International Conference on Scientific Computation and Differential Equations
July 15 - 19, 2024 @ National University of Singapore, Singapore
Invited Plenary Speakers
Ann ALMGREN
Lawrence Berkeley National Laboratory, USA
Elena CELLEDONI
Norwegian University of Science and Technology, Norway
Qianxiao LI
National University of Singapore, Singapore
Jianfeng LU
Duke University, USA
Carola-Bibiane SCHÖNLIEB
University of Cambridge, UK
Jie SHEN
Eastern Institute of Technology, Ningbo, China
Gilles VILMART
University of Geneva, Switzerland
Lei ZHANG
Peking University, China
Ann ALMGREN, Lawrence Berkeley National Laboratory, USA
Title: Adaptive Mesh Refinement: Algorithms and Applications
Abstract: Adaptive mesh refinement (AMR) is one of several techniques for dynamically modifying the spatial resolution of a simulation in particular regions of the spatial domain. Block-structured AMR specifically refines the mesh by defining locally structured regions with finer spatial, and possibly temporal, resolution. This combination of locally structured meshes within an irregular global hierarchy is in some sense the best of both worlds in that it enables regular local data access while enabling greater flexibility in the overall computation.
Originally, block-structured AMR was designed for solving hyperbolic conservation laws with explicit time-stepping; in this case the changes to solution methodology in transforming a single-level solver to an AMR-based solver are relatively straightforward. AMR has come a long way, however, and the more complex the simulation, the more complex the changes required to effectively use AMR. One can even consider whether to use different physical models at different levels of resolution. In this talk I will give an overview of block-structured AMR for different types of applications and will focus on a few key exemplars for how to think about adaptivity for multiphysics simulations.
Biography: Ann Almgren is a Senior Scientist and the Department Head of the Applied Mathematics Department in the Applied Mathematics and Computational Research (AMCR) Division of Lawrence Berkeley National Laboratory. Her primary research interest is in computational algorithms for solving partial differential equations (PDEs) in a variety of application areas. Her current projects include the development and implementation of new multiphysics algorithms in high-resolution adaptive mesh codes that are designed for the latest hybrid architectures. She is a SIAM Fellow, the Deputy Director of the ECP AMReX Co-Design Center, and serves on the editorial boards of CAMCoS and IJHPCA. In 2023 she was awarded the LBNL Director's Award for Exceptional Scientific Achievement. Prior to coming to LBL she worked at the Institute for Advanced Study in Princeton, NJ, and at Lawrence Livermore National Lab.
Elena CELLEDONI, Norwegian University of Science and Technology, Norway
Title: Deep learning of diffeomorphisms with application to shape analysis
Abstract: In this talk we discuss structure preservation and deep learning with applications to shape analysis. This is a framework for treating complex data and obtain metrics on spaces of data. Examples are spaces of unparametrized curves, time-signals, surfaces and images. A computationally demanding task for estimating distances between shapes, e.g. in object recognition, is the computation of optimal reparametrizations. This is an optimisation problem on the infinite dimensional group of orientation preserving diffeomorphisms.
We approximate diffeomorphisms with neural networks and use the optimal control and dynamical systems point of view to deep learning. We will discuss useful geometric properties in this context e.g. reparametrization invariance of the distance function, invertibility and contractivity of the neural networks. We will consider theory and applications of these ideas.
Biography: Elena Celledoni is a professor at the Department of Mathematical Sciences at the Norwegian University of Science and Technology (NTNU). She completed her undergraduate studies at the University of Trieste, Italy and her PhD at the University of Padua, Italy. She was a postdoctoral fellow at Department of Applied Mathematics and Theoretical Physics, Cambridge, UK, at the Mathematical Sciences Research Institute, Berkeley, California and at NTNU. She works on numerical analysis of differential equations and in particular the theory and applications of structure preserving algorithms. These methods are of use in industry for the simulation and control of rigid body dynamics, of slender structures, and of mechanical systems in general. She is also interested in the analysis and design of neural networks and their interplay with numerical analysis. This includes methods of shape analysis on Lie groups applied to activity recognition, with techniques both for curves and surfaces.
Qianxiao LI, National University of Singapore, Singapore
Title: On dynamics and learning
Abstract: In this talk, we discuss some interesting problems on the interface of machine learning and dynamical systems. The first part concerns the connections between deep residual networks and control systems. The second part involves the correspondence between sequence modelling and partially observed dynamics. Both lines of investigation point to new mathematical results and practical insights that arise when learning meets dynamics.
Biography: Qianxiao Li is an assistant professor in the Department of Mathematics, and a principal investigator in the Institute for Functional Intelligent Materials, National University of Singapore.
He graduated with a BA in mathematics from the University of Cambridge and a PhD in applied mathematics from Princeton University.
His research interests include the interplay of machine learning and dynamical systems, control theory, stochastic optimisation algorithms and data-driven methods for science and engineering.
