2022-06-22:
2022-06-15: Per Ljung, Chalmers & GU
2022-06-08: Gilles Vilmart, University of Geneva
Superconvergent methods inspired by the Crank-Nicolson scheme in the context of diffusion PDEs
Abstract:
In this talk, we present two different situations where the Crank-Nicolson method is surprisingly more accurate than one could expect and inspires the design of new efficient numerical integrators:
-in the context of splitting methods for deterministic parabolic PDEs with inhomogeneous general oblique boundary conditions, where order reduction phenomena can be avoided,
-in the context of ergodic parabolic stochastic PDEs, where high order can be achieved for sampling the invariant distribution, in spite of the low regularity of the solution.
This talk is based on joint works with Assyr Abdulle, Ibrahim Almuslimani, Guillaume Bertoli, Christophe Besse, and Charles-Edouard Bréhier.
2022-06-01: Andrii Dmytryshyn, Örebro University
Versal deformations of matrices
Abstract:
Jordan canonical form for matrices is well known and studied with various purposes but reduction to this form is an unstable operation: both the corresponding canonical form and the reduction transformation depend discontinuously on the entries of an original matrix. This issue complicates the use of the canonical form for numerical purposes. Therefore V.I. Arnold introduced a normal form to which an arbitrary family of matrices A' close to a given matrix A can be reduced by similarity transformation smoothly depending on the entries of A’. He called such a normal form a versal deformation of A.
In this presentation we will discuss versal deformations and their use in investigation of possible changes in canonical forms (eigenstructures), reduction of unstructured perturbations to structured perturbations, and codimension computations.
2022-05-25: Mohammad Asadzadeh, Chalmers & GU
On HP-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system
Abstract:
We study stability and convergence of hp-streamline diffusion (SD) finite element, and Nitsche's schemes
for the three dimensional, relativistic (3 spatial dimension and 3 velocities), time dependent Vlasov-Maxwell system
and Maxwell's equations, respectively. For the hp scheme for the Vlasov-Maxwell system, assuming that the exact
solution is in the Sobolev space H^{s+1}, we derive global a priori error bound of order O(h/p)^{s+1/2},
where h(= maxK h_K) is the mesh parameter and p(= max_K p_K) is the spectral order. This estimateis based on the local version with h_K = diam K being the diameter of the phase-space-time element K and p_K is the spectral order (the degree of approximating finite element polynomial) for K. As for the Nitsche's scheme, by a simple calculus of the field equations, we convert the Maxwell's system to an elliptic type equation. Then, combining the Nitsche's method for the spatial discretization with a second order time scheme, we obtain optimal convergence of O(h^2 +k^2), where h is the spatial mesh size and k is the time step.Here, as in the classical literature, the second order time scheme requires higher order regularity assumptions. Numerical justifcation of the results, in lower dimensions, is presented.
This is a joint work with P.Kowalczyk and C. Standar (appeared in KRM, 2019)
2022-05-18: Vidar Thomée, Chalmers & GU
On Positivity Preservation in Finite Element Methods for the Heat Equation
Abstract:
We consider the initial boundary value problem for the homogeneous
heat equation, with homogeneous Dirichlet boundary
conditions. By the maximum principle the solution is
nonnegative for positive time if the initial data are nonnegative.
We study to what extent this property carries over to some
finite element discretizations.
2022-05-11: Kent-Andre Mardal, Simula
Multi-physics problems related to brain clearance
Abstract:
Recent theories suggest that a fundamental reason for sleep is simply clearance of metabolic waste produced during the activities of the day. In this talk we will present multi-physics problems and numerical schemes that target these applications. In particular, we will be lead from basic applications of neuroscience into multi-physics problems involving Stokes, Biot and fractional solvers at the brain-fluid interface.
2022-05-04: Sebastian Reich, University of Potsdam
Robust parameter estimation using the ensemble Kalman filter
Abstract:
Estimating the parameters of a stochastic differential equation (SDE) from
continuous time observations of the process constitutes a classical inverse problem.
It is well-known that the maximum likelihood solution to this inverse problem does not
depend continuously on the data, i.e., is not robust, under the standard topology of continuous functions.
