Schedule

Christoph Strössner: Functional Tucker approximation using Chebyshev interpolation

Abstract: This talk is concerned with approximating a trivariate function defined on a tensor-product domain via function evaluations. Combining tensorized Chebyshev interpolation with a Tucker decomposition of low multilinear rank yields function approximations that can be computed and stored very efficiently.

The existing Chebfun3 algorithm [Hashemi and Trefethen, SIAM J. Sci. Comput., 39 (2017)] uses this format but the construction of the approximation proceeds indirectly, via a so called slice-Tucker decomposition. As a consequence, Chebfun3 sometimes uses unnecessarily many function evaluations. We propose a novel algorithm Chebfun3F that utilizes univariate fibers instead of bivariate slices to construct the Tucker decomposition.

Joao Paixao: Algorithms in Graphical Linear Algebra

Abstract: We showcase a modular, graphical language—graphical linear algebra—and use it as high-level language to reason calculationally about linear algebra. We propose a minimal framework of six axioms that highlight the dualities and symmetries of linear algebra, and use the resulting diagrammatic calculus as a convenient tool to prove a number of diverse theorems and recursive algorithms. Our work develops a relational approach to linear algebra, closely connected to classical relational algebra.

Julian Hall and Ivet Galabova: HiGHS - high performance software for linear optimization

Abstract: The need to solve large scale linear optimization problems is fundamental to organised decision-making, and the underlying computational challenge is dominated by issues in sparse numerical linear algebra. This talk will introduce our open-source suite of solvers, HiGHS (www.highs.dev), that is built on multiple prize-winning academic research, but largely funded by industrial consultancy. The numerical linear algebra challenge will be identified and discussed. Issues relating to publishing, developing and maintaining HiGHS, as well as our consultancy experiences, will also be addressed. A personal insight into carrying out consultancy and helping to develop HiGHS as a PhD student will be given.

Ieva Daužickaitė: Randomised preconditioning in variational data assimilation

Abstract: The solution of a variational data assimilation method weak constraint 4D-Var can be approximated by solving series of large sparse symmetric positive definite linear systems using conjugate gradient (CG) method. Its convergence can be improved by employing limited memory preconditioners that are constructed with approximations of eigenvalues and eigenvectors of the coefficient matrix. When the coefficient matrix varies significantly between the systems standard methods for eigenvalue decomposition that exploit the Lanczos and CG connection are not efficient. We explore using easy to parallelise randomised methods for low rank eigenvalue decomposition and obtain encouraging results.

Kirandeep Kour: TT-CP: An approach to efficient machine learning model

Abstract: An increasing amount of collected data are high-dimensional and efficient learning algorithms must exploit the tensorial structure as much as possible. The ever-present curse of dimensionality for high dimensional data and the loss of structure when vectorizing the data motivates the use of tailored low-rank tensor methods. We introduce an exact TT-CP expansion as a decomposition method that combines the simplicity of Canonical Polyadic (CP) with the robustness of the Tensor Train (TT) decomposition. The TT-CP factorization uses the Uniqueness enforcing TT-SVD algorithm to compute TT factors along with an important constraint, named, the norm equilibration. For demonstrating numerical results, we embed this approach into the kernel method and show that the TT-CP decomposition leads to more reliable, computationally, less sensitive to tuning parameters, and higher prediction accuracy, classification machine learning model.

André Gaul (CEO EMS Press)

Abstract: TBA

Spyridon Pougkakiotis, Preconditioning for an Interior Point-Proximal Method of Multipliers

Abstract: We present an infeasible Interior Point Method (IPM) combined with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving convex optimization problems We have proven polynomial complexity of the algorithm, under standard assumptions, for problems expressible by linear semi-definite programming. The theory holds also in the case where the associated Newton systems are solved inexactly. In turn this allows us to employ iterative linear algebra for the solution of these systems, and we discuss some preconditioning strategies for general-purpose convex problems. When building preconditioners, we exploit the numerical stability introduced by the use of PMM. Focusing on linear and quadratic programming, we demonstrate the benefits of using regularization in IPMs as well as the reliability of the approach.

