Scientific Computing Seminar
Division of Applied Mathematics
Brown University
Fall 2024
Date: Friday, November 1 11am-noon OK; noon-1pm NOT OK (room is booked)
Speaker:
Date: Friday, October 25, 2024
Speaker: Wei Wang (University of Florida)
Title: Multiscale Discontinuous Galerkin Method for Schrodinger Equations
Abstract: In this talk, we will introduce a high-order multiscale discontinuous Galerkin (DG) method for stationary Schrodinger
equations in quantum transport. Due to the oscillatory nature of the solutions, traditional numerical methods require extremely
refined meshes to capture the small scale structure, resulting in the high computational cost. Our multiscale method applies WKB asymptotic to construct non-polynomial multiscale basis functions so that it can capture the solution better on coarse meshes. We will first present our approach for the 1D case and then extend our approach to a special 2D case where the solution exhibits oscillations mainly in one direction. Numerical results and error estimates will also be provided.
Joint work with Prof. Bo Dong and Prof. Chi-Wang Shu.
Date: Friday, October 11, 2024
Speaker: Andreas Meister of University of Kassel
Title: Asymptotic Analysis and Numerical Methods for Compressible Flows at all Mach Numbers
Abstract: We will present a comprehensive study of a finite volume method for inviscidand 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. To overcome the well-known failure of compressible numerical method in the low Mach number regime we combine the numerical flux function with a preconditioned formulation. Both,a wide variety of trans-, super-, hyper-, and subsonic realistic test cases as wellas a formal discrete asymptotic analysis are employed in order to prove the validity of the derived numerical method from hypersonic to low Mach numberfluid flow.
Date: Friday, September 13, 2024
Speaker: Adi Ditkowski
Title: Department of Applied Mathematics School of Mathematical Sciences, Tel Aviv University,
Abstract: Block Finite Difference methods (BFD) are Finite Difference (FD) methods in which the domain is divided into blocks, or cells, containing two or more grid points with a different stencil used for each grid point, unlike the standard FD method, where the same stencil is used for every grid point. Using this approach, we can design the scheme such that the leading term of the truncation error lies in a different subspace than the solution. Different dynamics can then be assigned to the error and the solution. Therefore, we can construct high-performance schemes.
In this talk, we present BFD for the heat and advection equations in one and multiple dimensions. For the heat equations, we show that we can use the scheme's dissipation to get a fifth-order convergence rate from a third-order truncation scheme. Furthermore, a six-order convergence rate can be obtained using a post-processing filter.
We can obtain a six-order phase error for the advection equation while maintaining the rest of the errors bounded in time.
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We also show that these BFD schemes are highly accurate, nodal-based Discontinuous Galerkin Methods.
This work was conducted in collaboration with Anne Le Blanc and Chi-Wang Shu.
Fall 2023
Date: Friday, October 6, 2023
Speaker: Jesse Chan Rice University
Title:
Abstract:
Date: Friday, November 10, 2023 AT Noon!
Speaker: Tarik Dzanic
Title: Constructing provably robust, constraint-satisfying finite element methods for computational physics
Abstract: Finite element methods (FEM) have been widely used for simulating a broad range of multi-scale, multi-physics phenomena due to their accuracy and computational efficiency. However, when approximating systems that must abide by strict physical constraints, their robustness can be severely degraded as the predicted solutions may not satisfy these constraints, resulting in physically inconsistent predictions or the failure of the schemes altogether. In this talk, I will present provably robust, constraint-satisfying FEM approaches for computational physics applications, particularly focusing on discontinuous FEM-type schemes for hyperbolic and mixed systems. The proposed methodology relies on constructing and enforcing specific constraints on the solution such that the schemes can: 1) guarantee that the predicted solution remains physically admissible; 2) robustly and accurately resolve features that may otherwise cause numerically ill-behaved solutions; and 3) retain the high fidelity, high-order accuracy, and computational efficiency of standard FEM approaches. This talk will discuss how physical admissibility constraints can be paired with more restrictive entropy conditions that ensure well-behaved numerical solutions without sacrificing accuracy and how these conditions can be enforced through nonlinear filtering methods, which can be efficiently implemented on modern massively-parallel computing architectures. The resulting approach will be shown across a variety of application areas, ranging from multi-species gas dynamics and viscous fluid flows to magnetohydrodynamics and statistical thermodynamics.