Here we revisit this problem from the perspective of continuous-time gradient descent and
continuous-time ensemble Kalman filtering. Both approaches lead SDEs in the unknown parameters
which are driven by the given observations in a multiplicative manner. This perspective allows
us to clearly identify the source of the non-robustness via a rough path analysis as well as to
identify two possible solutions to the non-robustness issue.
2022-04-27: Johan Hoffman, KTH
Matrix Schur factorization and the structure of turbulence
Abstract:
Any matrix is unitary equivalent to an upper triangular matrix, expressed as a Schur factorization. Analogously, a spectral theorem states that any normal matrix is unitary equivalent to a diagonal matrix. Therefore, a matrix can be decomposed into a sum of a normal matrix corresponding to the diagonal part of the upper triangular matrix of the Schur factorization, and a non-normal matrix corresponding to the remaining non-diagonal part of the upper triangular matrix. The normal matrix may then be further decomposed into the sum of a symmetric and a skew-symmetric matrix. Hence, the result is a triple decomposition of a general matrix into a sum of a symmetric matrix with real eigenvalues, a skew-symmetric matrix with purely imaginary eigenvalues, and a non-normal matrix. In fluid mechanics, it is common to separate straining flow from rotating flow by a decomposition of the velocity gradient tensor into a symmetric and a skew-symmetric part. This double decomposition, however, does not distinguish shear flow from any of the two flow components. In contrast, a triple decomposition of the velocity gradient tensor leads to a decomposition of any flow field into pure straining flow, rigid body rotational flow, and shear flow. First proposed as a flow visualization technique for improved vortex identification, recent interest in the triple decomposition has focused e.g. on improved analysis of turbulence simulations, models for thrombosis in blood flow simulations, and stability analysis flow structures in turbulence.
2022-04-21: Ozan Öktem, KTH
Microlocal analysis and deep learning for tomographic reconstruction
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.
2022-04-20: Efthymios Karatzas, National Technical University of Athens
Random geometries and a Unified Reduced Order Basis for Parametrized PDEs based on Embedded Finite Element Methods and applications
Abstract:
We consider parametrized PDEs, geometrical randomly deformed systems, and we present a new beneficial approach for reduced basis construction based on level set geometry descriptions and fixed background geometries. This unified Reduced Order Basis employs a background mesh, solves efficiently with less computational cost, and it is independent of any random parameter which affects the physics of the PDE model. We will discuss results related to unfitted finite element methods for parameterized partial differential equations enhanced by a proper orthogonal decomposition method. This approach achievements are twofold. Firstly, we reduce much the computational effort since the unfitted mesh method allows us to avoid remeshing when updating the parametric domain. Secondly, the proposed reduced order model technique gives an implementation advantage considering geometrical parametrization. Computational efforts are even exploited more efficiently since the mesh is computed once and the transformation of each geometry to a reference geometry is not required. These combined advantages allow to solve many PDE problems faster and “cheaper” and provide the capability to find solutions in cases that could not be resolved in the past.
Acknowledgments. This work has received funding from the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under grant agreement No[1115], and the ”First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment” grant 3270.
2022-04-13: Marlis Hochbruck, Karlsruhe Institute of Technology
Exponential integrators for quasilinear hyperbolic evolution equation
Abstract:
In this talk we propose two exponential integrators of first and second order applied to quasilinear hyperbolic evolution equations. We work in an analytical framework which is an extension of the classical Kato framework and covers quasilinear Maxwell’s equations in full space and on a smooth domain as well as a class of quasilinear wave equations.
In contrast to earlier works, we do not assume regularity of the solution but only on the data. From this we deduce a well-posedness result upon which we base our error analysis.