Davide Palitta: The Short-term Rational Lanczos Method and Applications

Abstract: TBA

Jemima Tabeart: New Preconditioners for the Saddle Point Formulation of Weak-constrained 4D-Var

Abstract: Data assimilation algorithms for numerical weather prediction are increasingly high dimensional, and complicated with the widespread use of correlated observation error covariance matrices. Fast convergence is essential, making the saddle point formulation of weak-constraint 4D-Var desirable due to its potential for parallelization. We consider new preconditioners which better approximate the model and observation error terms in the block diagonal and inexact constraint preconditioners. Previous work has found that correlated observation errors, which are often neglected by standard preconditioners, can lead to ill-conditioned data assimilation problems. We show for the heat equation that accounting for observation error correlations in the preconditioners is necessary for fast convergence. Our new approach results in faster convergence in serial than standard approaches, and can be easily adapted to exploit parallel computer architectures.

Abstract:

Fouzhan Hosseini: overview of NAG and the role of Women in HPC

Eleni Vlachopoulou: One solver to rule them all: Designing Krylov subspace solvers to target different architectures

Heterogeneous programming models and C++ see increasing interest in numerical computing. As a result, models that enable us exploit different architectures for numerical linear algebra kernels have gained popularity. Why are these models important and how can we use them to improve usability and increase productivity? Is C++ the way to go to keep our solvers relevant to a continuously evolving setting? In this talk we will touch on the motivation behind those efforts and, also, give an example of implementing Krylov subspace methods for solving large and sparse linear systems in an architecture-agnostic way.

Stefano Massei: Maximum volume and cross approximation of symmetric semidefinite matrices: theory and algorithms

Abstract: Selecting the r×r submatrix of maximum volume of a matrix A ∈ R^{n×n} is an NP hard problem that arises in a variety of applications. For example, it yields a quasi-best low-rank approximation constructed from the rows and columns of A. We show that the maximum volume submatrix can always be chosen to be a principal submatrix if A is symmetric positive semidefinite (SPSD) and we propose a new greedy algorithm of cost O(n). In the second part of the talk we use a probabilistic argument to show that any SPSD matrix admits a cross approximation, built on a principal submatrix, whose approximation error is bounded by (r + 1) times the error of the best rank r approximation in the nu- clear norm. Relying on a derandomization technique, we derive a deterministic algorithm which is capable to retrieve a quasi optimal cross approximation.

Estelle Massart: Overview of a quotient geometry with simple geodesics for the manifold of fixed-rank positive-semidefinite matrices

Abstract: We describe the main geometric tools required to work on the manifold of fixed-rank symmetric positive-semidefinite matrices: we present expressions for the Riemannian logarithm and the injectivity radius, to complement the already known Riemannian exponential. This manifold is particularly relevant when dealing with low-rank approximations of large positive-(semi)definite matrices. The manifold is represented as a quotient of the set of full-rank rectangular matrices (endowed with the Euclidean metric) by the orthogonal group. Our results allow understanding the failure of some curve fitting algorithms, when the rank of the data is overestimated. We illustrate these observations on a dataset made of covariance matrices characterizing a wind field.

Theo Mary: Mixed Precision Low Rank Compression of Data Sparse Matrices

Abstract: Modern hardware increasingly supports low precision arithmetics, that provide unprecedented speed, communication, and energy benefits. Mixed precision algorithms are being developed that combine these lower precision arithmetics with higher precision ones, so as to avoid compromising the accuracy of the computations. In this talk, we present an approach to compute low rank approximations with multiple precisions. For low rank matrices with rapidly decaying singular values, we show that only a fraction of the entries need to be stored in the higher, target precision. We apply this idea to the compression of data sparse matrices, which possess a block low rank structure, and obtain significant gains in storage for a variety of applications. We conclude by discussing ongoing research on how to factorize data sparse matrices in mixed precision.

Pawan Goyal: Data-Driven Identification of port-Hamiltonian Systems

Abstract: Port-Hamiltonian systems have gained much attention in recent years due to their inherent valuable properties such as stability and passivity. In this work, we are interested in constructing linear port-Hamiltonian systems from input-output data in the frequency domain (frequency response). In particular, we study the identification problem of a passive system in the port- Hamiltonian form using tangential interpolation data. We present a simple construction approach based on the Mayo-Antoulas generalized realization theory that automatically yields a port-Hamiltonian realization for every strictly passive system with simple spectral zeros. We also discuss inferring frequency response using the time-domain input-output data. We illustrate the proposed method by means of several examples.