Spring 2024
Date: Friday, February 2, 2024
Speaker: Paris Perdrekis (U-PENN)
Title: A Unifying Framework for Operator Learning via Neural Fields
Abstract: Operator learning is an emerging area of machine learning which aims to learn mappings between infinite dimensional function spaces and has led to the development of new architectures such as the Fourier Neural Operator, the DeepONet, and their extensions. In this talk I will uncover a previously unrecognized connection between existing operator learning architectures and conditioned neural fields used in computer vision. This results in a unified framework for explaining differences between popular operator learning architectures, and creates a bridge for adapting well-developed tools from computer vision for operator learning. In particular, we find all existing operator learning architectures are neural fields whose conditioning mechanisms are restricted to use only pointwise and/or global information from their inputs. This motivates us to design new architectures which make use of a hierarchy of scales for conditioning a base neural field. By making use of multi-scale conditioning, we observe consistent performance gains and obtain state of the art results across a collection of challenging benchmarks in climate modelling and fluid dynamics.
*joint work with Jacob Seidman, Hanwen Wang, Shyam Sankaran and George Pappas
Date: Friday, February 9, 2024
Speaker: Andre Nachbin, Worcester Polytechnic Institute
Title: Solitary Water Waves on Graphs
Abstract: We have deduced a weakly nonlinear, weakly dispersive Boussinesq system for water waves on a 1D branching channel, namely on a graph. The model required a new compatibility condition at the graph’s node, where the main reach bifurcates into two reaches. The new nonlinear compatibility condition generalizes that presented by Stoker (1957) and includes forking angles. We present numerical simulations comparing solitary waves on the 1D (reduced) graph model with results of the (parent) 2D model, where a compatibility condition is not needed. We will comment on some new problems that arise.
Date: Friday, February 16, 2024
Speaker: Daniel Appelo (Virginia Tech)
Title: WaveHoltz: Parallel and Scalable Solution of the Helmholtz Equation via Wave Equation Iteration
Abstract: We introduce a novel idea, the WaveHoltz iteration, for solving the Helmholtz equation. The method is inspired by recent work on exact controllability (EC) methods and as in EC methods we make use of time domain methods for wave equations to design frequency domain Helmholtz solvers, but unlike EC methods we do not require adjoint solves. We show that the WaveHoltz iteration is symmetric and positive definite (compared to the indefinite Helmholtz equation). We present numerical examples, using various discretization techniques, that show that our method can be used to solve problems with rather high wave numbers.
Date: Friday, February 23, 2024
Speaker: Bamdad Hosseini, Washington University
Title: Kernel Methods for Solving and Learning PDEs: Algorithms and Error Analysis
Abstract: Abstract: In this talk I will present a kernel collocation method for solving PDEs and inverse problems that is inspired by advances in the machine learning literature and in particular the theory of Reproducing Kernel Hilbert Spaces (RKHSs) and Gaussian Processes (GPs). The proposed methodology has a lot of desirable properties such as a dimension benign implementation and relatively straightforward error analysis. I will discuss the main idea behind this error analysis and how it reveals a broader framework for the analysis of other PDE solvers including neural net techniques such as Physics Informed Neural Nets (PINNs).
Date: Friday, March 8, 2024
Speaker: . Yingda Cheng, Virginia Tech
Title: Implicit time and rank adaptive method for time-dependent PDEs
Abstract: In this work, we develop implicit time- and rank-adaptive schemes for stiff time-dependent matrix differential equations. The idea of dynamic low rank approximation (DLRA) is to capture the dynamic evolution of the low rank matrices by a numerical procedure. Our scheme is based on a three-step procedure similar to the unconventional robust integrator (BUG integrator). First, a prediction step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using a base implicit solve for the small core matrix. Finally, a truncation is made according to the error threshold. However, compared with the BUG integrator, which has convergence issues for some PDEs with cross terms, our scheme has two important features which enhance the robustness for convergence. We use the row and column spaces from the explicit step truncation method in the prediction step to alleviate the tangent projection error associated with the dynamic low rank approximation. In addition, a time adaptive strategy is proposed in conjunction with rank adaptivity. We benchmark the scheme in several tests such as anisotropic diffusion to show robust convergence properties.
