numerical methods

Trees, Algebras, and Differential Equations: Extending the B-series.jl package for numerical analysis of initial value problems

In this project we seek to extend the capabilities of bseries.jl to new classes of methods and/or new kinds of analysis. There are a number of possible directions and the specific one chosen will depend on the interests and knowledge of the student. Possibilities include extensions to:- Multi-derivative methods - Partitioned methods (e.g. for Hamiltonian systems) - General linear (multistep, multistage) methods - Exponential methods - Alternative bases for order conditions - Application of simplifying assumptions in method design - Generalized additive Runge-Kutta methods - Characterization of energy-preserving B-series - Extensions of B-series, such as aromatic B-series, exotic B-series, and S-series.

Applied Mathematics School 2024

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The KAUST Applied Mathematics School is a 1-week visiting program aiming to introduce undergraduate and master's students to special research topics focusing on key areas in applied mathematics and computational science.

Numerical Mathematics Research Group

Professor David I. Ketcheson leads the NUMERICS research group, which stands for Numerical Mathematics Research group areas of expertise and current scientific interests: numerical methods, partial differential equations, nonlinear waves Research within our group focuses on the design, analysis, and implementation of numerical methods for ordinary and partial differential equations, as well as the application of numerical methods to problems in nonlinear wave propagation. Meet the Team Get to know NUMERICS Team Events Learn more about NUMERICS events Projects Learn more about NUMERICS projects

Welcome at Kaust UQ School On Numerical Methods For Direct And Inverse Problems 2016 22-28 May

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B9 H1 R2322

KAUST UQ SCHOOL 2016 is an annual base thematic conference at King Abdullah University of Science and Technology held by Raul Tempone, Professor of Applied Mathematics and Computational Sciences at CEMSE (Computer, Electrical and Mathematical Sciences & Engineering Division). Tempone’s interests in the mathematical foundation of computational science and engineering are reflected in this summer school. The school’s goal is to provide participants with an overview on the most recent research progress in the field of uncertainty quantification, with emphasis to • Multi-Level and Multi-Index

Efficient Numerical Methods for High-Dimensional Approximation Problems

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B4 L5 R5220

I​n the field of uncertainty quantification, the effects of parameter uncertainties on scientific simulations may be studied by integrating or approximating a quantity of interest as a function over the parameter space. If this is done numerically, using regular grids with a fixed resolution, the required computational work increases exponentially with respect to the number of uncertain parameters -- a phenomenon known as the curse of dimensionality.

Mini Course Part 2-3 - Modelling and numerical methods for diffuse interface models in compressible multi-phase flows

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B1 L3 R3119

In this course we consider multi-phase flows, i.e., flows of one substance which is present as liquid as well as vapor. We focus on models that resolve individual bubbles/droplets and that treat both phases as compressible. We will also discuss incompressible/low Mach limits, since in most applications the liquid is nearly incompressible. Understanding and simulating such small-scale models is important in order to obtain information which can be used in larger scale models for e.g. sprays which play important roles in processes of practical interest as diverse as combustion, chemical engineering, and cloud formation

Mini Course Part 3-3 - Modelling and numerical methods for diffuse interface models in compressible multi-phase flows

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B1 L3 R3119

In this course we consider multi-phase flows, i.e., flows of one substance which is present as liquid as well as vapor. We focus on models that resolve individual bubbles/droplets and that treat both phases as compressible. We will also discuss incompressible/low Mach limits, since in most applications the liquid is nearly incompressible. Understanding and simulating such small-scale models is important in order to obtain information which can be used in larger scale models for e.g. sprays which play important roles in processes of practical interest as diverse as combustion, chemical engineering, and cloud formation

Mini Course Part 1-3 - Modelling and numerical methods for diffuse interface models in compressible multi-phase flows

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B1 L3 R3119

In this course we consider multi-phase flows, i.e., flows of one substance which is present as liquid as well as vapor. We focus on models that resolve individual bubbles/droplets and that treat both phases as compressible. We will also discuss incompressible/low Mach limits, since in most applications the liquid is nearly incompressible. Understanding and simulating such small-scale models is important in order to obtain information which can be used in larger scale models for e.g. sprays which play important roles in processes of practical interest as diverse as combustion, chemical engineering, and cloud formation

Numerical methods for stochastic conservation laws with multiplicative rough drivers in the flux