Jianfeng LU, Duke University, USA
Title: Analysis of flow-based generative models
Abstract: We will discuss recent progress on mathematical analysis of flow based generative models, a highly successful paradigm for learning a probability distribution from data and generating further samples. We will talk about recent results in convergence analysis of diffusion models and related flow-based methods. In particular, we established convergence of score-based diffusion models applying to any distribution with bounded 2nd moment, relying only on a $L^2$-accurate score estimates, with polynomial dependence on all parameters and no reliance on smoothness or functional inequalities. We will also discuss convergence analysis of flow-based generative models based on tools from optimal transportation, viewing the forward process as a proximal gradient descent under Wasserstein metric. Time permits, we will also discuss some applications of this paradigm to scientific machine learning problems.
Biography: Jianfeng Lu is a Professor of Mathematics, Physics, and Chemistry at Duke University. Before joining Duke University, he obtained his PhD in Applied Mathematics from Princeton University in 2009 and was a Courant Instructor at New York University from 2009 to 2012. He works on mathematical analysis and algorithm development for problems and challenges arising from computational physics, theoretical chemistry, materials science, high-dimensional PDEs, and machine learning. He is a fellow of AMS. His work has been recognized by a Sloan Fellowship, a NSF Career Award, the IMA Prize in Mathematics and its Applications, and the Feng Kang Prize.
Title: Mathematical imaging: From geometric PDEs and variational modelling to deep learning for images
Abstract: Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of nonlinear partial differential equations, inverse problems, harmonic, stochastic and statistical analysis, and optimisation.
In this talk we will learn about some of these mathematical problems, about variational models and PDEs for image analysis and inverse imaging problems as well as recent advances where such mathematical models are complemented by deep neural networks.
The talk is furnished with applications to art restoration, forest conservation and cancer research.
Biography: Carola-Bibiane Schönlieb is Professor of Applied Mathematics at the University of Cambridge. There, she is head of the Cambridge Image Analysis group and co-Director of the EPSRC Cambridge Mathematics of Information in Healthcare Hub. Since 2011 she is a fellow of Jesus College Cambridge and since 2016 a fellow of the Alan Turing Institute, London. She also holds the Chair of the Committee for Applications and Interdisciplinary Relations (CAIR) of the EMS. Her current research interests focus on variational methods, partial differential equations and machine learning for image analysis, image processing and inverse imaging problems. She has active interdisciplinary collaborations with clinicians, biologists and physicists on biomedical imaging topics, chemical engineers and plant scientists on image sensing, as well as collaborations with artists and art conservators on digital art restoration.
Her research has been acknowledged by scientific prizes, among them the LMS Whitehead Prize 2016, the Philip Leverhulme Prize in 2017, the Calderon Prize 2019, a Royal Society Wolfson fellowship in 2020, a doctorate honoris causa from the University of Klagenfurt in 2022, and by invitations to give plenary lectures at several renowned applied mathematics conferences, among them the SIAM conference on Imaging Science in 2014, the SIAM conference on Partial Differential Equations in 2015, the SIAM annual meeting in 2017, the Applied Inverse Problems Conference in 2019, the FOCM 2020 and the GAMM 2021.
Carola graduated from the Institute for Mathematics, University of Salzburg (Austria) in 2004. From 2004 to 2005 she held a teaching position in Salzburg. She received her PhD degree from the University of Cambridge (UK) in 2009. After one year of postdoctoral activity at the University of Göttingen (Germany), she became a Lecturer at Cambridge in 2010, promoted to Reader in 2015 and promoted to Professor in 2018.
Jie SHEN, Eastern Institute of Technology, Ningbo, China
Title: A new class of higher-order stiffly stable schemes with application to the Navier-Stokes equations
Abstract: How to construct higher-order decoupled, and stable schemes for the Navier-Stokes equations has been a long-standing problem. More precisely, only the decoupled schemes with first-order pressure extrapolation have been proven to be stable and convergent.
To overcome this difficulty, we first construct a new class of higher-order stiffly stable schemes for parabolic equations. Different from traditional time discretization schemes which are usually based on Taylor expansions at $t_{n+\beta}$ with $\beta\in [0,1]$ and whose stability regions decrease as their order of accuracy increase, we construct new schemes based on Taylor expansion at $t_{n+\beta}$ with $\beta>1$ as a parameter, and show that their stability regions increase with $\beta$, thus allowing us to choose $\beta$ according to the stability and accuracy requirement. We shall provide a rigorous stability and error analysis for this new class of schemes.
Then, we show that by choosing suitable $\beta$, we can construct unconditionally stable (in H^1 norm), decoupled consistent splitting schemes up to fourth-order for the time-dependent Stokes problem. Finally, by combining the generalized SAV approach with the new consistent splitting schemes, we can construct unconditionally stable and totally decoupled schemes of second- to fourth order for the Navier-Stokes equations, and derive uniform optimal error estimates. We shall also present ample numerical results to show the computational advantages of these schemes.