Date: Friday, March 29, 2024
Speaker: Thomas Surowiec, Simula Research Laboratory
Title: Proximal Galerkin: A structure-preserving numerical method for constrained variational problems. From the obstacle problem to fully nonlinear PDEs
Abstract: Bound constraints are one of the most natural features to include in a physical model. Water should not flow through solid rock, warm plastic bends to its mold, and granular material like snow and sand has a natural angle of repose. A computational model can account for these properties by imposing bound constraints on the space of solutions. However, this natural modeling assumption has a steep price. At the theoretical level, it propels us from the realm of partial differential equations (PDEs) to the more intricate domain of infinite-dimensional variational inequalities (VIs). Meanwhile, at the computational level, we meet difficulties overcoming challenges with mesh-dependence, high iteration complexity, and low accuracy that do not appear in unconstrained numerical models.
By combining notions from convex analysis, mathematical optimization, and mixed finite element methods, we propose a new type of nonlinear, geometry-preserving finite element method for variational inequalities, optimal control problems, and several challenging classes of fully nonlinear PDEs. We call this approach the "Proximal Galerkin Method."
With the classical obstacle problem serving as the canonical example for pointwise bound-constrained elliptic variational inequalities, we systematically motivate the method from the perspective of solvers for mixed complementarity problems arising in optimization. After introducing two classes of specially tailored finite element spaces, we investigate the method's performance over various meshes and polynomial degrees, where we observe strong mesh-independence, low iteration complexity, and high order accuracy. We also consider the method's behavior on degenerate problems and its ease of implementation, even when using higher-degree basis functions. These results are complemented by examples in topology optimization and advection-diffusion. The second part of the talk is focussed on both the theoretical underpinnings of the method as well as extensions to fully nonlinear PDEs; in this case, the Eikonal and Monge-Ampère equations.
The proximal Galerkin method was discovered in collaboration with Brendan Keith (Brown).
The results of this talk are part of joint, ongoing work with Brendan Keith, Jørgen Dokken (Simula Research Laboratory), and Patrick E. Farrell (Oxford).
Date: Friday, April 12, 2024
Speaker: ROOM IS BOOKED FOR DEFENSE
Title:
Abstract:
Date: Friday, April 19, 2024
Speaker: Paula Chen (Naval Air Warfare Center Weapons Division (NAWCWD), China Lake)
Title: Recent Advancements in Designing FPGA Implementations for Real-Time Optimal Control
Abstract: Field programmable gate arrays (FPGAs) represent a promising hardware architecture for real-time, embedded applications. In appropriate settings, FPGAs have been shown to be able to achieve 10-100x speedup over CPUs and GPUs, while using less size, weight, and power. In this talk, we provide an overview of recent advancements and open challenges in designing real-time FPGA implementations for optimal control algorithms. In particular, we discuss how high-level synthesis software tools can be leveraged to facilitate this design process.
Date: Friday, April 26, 2024:
Speaker: Charlie Parker (Oxford University)
Title: Computing H2-conforming finite element approximations without having to implement C1-elements
Abstract: Fourth-order elliptic problems arise in a variety of applications from thin plates to phase separation to liquid crystals. A conforming Galerkin discretization requires a finite dimensional subspace of H2, which in turn means that conforming finite element subspaces are C1-continuous. In contrast to standard H1-conforming C0-elements, C1-elements, particularly those of high order, are less understood from a theoretical perspective and are not implemented in many existing finite element codes. In this talk, we address the implementation of the elements. In particular, we present algorithms that compute C1 finite element approximations to fourth-order elliptic problems and which only require elements with at most C0-continuity. We also discuss preconditioners and illustrate the method on a number of representative test problems.