This is joint work with Benjamin Dörich, KIT
Reference
B. Dörich and M. Hochbruck. Exponential integrators for quasilinear wave-type equations. CRC 1173 Preprint 2021/12, Karlsruhe Institute of Technology, 2021, to appear in SIAM J. Numer. Anal.
https://www.waves.kit. edu/downloads/CRC1173_Preprint_2021-12.pdf
2022-04-12: Andreas Rosén (Chalmers and GU)
Dirac Integral equations, plasmonics and eddy currents
Abstract:
A well-known problem in Maxwell scattering is how to handle the divergence-free constraints on the fields. We present a recently developed integral equation reformulation of the Maxwell transmission problem, which solves this problem by embedding Maxwell's equations in a more stable 8/8 Dirac system. In the talk we discuss the Dirac integral equation obtained from a Cauchy representation of the fields, and how to optimize its numerical performance by tuning its 12 free parameters. We also demonstrate numerical results: an efficient solver for dielectric, plasmonic, and eddy current scattering without any false eigenwavenumbers or low-frequency breakdown. This is joint work with Johan Helsing and Anders Karlsson, Lund.
2022-04-06: Andreas Petersson, UiO
Numerical approximation of the heat modulated infinite dimensional Heston model
Abstract:
The HEat modulated Infinite DImensional Heston (HEIDEH) model
and its numerical approximation are introduced and analysed. This is a
special case of the infinite dimensional Heston stochastic volatility
model of (F.E. Benth, I. C. Simonsen '18). Therein, the authors consider
a potential model for risk-neutral forward prices of
commodity-delivering contracts. The model consists of a one-dimensional
stochastic advection equation coupled with a stochastic volatility
process, defined as a Cholesky-type decomposition of the tensor product
of a Hilbert-space valued Ornstein-Uhlenbeck process. In this work the
Ornstein-Uhlenbeck process is specified to be the solution to a
stochastic heat equation and the resulting HEIDEH model is studied in a
fractional Sobolev space setting.
In the talk, a
description and motivation of the model will be given. Then, a class of
covariance kernels are described that give rise to admissible Q-Wiener
processes. Regularity of the model is discussed under this class of
kernels. Finally, an approximation based on a combination of an explicit
finite difference scheme, a finite element method, the backward Euler
scheme and the circulant embedding method is presented. Convergence
rates are derived with the error measured pointwise, in a mean square
sense, in time and space. The resulting rates are higher than what can
be obtained from a standard Sobolev embedding technique.
This is joint work with Fred Espen Benth, Gabriel Lord and Giulia Di Nunno.
2022-03-30: Christian Lubich, University of Tübingen
A large-stepsize modified Boris method for charged-particle dynamics in a strong nonhomogeneous magnetic field
Abstract:
Efficient numerical integrators for charged-particle dynamics are of substantial interest in the context of particle methods for the partial differential equations arising in plasma physics. The standard integrator in this field is the Boris method but that method requires tiny step sizes in the presence of strong magnetic fields. We give an error analysis of a remarkably simple modification of the Boris method that was recently proposed by Xiao and Qin (Computer Physics Comm., 2021). We show that the guiding center motion of a charged particle in a nonhomogeneous magnetic field of size inversely proportional to a small parameter $\eps\ll 1$ is approximated on fixed time intervals with a second-order error $O(h^2)$ for step sizes $h$ that satisfy $h^2 \ge \eps$, as opposed to the standard Boris method that would require tiny step sizes $h \ll \eps$. The proof is based on comparing the modulated Fourier expansions of the exact solution, which was studied by Hairer and Lubich (Numer. Math., 2020), and of the numerical solution of the modified Boris method.
The talk is based on joint work with Yanyan Shi.
2022-03-23: Sergio Blanes, Valencia Polytechnic University
Positivity-preserving methods for ordinary differential equations
Abstract:
Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge--Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.
2022-03-16: Max Jensen, Sussex University
Finite Element Approximation of Hamilton-Jacobi-Bellman equations with nonlinear mixed boundary conditions
Abstract:
We show uniform convergence of monotone P1 finite element methods to the viscosity solution of isotropic parabolic Hamilton-Jacobi-Bellman equations with mixed boundary conditions on unstructured meshes and for possibly degenerate diffusions. Boundary operators can generally be discontinuous across face-boundaries and type changes. Robin-type boundary conditions are discretised via a lower Dini derivative. In time the Bellman equation is approximated through IMEX schemes. Existence and uniqueness of numerical solutions follows through Howard’s algorithm. We show how equations of this type naturally appear in models of mathematical finance.