Anastasia Borovykh: To interact or not? The benefits of interacting particles: convergence properties and variance reduction

Abstract: A good machine learning model can be characterized by i) convergence (close) to the optimum, ii) good generalization performance. The conventional way of optimizing is to run one instance of an optimizer. The question we address here is whether it is beneficial to run multiple instances, or particles, and allow these to interact. Working in the general context of stochastic mirror descent, we present convergence results which show that by controlling the interaction the variance of stochastic gradients can be reduced. At the same time interactions can help in converging to flatter minima which have been shown to generalize better.

Jamie Haddock: Greedy and randomized projection methods

Abstract: Stochastic iterative algorithms have gained recent interest for solving large-scale systems of equations, Ax=y. One such example is the Randomized Kaczmarz (RK) algorithm, which acts only on single rows of the matrix A at a time. While RK randomly selects a row, Motzkin's algorithm employs a greedy row selection; the Sampling Kaczmarz-Motzkin (SKM) algorithm combines these two strategies. In this talk, we present a convergence analysis for SKM which interpolates between RK and Motzkin's algorithm.

Gerhard Kirsten: A matrix-oriented POD-DEIM algorithm applied to nonlinear differential matrix equations

Abstract: We are interested in approximating the numerical solution of large dimensional nonlinear matrix differential equations. In the framework of the Proper Orthogonal Decomposition (POD) methodology and the Discrete Empirical Interpolation Method (DEIM), we derive a novel matrix-oriented reduction process leading to an effective, structure aware low order approximation of the original problem. The nonlinear term is also reduced by means of a fully matricial interpolation using left and right projections onto two distinct reduction spaces, giving rise to a new two-sided version of DEIM. Several numerical experiments based on typical benchmark problems illustrate the effectiveness of the new matrix-oriented setting.

Elisa Riccietti: The extended normal equations: conditioning and iterative solution

Abstract: The extended normal equations are linear systems of the form $A^T Ax=A^T b+c$, which occur for example in optimization (in some penalty function approaches and in multilevel methods). The occurrence of $c$ gives rise to a problem with a different conditioning from the standard normal equations and prevents direct application of standard methods for least squares. We obtain an explicit formula for the structured condition number that can be used to compute a more accurate estimate of the forward error than the standard one used for generic linear systems, which does not take into account the structure of the problem. Then, we propose a modification of CGLS tailored for this problem and we show its increased robustness and accuracy compared with standard iterative methods.

Taewon Cho: Hybrid Projection Methods for Large-scale Inverse Problems with Mixed Gaussian Priors

Abstract: When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed before starting the solution process. We develop hybrid projection methods for inverse problems with mixed Gaussian priors where the prior covariance matrix is a convex combination of matrices and the mixing parameter and the regularization parameter do not need to be known in advance. Such scenarios may arise when data is used to generate a sample prior covariance matrix or when different priors are needed to capture different qualities of the solution. The proposed hybrid methods are based on a mixed Golub-Kahan process, which is an extension of the generalized Golub-Kahan bidiagonalization, and a distinctive feature of the proposed approach is that both the regularization parameter and the weighting parameter for the covariance matrix can be estimated automatically during the iterative process.

Monica Pragliola: Automatic model reduction via hierarchical Bayesian modeling

Abstract: Solving inverse problems with sparsity promoting regularizing penalties can be recast in the Bayesian framework as finding a maximum a posteriori estimate with sparsity promoting priors. In the latter context, a computationally convenient choice of prior is the family of conditionally Gaussian hierarchical models for which the prior variances of the components of the unknown are independent and follow a hyperprior from a generalized gamma family. In this talk we will show how the selected densities can be fruitfully coupled to Krylov subspace iterative linear solvers, yielding a quick identification of the support of the solution and reducing the effective dimensionality of the problem. Computed examples will assess the performance of the algorithm highlighting its efficiency.