Date: Friday, May 3, 2024:
Speaker: Florian Schaefer (Georgia Tech)
Title: Models, Solvers, Learners: Statistical Inspiration for Scientific Computing
Abstract: The convergence of scientific computing with statistics and machine learning is an exciting recent development. In this talk, I will present two lines of work that blur the line between statistical inference and numerical computation.
The first and main part of the talk uses ideas from semidefinite programming and information geometry to efficiently simulate gas dynamics in the presence of shock waves. The latter cause severe numerical challenges for classical and learning-based solvers.
The talk begins by observing that shock formation arises from the deformation map reaching the boundary of the manifold of diffeomorphisms. This motivates using the log-determinant barrier function of semidefinite programming to modify the geometry of the manifold such that the deformation map approaches but never reaches its boundary. This information geometric regularization (IGR) preserves the original long-time behavior without forming singular shocks, greatly simplifying numerical simulation. The modified geometry on the diffeomorphism manifold is also the information geometry of the mass density. I will show how this observation motivates information geometric mechanics that views the solutions of continuum mechanical equations as parameters of probability distributions to be evolved on a suitable information geometry, promising far-reaching extensions of IGR.
The second part of the talk derives fast solvers for elliptic PDEs by relating Cholesky factorization to the conditional distributions of a Gaussian process. The resulting algorithms achieve the new state of the art in terms of rigorous accuracy-vs-cost guarantees, are readily parallelizable, and performant in practice.
Date: Friday,May 10, 2024:
Speaker: Leszek Demkowicz (UT Austin)
Title: Separation of Variables with Non-Self Adjoint Operators with Applications to Analysis of Waveguides
Abstract: I have been teaching separation of variables for over 30 years, and I have always been instructing students
to look for a self-adjoint problem first. We know then that the separation constant (the eigenvalue) is real
and, if the operator happens to be positive definite, we also know that the constant is positive.
This a-priori knowledge about the separation constant simplifies greatly the solution of the problem.
More importantly, by the Sturm-Liouville theory (Spectral Theorem for Self-Adjoint Operators), we know
that the eigenvectors form simultaneously a basis for the L2 as well H1 space, which opens up the way
for a rigorous well-posedness proof.
Only recently, when studying acoustical and electromagnetic waveguides, we have run into a situation where
one needs to perform the separation of variables with a non-self adjoint operator. This has led me to
a half-year long study of excellent and fundamental book by Gohberg and Krein [1] and a number of fundamental
results that I have learned (and should have known a long time ago).
In the talk, I will illustrate the deep theory for non-self adjoint operators with a simple model acoustic
waveguide problem with impedance boundary conditions.
[1] I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert
Space (Translations of Mathematical Monographs). American Mathemtical Society, 1965
(Russian edition), vol. 18.
[2] M. Melenk, L. Demkowicz, and S. Henneking, ‘‘Stability analysis for electromagnetic
waveguides. Part 1: Acoustic and homogeneous electromagnetic waveguides.,’’ Oden
Institute, The University of Texas at Austin, Austin, TX 78712, Tech. Rep. 2, 2023, in review.
[3] L. Demkowicz, M. Melenk, J. Badger, and S. Henneking, ‘‘Stability analysis for acoustic and
electromagnetic waveguides. Part 2: Non-homogeneous waveguides.,’’ Advances in
Computational Mathematics, 2024, accepted, see also Oden Institute Report 2023/3.
Spring 2023
Date: Friday, Feb., 10, 2023
Speaker: Stefanie Guenther, Lawrence Livermore National Laboratory
Title: Optimal control of open quantum systems on HPC platforms
Abstract:
The fundamental operations in current superconducting quantum computers are realized by applying electromagnetic pulses that manipulate qubit states. Those pulses provide the interface between the quantum compiler and the quantum hardware, for example to initialize quantum algorithms, realize logical operations, or create non-classical states.
However, designing pulse shapes that drive the qubits to a desired target with high fidelity and in short turnaround times is a major challenge that often requires domain experts to perform experimental searches guided by physical expertise.