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B1 L4 R4214

Abstract Stochastic conservation laws (SCL) with quasilinear multiplicative``rough'' path dependence in the flux arise in modeling of mean field games. An impressive collection of theoretical results has been developed for SCL in recent years by Gess, Lions, Perthame, and Souganidis. We present the first fully computable numerical methods for pathwise solutions of scalar SCL with, for instance, "rough" paths in the form of Wiener processes. Convergence rates are derived for the numerical methods and we show that for strictly convex flux functions, "rough" path oscillations lead to

Energy and Entropy in Numerical Methods

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B9 L2 H2

Secondary quantities such as energy and entropy can be very important for numerical methods. Firstly, preserving these quantities can ensure that non-physical behavior is excluded. Secondly, preserving such quantities can result in stability estimates. Finally, preserving the correct energy/entropy evolution in time can result in additional desirable properties such as lower numerical errors. In this talk, a brief overview of some recent advances concerning energy and entropy preserving numerical methods for ordinary and partial differential equations will be given, together with an outlook on future research directions and applications.

Deterministic particle approximation for nonlinear transport equations

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Building L3 R3119

Approximating the solution to an evolutionary partial differential equation by a set of "moving particles" has several advantages. It validates the use of a continuity equation in an "individuals-based" modeling setting, it provides a link between Lagrangian and Eulerian description, and it defines a "natural" numerical approach to those equations. I will describe recent rigorous results in that context. The main one deals with one-dimensional scalar conservation laws with nonnegative initial data, for which we prove that the a suitably designed "follow-the-leader" particle scheme approximates entropy solutions in the sense of Kruzkov in the many particle limit. Said result represents a new way to solve scalar conservation laws with bounded and integrable initial data. The same method applies to second order traffic flow models, to nonlocal transport equations, and to the Hughes model for pedestrian movements.

Mean Field Games: Numerical Methods and Applications

Mathieu Laurière, Postdoc, Department of Operations Research and Financial Engineering, Princeton University

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KAUST

In this talk, several numerical methods will be presented and illustrated on examples. Borrowing tools from stochastic analysis, optimization, partial differential equations and machine learning, these methods enable us to solve mean field games with possibly complex sources of noise or high dimensional state variables.

International Seminar on Mathematical Models and Numerical Methods for Flow and Transport in Porous Media

Speakers from KAUST, CEMSE, PSE, G-CSC, IBRAE and INM RAS

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KAUST

This workshop is devoted to numerical simulation of groundwater flow and subsurface contamination transport, as well as related problems. The main topics are mathematical modeling of the processes in porous media and the numerical methods for discretization, solution of the discretized systems and numerical treatment of inverse problems. In particular, fractured porous media and partially saturated aquifers will be concerned.

Asymptotic Analysis and Numerical Methods for Compressible Flows at Low Mach Numbers

Prof. Dr. Andreas Meister, Professor, Department of Mathematics and Natural Sciences, University of Kassel

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B1 L4 R4102

We will present a comprehensive study of a finite volume method for inviscid and viscous flow fields at high and low speeds. Thereby, the results of a formal asymptotic low Mach number analysis are used to extend the validity of the numerical method from the simulation of compressible flow fields at transonic as well as supersonic speed to the low Mach number regime.

Numerical Methods for Energy Minimisation: Application to Machine Learning Algorithms

Prof. Charalambos Makridakis, Director of the Institute of Applied and Computational Mathematics of FORTH, Greece

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B2 L5 R5220

In this talk, we discuss problems and numerical methods arising in the calculus of variations and energy minimization. Among numerous applications, energy minimization is a core element of Machine Learning algorithms. Within the field of nonlinear PDEs, the calculus of variations has received a lot of attention from the analysis point of view. Although quite interesting and challenging, the numerical analysis of these problems is much less developed.

Multiscale numerical methods. Theory and Practice

Yalchin Efendiev, Professor, Institute for Scientific Computation, Texas A and M University

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B9 L3 R3128

In this talk, I will talk about general approaches for multiscale modeling (closely related to porous media applications). I will mainly focus on numerical approaches, where multiscale finite element basis functions are constructed and used in approximating the solution. In these approaches, macroscopic equations are formed via some variational formulations of the problem. I will discuss how these approaches are used in deriving, so called upscaling techniques and the relation to well known upscaling methods. The concepts discussed in the talk are used for linear and nonlinear problems. I will discuss some applications.