2022-03-09: Tony Stillfjord, Lund University
SRKCD: stabilized Runge-Kutta methods for stochastic optimization
Abstract:
I will introduce a family of stochastic optimization methods that we call SRKCD, which are based on the Runge-Kutta-Chebyshev (RKC) schemes. The RKC methods are explicit methods originally designed for solving stiff ordinary differential equations by ensuring that their stability regions are of maximal size. In the optimization context, this allows for larger step sizes (learning rates) and better robustness compared to e.g. the popular stochastic gradient descent method.
Our main contribution is a convergence proof, not only for SRKCD but for essentially any stochastic optimization method based on an explicit Runge-Kutta scheme. This shows convergence in expectation with an optimal sublinear rate, under standard assumptions of strong convexity and Lipschitz-continuous gradients. For non-convex objectives, we get convergence to zero in expectation of the gradients. I will illustrate the improved stability properties of SRKCD on both a small-scale test example and on a problem arising from an image classification application in machine learning.
2022-03-02: Anna Persson, KTH
Superconvergence of a multiscale method for ground state computations of Bose-Einstein condensates
Abstract:
Bose-Einstein condensates are formed when a gas of Bosons are cooled to ultra-low temperatures. At this point, most of the Bosons occupy the same quantum state and behave like one giant ``macro particle''. This is classified as a new state of matter, where some quantum mechanical phenomena can be observed macroscopically. One such phenomenon is superfluidity, which describes a flow without inner friction.
To compute ground states of Bose-Einstein condensates the stationary Gross-Pitaevskii equation (GPE) is often used. In this talk we revisit a two-level multiscale technique introduced in (P.Henning, A.Målqvist, D. Petersim '14) for computing numerical approximations to the GPE. The method is based on localized orthogonal decomposition of the solution space, which exhibit high approximation properties and improves the convergence rates compared to classical finite element methods. This reduces the computational cost for computing the ground states significantly.
In the previous paper, high order convergence of the method was proven, but even higher orders were observed numerically. In this talk we show how to improve the analysis to achieve the observed rates, which are as high as O(H^6) for the eigenvalues. We also show some numerical experiments for both smooth and discontinuous potentials.
2022-02-23: Dan Wang, Hong Kong University of Science and Technology
Phase Analysis of MIMO LTI Systems
Abstract:
In this work, we define the phases of a special class of complex matrices, called sectorial matrices. Various properties of matrix phases have been studied. In particular, a majorization relation between the phases of the eigenvalues of AB and the phases of A and B is established. With the concept of matrix phases, we define the phase response of MIMO LTI systems, which, together with magnitudes response, gives a complete MIMO Bode plot. The phase concept subsumes the well-known notions of positive real systems and negative imaginary systems. We also formulate a small phase theorem for feedback stability, as a counterpart to the celebrated small gain theorem.
2022-02-16: Anders Logg, Chalmers and GU
Mesh generation for large-scale city modeling and simulation
Abstract:
Digital Twins are digital replicas of physical things. Digital twins
have traditionally been used in the manufacturing industry to model
products and production processes. In recent years, Digital Twins have
extended to new domains and are now becoming reality for buildings and
whole cities.
At the Digital Twin Cities Centre, we are developing new methods for
large-scale city modeling and simulation. An important first challenge
is how to automatically and efficiently generate computational meshes
from raw data. In this talk, I will present and demonstrate a new
method for 3D mesh generation from publicly available data in the form
of point clouds and cadastral maps.
This is joint work together with Vasilis Naserentin.
2022-02-09: Daniele Boffi, KAUST
On the inf-sup condition for a mixed formulation
Abstract:
The stability and convergence analysis of finite element
discretizations arising from mixed formulations is based on the
validity of suitable inf-sup conditions. It is well known that the
inf-sup conditions are sufficient and necessary, in a suitable sense,
for the optimal behavior of a discrete scheme.