At LLNL, we have developed a framework for automated design of those pulses through numerical optimal control techniques on classical High-Performance Computing (HPC) platforms. In this talk, I will introduce the main components of this framework, from the numerical simulation of quantum dynamics in superconducting quantum devices, to the adjoint-based optimization scheme that designs the control pulses on HPC platforms. I will demonstrate various application scenarios such as the challenge of unconditional quantum reset and logical quantum gate transformations. I will include a short introduction to the main principles, advantages and challenges of current quantum computing so that this seminar can be self-contained and approachable for a wide audience.
Short Bio:
Stefanie's research focusses on numerical optimal control for differential equations on High-Performance Computing platforms. She joined the Lawrence Livermore National Laboratory (LLNL) in 2019 on the Sidney Fernbach Fellowship, where she worked on optimal control techniques to improve and accelerate scientific machine learning, as well as optimal control for quantum computing to improve fundamental quantum operations, and is now a staff member at the Center for Applied Scientific Computing (CASC) at LLNL since Jan 2022. Prior, Stefanie has received her PhD in 2017 from RWTH Aachen University, Germany, researching simultaneous optimization methods for time-dependent partial differential equations with applications in computational fluid dynamics, including chaotic dynamics in collaboration with Prof. Qiqi Wang at MIT, Cambridge.
Date: Friday, Feb. 17, 2023
Speaker: Xiangxiong Zhang, Purdue University
Title: Recent Progress on Q^k Spectral Element Method: Accuracy, Monotonicity and Applications on Compressible Navier-Stokes equations
Abstract:
Spectral element methods usually refer to finite element methods with high order polynomial basis. The Q^k spectral element method has been a popular high order method for solving second order PDEs, e.g., wave equations, for more than three decades, obtained by continuous finite element method with tenor product polynomial of degree k basis and with at least (k+1)-point Gauss-Lobatto quadrature. In this talk, I will present some brand new results of this classical scheme, including its accuracy, monotonicity (stability), and examples of using monotonicity to construct high order accurate bound (or positivity) preserving schemes in various applications including the Allen-Cahn equation coupled with an incompressible velocity field, Keller-Segel equation for chemotaxis, nonlinear eigenvalue problem for Gross–Pitaevskii equation, and especially compressible Navier-Stokes equations.
1) Accuracy: when the least accurate (k+1)-point Gauss-Lobatto quadrature is used, the spectral element method is also a finite difference (FD) scheme, and this FD scheme can sometimes be (k+2)-th order accurate for k>=2. This has been observed in practice and well known but never proven before as rigorous a priori error estimates in multiple dimensions. We are able to prove it for linear elliptic, wave, parabolic and Schrödinger equations for Dirichlet boundary conditions. For Neumann boundary conditions, (k+2)-th order can be proven if there is no mixed second order derivative. Otherwise, only (k+3/2)-th order can be proven and some order loss is indeed observed in numerical tests. The accuracy result also applies to the spectral element method on any curvilinear mesh that can be smoothly mapped to a rectangular mesh, e.g., solving a wave equation on an annulus region with a curvilinear mesh generated by polar coordinates.
2) Monotonicity: consider solving the Poisson equation, then a scheme is called monotone if the inverse of the stiffness matrix is entrywise non-negative. It is well known that second order centered difference or P1 finite element method can form an M-matrix thus they are monotone, and high order accurate schemes in general are not monotone. But on structured meshes, high order accurate schemes can be monotone, though they do not form M-matrices. In particular, we have proven that the fourth order accurate FD scheme (Q^2 spectral element method) is a product of two M-matrices thus monotone for a variable coefficient diffusion operator: this is the first time that a high order accurate scheme is proven monotone for a variable coefficient operator. We have also proven the fifth order accurate FD scheme (Q^3 spectral element method) is a product of three M-matrices thus monotone for the Poisson equation: this is the first time that a fifth order accurate discrete Laplacian is proven monotone in two dimensions (all previously known high order monotone discrete Laplacian in 2D are fourth order accurate). For solving compressible Navier-Stokes equations, the monotonicity of Q^2 and Q^3 elements can be used to construct semi-implicit positivity-preserving schemes.