Mirjeta Pasha: Modulus-based iterative methods for constrained $\ell_p$-$\ell_q$ minimization

Abstract: The need to solve discrete ill-posed problems arises in many areas of science and engineering. Solutions of these problems, if they exist, are very sensitive to perturbations in available data. Regularization replaces the original problem by a nearby regularized problem, whose solution is less sensitive to the error in the data. The regularized problem contains a fidelity term and a regularization term. Recently, the use of a $p$-norm to measure the fidelity term and a $q$-norm to measure the regularization term has received considerable attention. The balance between these terms is determined by a regularization parameter. In many applications, such as in image restoration, the desired solution is known to live in a convex set, such as the nonnegative orthant. It is natural to require the computed solution of the regularized problem to satisfy the same constraint(s). In this talk we will show that this procedure induces a regularization method and describes a modulus-based iterative method for computing a constrained approximate solution of a smoothed version of the regularized problem. Convergence of the iterative method is shown, and numerical examples that illustrate the performance of the proposed method are presented.

Stephen Moore: Space-Time Multipatch Discontinuous Galerkin Isogeometric Analysis for Parabolic Evolution Problems

Abstract: We present and analyze a stable space-time multipatch discontinuous Galerkin isogeometric analysis (dGIGA) scheme for the numerical solution of parabolic evolution equations in deforming space-time computational domains. Here, we use a time-upwind test function and apply multipatch dGIGA methodology for discretizing the evolution problem both in space and in time. This yields a discrete bilinear form which is elliptic on the IGA space with respect to a space-time dG norm. This property together with a corresponding boundedness property and consistency and approximation results for the IGA spaces yields an a priori discretization error estimate with respect to the space-time dG norm. The theoretical results are confirmed by several numbers examples.

Stefano Pozza: A Lanczos-like method for the time-ordered exponential

Abstract: The time-ordered exponential is defined as the function that solves a system of coupled first-order linear differential equations with generally non-constant coefficients. In spite of being at the heart of much system dynamics, control theory, and model reduction problems, the time-ordered exponential function remains elusively difficult to evaluate. In the talk, we present a Lanczos-like algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. The algorithm is presented in a theoretical setting. Nevertheless, a strategy for its numerical implementation is also outlined and will be subject of future investigation.

Philip Saltenberger: On a generic canonical form for J-normal matrices

Abstract: This talk focuses on structured matrices in indefinite inner product spaces and their structure-preserving transformation to simpler forms. We consider matrices that are normal with respect to the symplectic indefinite inner product x*J_{2n}y and present a new 'generic' canonical form for J-normal matrices under symplectic similarity transformations. This canonical form is sparse, clearly structured and reveals the eigenvalues of the matrix at hand. Moreover, it exists for an open and dense subset of J-normal matrices and so is can be seen as a (topologically) generic canonical form for J-normal matrices under symplectic similarity.

Carolin Penke: Stable Computation of Generalized Polar Decompositions

Abstract: The QDWH algorithm computes the polar decomposition of a matrix in a stable and efficient way. We generalize this method in order to compute generalized polar decompositions with respect to signature matrices. Here, the role of the QR decomposition is played by the hyperbolic QR decomposition. However, it doesn't show the same favorable properties concerning stability as its orthogonal counterpart. Remedies are found by exploiting connections to the LDL^T factorization and by employing well-conditioned permuted graph bases. Applications are found in computational quantum physics, where eigenvalues and eigenvectors describe optical properties of condensed matter or molecules.

Matthew Colbrook: Computing spectral measures of self-adjoint operators

Abstract: Spectral measures of self-adjoint operators arise in numerous applications, providing an analogue of diagonalisation through the spectral theorem. Using the resolvent operator (solving shifted linear systems), we develop an algorithm for computing smoothed approximations of spectral measures. The algorithm can be applied to differential, integral, and lattice operators, achieving arbitrarily high-order convergence in terms of a smoothing parameter. Explicit pointwise and $$L^p$$-error bounds are derived. We provide numerical examples (using state-of-the-art spectral methods), including PDEs, and compute 1000 eigenvalues of a Dirac operator to machine precision without spectral pollution. This is joint work with Andrew Horning and Alex Townsend.