Starting from an example coming from linear elasticity, we discuss a
non standard mixed formulation for the approximation of elliptic
problems. We will show that, although the inf-sup condition is not
satisfied, the method can be successfully used if certain regularity
assumptions are met.
This is a joint work with Fleurianne Bertrand.
2022-02-02: Ingrid Lacroix-Violet, Institut Élie Cartan de Lorraine
Linearly implicit numerical methods
Abstract:
In this talk, I will present a new class of numerical methods for the time integration of evolution equations. The systematic design of these methods mixes the Runge-Kutta collocation formalism with collocation techniques in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so. After the introduction of the methods, I will set suitable definitions of consistency and stability for these methods, prove their convergence and order on the ODE case and finally I will present some numerical simulations.
2022-01-26: Axel Ringh, Chalmers and GU
Multi-marginal graph-structure optimal transport: Modeling, applications, and computational methods
Abstract
Optimal transport has seen rapid development over the last decades, and recently there has been a surge of research related to computational methods for numerically solving the problem. The most well-known algorithm is the Sinkhorn algorithm, in which the transport plan is iteratively updated so that its projections match the given marginals.
In this talk I will consider a generalization of optimal transport, namely multi-marginal optimal transport. In particular, I will introduce graph-structured multi-marginal optimal transport and show that this type of structured problems can be used to model and solve a variety of different problems. This includes, e.g., barycenter problems, displacement interpolation problems, and multi-species density control problems. Moreover, while the Sinkhorn algorithm extends to the multi-marginal setting, it only partially alleviates the computational difficulty for the multi-marginal problem. This is because computing the corresponding projections needed in the algorithm still scales exponentially in the number of marginals. However, in this talk I will present how these projections can be efficiently compute when the underlying graph structure is a trees, as well as for some other simple graph structures.
2022-01-19: Stig Larsson, Chalmers & GU
What is DeepONet?
Abstract:
In this talk I will review the paper ``DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators'' by Lu Lu, Pengzhan Jin, and George Em Karniadakis, arXiv:1910.0319
2022-01-12: Adam Andersson, Chalmers & GU and Saab
Local mesh refinement, however, can cause a severe bottleneck for the LF method, or any other standard explicit scheme, due to the stringent CFL stability condition on the time-step imposed by the smallest element in the mesh. Even when the locally refined region consists of a few small elements only, those elements will dictate a tiny time-step throughout the entire computational domain for stability. To overcome the crippling effect of a few small elements, without sacrificing its
high efficiency or explicitness elsewhere, a leapfrog based explicit local time-stepping (LF-LTS) method was proposed for the time integration of second-order wave equations. Recently, optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step depends on the smallest elements in the mesh. In general one cannot improve upon that stability constraint, as the LF-LTS method may become unstable at certain
discrete values of the time-step. To remove those critical values for the time-step, we apply a slight modification to the original LF-LTS method which nonetheless preserves its desirable properties: it is fully explicit, second-order accurate,
satisfies a three-term (leapfrog like) recurrence relation, and conserves the energy. The new stabilized LF-LTS method
also yields optimal convergence rates for a standard conforming FE discretization, yet under a CFL condition
where the time-step no longer depends on the mesh size inside the locally refined region.
Joint work with: Simon Michel (Univ. of Basel) and Stefan Sauter (Univ. of Zurich)
2021-06-09: Ulrik Skre Fjordholm, University of Oslo
Numerical methods for conservation laws on graphs
Abstract:
We consider a set of scalar conservation laws on a graph. Based on a choice of stationary states of the problem – analogous to the constants in Kruzkhov's entropy condition – we establish the uniqueness and stability of entropy solutions. For rather general flux functions we establish the convergence of an easy-to-implement Engquist–Osher-type finite volume method.
This is joint work with Markus Musch and Nils Henrik Risebro (University of Oslo).