Biography: Professor Jie Shen received his B.S. in Computational Mathematics from Peking University in 1982, and his PhD in Mathematics from Universite de Paris-Sud (currently Paris Saclay) at Orsay in 1987. He worked at Indiana University (1987-1991), Penn State University (1991-2001), University of Central Florida (2001-2002) and Purdue University (2002-2023). He served as the Director of Center for Computational and Applied Mathematics at Purdue University from 2012 to 2022, and was ratified as Distinguished Professor of Mathematics at Purdue University in 2023. In May 2023, he joined Eastern Institute of Technology, Ningbo, China, as a Chair Professor and Dean of School of Mathematical Science.
He is a recipient of the Fulbright “Research Chair” Award in 2008 and the Inaugural Research Award of the College of Science at Purdue University in 2013, and an elected Fellow of AMS and SIAM.
He serves on editorial boards for several leading international research journals, and has authored/coauthored over 250 peer-reviewed research articles and two books with over 25,000 citations in Google Scholar.
His main research interests are numerical analysis, spectral methods and scientific computing with applications in computational fluid dynamics and materials science.
Gilles VILMART, University of Geneva, Switzerland
Title: Explicit stabilized integrators for stiff problems: the interplay of geometric integration and stochastic integration
Abstract: The preservation of geometric structures by numerical methods, such as the symplecticity of the flow for the long time solution of deterministic Hamiltonian systems, often reveals essential for an accurate numerical integration, and this is the objective of geometric numerical integration.
In this talk, we highlight the role that some key geometric integration tools originally introduced in the deterministic setting, such as modified differential equations, processing techniques, Butcher trees, B-series and their generalizations, play in the design of high-order stochastic integrators, in particular for sampling the invariant distribution of ergodic stochastic partial differential equations or high-dimensional ergodic stochastic systems that typically arise in Langevin dynamics in the context of molecular dynamics simulations.
We show that this approach reveals decisive in particular for the construction of efficient explicit stabilized integrators for stiff stochastic problems, which are a popular alternative to implicit methods to avoid the severe timestep restrictions faced by standard explicit integrators.
Geometric numerical integration, high-dimensional ergodic stochastic systems, Butcher trees, B-series, explicit stabilized integrators.
Biography: Gilles Vilmart is a Senior Lecturer at the University of Geneva (Switzerland), Section of Mathematics. He received his PhD in Mathematics in 2008 from the University of Rennes 1 (France, National Institute for Research in Digital Science and Technology) and the University of Geneva (double doctorate program). Before joining the University of Geneva in 2013, he was a post-doctoral researcher at the Swiss Federal Institute of Technology, Lausanne and agrégé-préparateur at the École Normale Supérieure de Rennes where he obtained his Research Habilitation in 2013.
His research focuses on the numerical analysis of geometric and multiscale methods for deterministic or stochastic (partial) differential equations, with special emphasis on geometric numerical integration methods with related algebraic structures, and numerical homogenization methods for highly oscillatory problems.
He received the Mathematics Young Researcher First Prize 2013 of the Region Bretagne (France).
Lei ZHANG, Peking University, China
Title: Construction of Solution Landscape for Complex Systems
Abstract: Energy landscape has been widely applied to many physical and biological systems. A long-standing problem in computational mathematics and physics is how to search for the entire family tree of possible stationary states on the energy landscape without unwanted random guesses? Here we introduce a novel concept “Solution Landscape”, which is a pathway map consisting of all stationary points and their connections. We develop a generic and efficient saddle dynamics method to construct the solution landscape, which not only identifies all possible minima, but also advances our understanding of how a complex system moves on the energy landscape. We then apply the solution landscape approach to study two problems: One is construction of the defect landscapes of confined nematic liquid crystals, and the other one is to find the transition pathways connecting crystalline and quasicrystalline phases.
Biography: Lei Zhang is Boya Distinguished Professor at Beijing International Center for Mathematical Research, Peking University. He is also a Principal Investigator at Center for Quantitative Biology, Center for Machine Learning Research. He obtained his PhD in Mathematics at Penn State University in 2009. His research is in the area of computational and applied mathematics and interdisciplinary science in biology, materials, and machine learning. He has published the papers in Phys. Rev. Lett., PNAS, Acta Numerica, Science journals, Cell journals, SIAM journals. He was awarded/funded by NSFC Innovation Research Group, NSFC Outstanding Youth Award, National Key Research and Development Program of China, NSFC Excellent Youth Award, Royal Society Newton Advanced Fellowship, etc. He serves as an Associate Editor for SIAM J. Appl. Math, Science China Mathematics, CSIAM Trans. Appl. Math, DCDS-B, The Innovation, and Mathematica Numerica Sinica.