José Garay: Additive Schwarz preconditioners for a Localized Orthogonal Decomposition method

Abstract: The solution of multiscale elliptic PDEs with non-separable scales and high contrast in the coefficients by standard Finite Element Methods (FEM) is typically prohibitively expensive since it requires the resolution of all characteristic lengths to produce an accurate solution. Numerical homogenization methods such as Localized Orthogonal Decomposition (LOD) methods provide access to feasible and reliable simulations of such multiscale problems. These methods are based on the idea of a generalized finite element space where the generalized basis functions are obtained by modifying standard coarse FEM basis functions to incorporate relevant microscopic information in a computationally feasible procedure. Using this enhanced basis one can solve a much smaller problem to produce an approximate solution whose accuracy is comparable to the solution obtained by the expensive standard FEM. We present a variant of LOD that uses domain decomposition techniques to compute the basis corrections and we also provide a two-level preconditioner for the resulting linear system.

Giovanni Barbarino: The limit empirical spectral distribution of complex matrix polynomials

Abstract: It is widely known that the roots of a random polynomial with degree k, i.i.d. entries with distributions under mild assumptions, and in the limit k → ∞, are uniformly distributed on the unit circle. Similarly, classical results in random matrix theory state that, when the entries of an n × n matrix are i.i.d. distributed random variables with mean 0 and variance 1/n, and in the limit n → ∞, then the eigenvalues are uniformly distributed on the unit disk.

In this work, we study the spectral behaviour of complex n × n matrix polynomials with degree k, and we obtain exact formulae for the almost sure limit of their Empirical Spectral Distributions (ESD) in two distinct scenarios: (1) n → ∞ with k constant and (2) k → ∞ with n constant. We also show how the results change when considering monic polynomials.

The main tools in the work are the replacement principle by Tao, Vu and Krishnapur, and the Girko approach with the logarithmic potential, altogheter with classical perturbations result in linear algebra. Along the way, we also develop some auxiliary results of potential independent interest, regarding the tail bound for the norm of the pseudoinverse of a non-zero mean matrix, and estimates on the singular values of certain structured random matrices.

Patrick Kürschner:Inexact low-rank methods for solving large matrix equations

Abstract: We discuss low-rank methods for approximately solving large matrix equations by means of low-rank matrices. The rational Krylov subspace method and the low-rank alternating directions implicit (ADI) iteration are two well established state-of-the-art algorithms for this purpose, especially for matrix equations of Lyapunov, Sylvester, and Riccati type. In every iteration step of these methods, a large and sparse linear system of equations has to be solved. In this talk we investigate what happens when these linear systems are only solved inexactly, e.g., via iterative methods. We present criteria which dictate the necessary accuracy for these inexact linear solves. If time permits, we will also briefly talk about inexactness in matrix-valued low-rank Krylov methods for general linear matrix equations.

Kasia Świrydowicz: Low-synchronization Gram-Schmidt algorithms and Arnoldi-QR factorization

Abstract: I will present recent developments that lead to complete reformulation of modified and reiterated classical Gram-Schmidt into two (and one) synchronization algorithms. The results were also extended to Arnoldi Gram-Schmidt and thus allowed us to formulate one synchronization stable GMRES.

Asma Farooq: How perturbations propagate along the solutions of linear ordinary differential equations: a relative error analysis

Abstract: In this talk, we are going to present how perturbations in initial value y0 or in the co-efficient matrix A propagate along the solutions of n-dimensional linear ordinary differential equations. In other words for a fixed t >= 0 and y0 ∈ R^{n}, we study the conditioning of the problem $(y_0,A) \mapsto e^{tA}y_0$ and an asymptotic analysis of condition numbers, as $t \rightarrow +\infty$, will be given. The analysis is accomplished for the case where A is normal matrix. We remark that our study is new one because it is different from known conditioning studies of the matrix exponential: we look attentively at the relative errors, taking into account the role of initial value relevant to a perturbation $\widetilde{y}_0$ of the initial value y0 or to a perturbation $\widetilde{A}$ of the matrix A rather than the relative error considered in many papers in literature.

Xiaoxing Liu: Pairwise-relaxing SPH: a conservative and consistent SPH scheme

Abstract: In the context of SPH, there exist several consistent schemes such as corrective SPH and reproducing kernel SPH. However, these consistent SPH schemes have difficulties in ensuring conservation property. In this talk, a conservative and consistent SPH scheme, called pairwise-relaxing SPH (PR-SPH) will be introduced. In PR-SPH, pairwise relaxing coefficients are introduced and determined pair-wisely throughout the entire calculation domain by enforcing the Taylor-series consistency condition. An underdetermined system of equations is created and solved for these pairwise relaxing coefficients. Several numerical tests will be presented to demonstrate the conservation and consistency property of PR-SPH.

