv.org/pdf/2203.02075.pdf.
Week 23
Time: Thu 2022-06-09 14.00-15.00
Speaker: Stefano Ottolenghi (SU)
Title, abstract and location: CANCELLED
Week 24
Title: Microlocal analysis and deep learning for tomographic reconstruction
Time: Thu 2022-06-16 14.00 - 15.00
Location: KTH, 3721, Lindstedsvägen 25
Speaker: Ozan Öktem (KTH)
Abstract: The talk outlines recent progress in developing domain adapted deep neural networks for the task of (a) extracting the wavefront set of an image from its shearlet coefficients and (b) inpainting the invisible part of the wavefront set in limited angle tomography. A key component in both tasks is to represent them as optimal non-randomised decision rules in statistical decision theory. The talk will also outline how to combine these two networks with a deep neural network for reconstruction, whose architecture is obtained by unrolling a suitable iterative scheme. Specifying the visible parts of the wavefront set relies on characterising the microlocal canonical relation of the deep neural network for reconstruction, which here inverts the ray transform. This results in a deep learning based approach for limited angle tomographic reconstruction that is aware of the microlocal canonical relation for the ray transform and also on the characterisation of visible part of the wavefront set.