2021-06-02: André Massing, NTNU
Stabilized Cut Discontinuous Galerkin Methods
Abstract:
To ease the burden of mesh generation in finite-element based simulation pipelines, novel so called unfitted finite element methods have gained much attention in recent years. The main idea is to embed the domain of interest into a structured but unfitted background mesh where the domain boundary can cut through the mesh in an arbitrary fashion. Unfitted finite-element based methods typically suffer from stability and conditioning problems caused by the presence of small cut elements. In this talk we develop both theoretically and practically a novel cut Discontinuous Galerkin framework (CutDG) by combining stabilization techniques from the cut finite element method with the classical interior penalty discontinuous Galerkin methods for elliptic and hyperbolic problems. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi-)norms associated with the relevant partial differential operators. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the both elliptic and stationary advection-reaction problems which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two-and three-dimensional test problems.
2021-05-26: Sven-Erik Ekström, Uppsala University
Matrix-less methods: approximating eigenvalues and eigenvectors without the matrix
Abstract:
Discretizing partial differential equations (PDE) typically gives rise to structured matrices. Exploiting these, often hidden, structures can be very beneficial when analysing these matrices or when constructing fast solvers.
In this talk we first present an overview of the theory of generalised locally Toeplitz (GLT) sequences; we introduce the notions of matrix sequences and symbols, and how one can approximate eigenvalues and singular values of matrices belonging to these sequences.
Then, we discuss the so-called matrix-less methods (not to be confused with matrix-free methods). These methods are extremely fast and efficient compared with conventional methods for computing eigenvalues.
Illustrative numerical examples will be presented, from model problems and PDE discretizations, both for Hermitian and non-Hermitian matrix sequences.
Finally, we show preliminary results for approximating the eigenvalues of discretization matrices of variable coefficient problems, and the computation of eigenvectors of Hermitian Toeplitz matrices.
2021-05-19: Sara Zahedi, KTH
A cut finite element method for incompressible two-phase flows
Abstract:
I will present a computational technique, a space-time Cut Finite Element Method (CutFEM), that can be used for simulations of two-phase flow problems. With CutFEM we develop a strategy for accurately approximating solutions to Partial Differential Equations (PDEs) in complex geometries that are arbitrarily located with respect to a fixed background mesh. We consider the time-dependent Navier-Stokes equations involving two immiscible incompressible fluids with different viscosities, densities, and with surface tension. Due to surface tension effects at the interface separating the two fluids and different fluid viscosities the pressure may be discontinuous and the velocity field may have a kink across the interface. I will address several challenges that computational methods for such simulations must handle, such as how to accurately capture discontinuities across an evolving interface and how to compute the mean curvature vector needed for the surface tension force with a convenient implementation allowing the interface to be arbitrary located with respect to a fixed mesh.
2021-05-12: Des Higham, The University of Edinburgh
Should We Be Perturbed About Deep Learning?
Abstract:
Many commentators are asking whether current AI solutions are sufficiently robust, resilient, and trustworthy; and how such issues should be quantified and addressed. I believe that numerical analysts can contribute to the debate. In part 1 of this talk I will look at the common practice of using low precision floating point formats to speed up computation time. I will focus on evaluating the softmax and log-sum-exp functions, which play an important role in many classification tools. Here, across widely used packages we see mathematically equivalent but computationally different formulations; these variations have been designed in an effort to avoid overflow and underflow. I will show that classical rounding error analysis gives insights into their floating point accuracy, and suggests a method of choice. In part 2 I will look at a bigger picture question concerning sensitivity to adversarial attacks in deep learning. Adversarial attacks are deliberate, targeted perturbations to input data that have a dramatic effect on the output; for example, a traffic "Stop" sign on the roadside can be misinterpreted as a speed limit sign when minimal graffiti is added. The vulnerability of systems to such interventions raises questions around security, privacy and ethics, and there has been a rapid escalation of attack and defence strategies. I will consider a related higher-level question: under realistic assumptions, do adversarial examples always exist with high probability? I will also introduce and discuss the idea of a stealth attack: an undetectable, targeted perturbation to the trained network itself.
Part 1 is joint work with Pierre Blanchard (ARM) and Nick Higham (Manchester).
Part 2 is joint work with Alexander Gorban and Ivan Tyukin (Leicester).
2021-05-05: Erik Jansson, Chalmers & GU