Yang Yu: Mask-wearing Face Recognition with Residual Networks

Abstract: With more people wearing masks to prevent the spread of the COVID-19 virus, the performance of many face recognition methods is challenged. Therefore, we established an approach to deal with face modeling and recognition with or without masks by adopting residual networks (ResNet). The proposed method deals with the detection problem with MTCNN and landmarks labeling with GBDT. Additional landmarks will be added to the edge of the mask, if exists, and be combined with the original ones for the alignment operation. Next we will draw an 128 dimension vector from the aligned landmarks by means of ResNet. And finally the face detection procedure could be carried out with the comparison of the Euclid distance between the vectors of different faces. Numerical experiments show that our method could recognize face shaded by several masks with relatively high accuracy. (No mask accuracy: 97%+, masked face accuracy: around 75%. Codes are still under implementation)

Elizabeth Newman: Train Like a (Var)Pro: Efficient Training of Neural Networks with Variable Projection

Abstract: Deep neural networks (DNNs) have achieved state-of-the-art performance across a variety of traditional machine learning tasks. The ability of DNNs to efficiently approximate high-dimensional mappings from input to target data has also motivated their use in scientific applications, e.g., to generate surrogate models. In this talk, we consider the supervised training of DNNs, which arises in many of the above applications. The central problem of optimizing the weights of the given DNN is notoriously challenging due to, e.g., the large number of weights and non-convexity.

To solve the optimization problem more efficiently, we propose the use of variable projection (VarPro), a method originally designed for separable nonlinear least-squares problems. We introduce our Gauss-Newton VarPro method (GNvpro) that extends the reach of the VarPro idea to non-quadratic objective functions, most notably, cross-entropy loss functions arising in classification. Through several numerical experiments, we demonstrate that GNvpro not only solves the optimization problem more efficiently but also yields DNNs that generalize better than commonly-used optimization schemes.

Samuel Hatfield:Constructing tangent-linear and adjoint models through automatic differentiation of neural networks

Abstract: Data assimilation refers to a class of algorithms that are used to provide state estimates of partially and/or noisily observed systems. A common application for data assimilation, and the focus for my research, is the construction of the initial conditions for numerical weather predictions. Many numerical weather prediction centres rely on the 4D-Var approach to data assimilation. With this algorithm, a linearisation of the forecast model, the tangent-linear model, is used to propagate model state perturbations forwards in time, and the adjoint of this linearised model is used to propagate gradients backwards in time. In this way, an objective function measuring the least-squares fit of a four-dimensional model trajectory to observations can be minimised efficiently. The construction and maintenance of these separate tangent-linear and adjoint models is usually done manually, a time-consuming, tedious and error-prone process. In this talk I will propose an alternative. By replacing subroutines in a weather model with an emulator based on a neural network, we can construct tangent-linear and adjoint models trivially by taking advantage of the by-design differentiability of the neural network. I will present results showing how an operational numerical prediction model using this scheme can produce forecasts competitive in skill with the original scheme.

Adem Kaya: Conditioning Analysis for Discrete Helmholtz Problems

Abstract:We examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the problem with the condition number. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations which have different characteristics. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions.

Nils-Arne Dreier: Hardware-Oriented Block Krylov Methods

Abstract: Krylov methods play an important role for solving systems of linear equations in high-performance computing. In this talk, we consider block Krylov methods for solving systems with multiple right-hand sides. We build up on a recently presented generic block Krylov framework. Beside the analysis of the convergence properties, we analyze the performance properties of the building blocks for modern high-performance software. Combining these results give insights for choosing optimal parameters in the block Krylov framework.

Alice Cortinovis: Randomized trace estimates for indefinite matrices with an application to determinants

Abstract: Randomized trace estimation is a popular technique to approximate the trace of a large-scale matrix A by computing the average of quadratic forms x^T*A*x for many samples of a random vector X. We present new tail bounds that apply to indefinite matrices A for Rademacher and Gaussian random vectors. Then we focus on the approximation of the determinant of a symmetric positive definite matrix B, which is equal to exp(trace(log(B))), where the matrix log(B) is indefinite in general. We study the combination of our bounds with the Lanczos method for the approximation of x^T*log(B)*x, improving and extending an existing result.