Fast Solution Methods for Wave Propagation Problems: From Classical Domain Decomposition Solvers to Learning

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B2 L5 R5220

Wave propagation and scattering problems are of huge importance in many applications in science and engineering - e.g., in seismic and medical imaging and more generally in acoustics and electromagnetics.

International Agenda for KAUST's SRI-UQ

1 min read ·
SIAM News published a piece on Professor Raul Tempone's Center for Uncertainty Quantification.

Conference on Advances in Uncertainty Quantification Methods, Algorithm and Applications (#UQAW2016)

1 min read ·
The 4th consecutive year that KAUST is holding the UQAW annual event dedicated to reporting on the state-of-the-art and latest advances in the methods used to quantify the calculation uncertainties. The aim of the discussion forum in exchanging information on innovative algorithm design as well as their practical applications and performances in experimental settings.

Understanding the COVID wave

1 min read ·
As countries around the world work to "flatten the curve," a reference to slowing the progression of COVID-19 cases, David Ketcheson is putting mathematical modeling to work to understand how transmission is playing out.

Longfei Gao

Dr. Longfei Gao was a Postdoctoral Research Fellow at Extreme Computing Research Center (ECRC) in King Abdullah University of Science and Technology (KAUST), working with Professor David Keyes in his research group. Education Profile PhD in Applied Mathematics and Computational Science, King Abdullah University of Science and Technology (KAUST), 2013 BS in Mathematics and Applied Mathematics, Jinan University (JNU), Guangzhou, China, 2010 Professional Memberships Postdoctoral Researcher, Extreme Computing Research Center, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

Lauri Mustonen

Lauri Mustonen is a Former visiting student from Aalto University at Professor Raul F. Tempone's Stochastic Numerics Research Group (STOCHNUM) at King Abdullah University of Science and Technology (KAUST). Research Interests Lauri's research interests include uncertainty Quantification, Inverse Problems, Numerical Methods. Education Profile M.Sc. (Tech.), Engineering Physics, and Mathematics, Aalto University, 2014. Professional Memberships http://math.aalto.fi/en/research/applied/inverseproblems/

Kadir Akbudak

Dr. Kadir Akbudak is a Consultant at Extreme Computing Research Center at KAUST, working with Professor David E. Keyes and Dr. Hatem Ltaief. Research Interests Kadir Akbudak's research interests include Combinatorial scientific computing, Graph and hypergraph partitioning, Locality-aware partitioning and scheduling, High performance kernels for sparse computations, Vectorization for irregular applications, Iterative numerical methods and Parallel numerical algorithms. Education Profile Ph.D., Computer Engineering, Bilkent University, 2015. M.S., Computer Engineering, Bilkent University, 2009

Hakan Bagci

Professor Bagci is a highly accomplished researcher in the field of computational electromagnetics (CEM). His research focuses on the theoretical and applied aspects of CEM.

Gabriel Wittum

Dr. Gabriel Wittum, professor of applied mathematics and computational science at KAUST, is an expert in modelling and simulating problems in empirical science.

David Ketcheson

Professor Ketcheson is an expert in numerical analysis and computational science, specializing in the study of numerical methods for differential equations and the modeling and analysis of time-dependent nonlinear wave phenomena.

André Gustavo Carlon

Postdoctoral Research Fellow, Stochastic Numerics Research Group
André Gustavo Carlon is a Postdoctoral Fellow at Stochastic Numerics Research Group under the supervision of Professor Raul F. Tempone at King Abdullah University of Science and Technology (KAUST). Research Interests André's research interests include the analysis of stochastic optimization methods, ​the application of stochastic gradient methods to engineering problems, and Bayesian optimal experimental design with non-linear models. Selected Publications​ A. G. Carlon, L. Espath, R. Tempone. Approximating Hessian matrices using Bayesian inference: a new approach for quasi-Newton methods in

Aku Jaakko Alexis Kammonen

Postdoctoral Research Fellow, Stochastic Numerics Research Group
Aku Jaakko Alexis Kammonen is a Postdoctoral Fellow at Stochastic Numerics Research Group under the supervision of Professor Raul F. Tempone at King Abdullah University of Science and Technology (KAUST). Research Interests Aku's research interests include Machine Learning, Neural Networks, Random Features, Spectral Bias, Adversarial Attacks, and Numerical Analysis. Selected Publications A. Kammonen, L. Liang, A. Pandey, and R. Tempone (2024, March 19–23). Comparing Spectral Bias and Robustness For Two-Layer Neural Networks: SGD vs Adaptive Random Fourier Features [Poster presentation]