Date: Friday, Feb. 24, 2023
Speaker: Hasnaa Zidani (INSA Rouen) and Olivier Bokanowski (Laboratoire Jacques-Louis Lions, Sorbonne universite)
Title: TBA
Abstract:
TBA.
Date: Friday, Mar. 24, 2023
Speaker: Tongtong Li (Math Department, Dartmouth College)
Title: Data Assimilation for Discontinuous State Variables
Abstract: Data assimilation is a method for combining available observations with a background from numerical model, to find the best estimate of the system, which is crucial for improving environmental variable prediction. However, commonly used Gaussian distribution assumption could introduce biases for state variables with discontinuous profiles, such as sea ice thickness with sharp features. In this talk, we focus on the design of non-Gaussian prior based on various statistics of the state variables. In particular, we adopt a covariance matrix, which is designed using the gradient information of the state variable, for a prior distribution in the data assimilation framework. This method is computationally efficien and is flexible to be applied in various data assimilation algorithms.
TBA.
Date: Wednesday, April 12, 2023
Speaker: Marie Elisabeth Rognes, Simula Research Laboratory
Title: Perivascular pathways and the dimension-2 gap
Abstract:
Your brain has its own waterscape: whether you are reading, thinking or sleeping, fluid flows through or around the brain tissue, clearing waste in the process. These biophysical processes are crucial for the well-being and function of the brain. In spite of their importance we understand them but little, and mathematical and computational modelling could play a crucial role in gaining new insight. In this talk, I will discuss mathematical, numerical and biophysical approaches to understand mechanisms underlying flow and transport in the human brain with an emphasis on coupled systems of partial differential equations with dimension gap > 1.
Short bio:
Marie E. Rognes is Research Professor in Scientific Computing and Numerical Analysis at Simula Research Laboratory, Oslo, Norway and a Visiting Scholar at the University of California San Diego. She joined Simula Research Laboratory in 2009, after received her Ph.D from the University of Oslo in the same year, led its Department for Biomedical Computing from 2012-2016, and currently leads a number of research projects focusing on mathematical modelling and numerical methods for brain mechanics including an ERC Starting Grant in Mathematics (2017-2023). She won the 2015 Wilkinson Prize for Numerical Software, the 2018 Royal Norwegian Society of Sciences and Letters Prize for Young Researchers within the Natural Sciences, is a member of the Norwegian Academy of Technological Sciences, and a member of the FEniCS Steering Council.
Date: Friday, April 14, 2023
Speaker: Alexandre Ern (Universite Paris-Est and INRIA)
Title: Hybrid high-order methods for the biharmonic problem
Abstract:
We start with a gentle introduction to the devising and analysis of hybrid high-order (HHO) methods for the Poisson model problem. Then, we address the biharmonic problem and we compare the proposed HHO methods to the literature, in particular to weak Galerkin methods. Finally, we briefly discuss how the error analysis can be carried out in the case of an exact solution with low regularity.
Date: Friday, May 5
Speaker: Bon Eisenberg ( Illinois Institute of Technology and Rush Medical School )
Title: From Maxwell to Mitochondria, a Kirchhoff Computation
Abstract: Current flow in circuits is the physical basis of computation
in computers and the nervous system. We show how the Maxwell equations
in circuits imply Kirchhoff's current law, but only when current includes
the universal displacement current ε_0 ∂E/∂t.
Mitochondria house the respiratory chain of proteins
that make ATP, that stores the chemical energy of life. Kirchhoff's
current law makes
possible computations of the 〖>10〗^18 atoms of the mitochondrion, as
it made possible the computation of the Atlantic cable (Kelvin Heaviside),
the nerve axon (Hodgkin Huxley), the lens of the eye (Mathias Huang),
and the glia of the optic nerve bundle (Huang Xu et al).
Date: Thursday, May 11, 2023 (B&H 751, 4:00 pm)
Speaker: Boyan Lazarov (Lawrence Livermore National Laboratory)
Title: Large scale density-based topology optimization with applications in mechanics, and coupled multiphysics problems
Abstract:
Topology optimization has gained the status of being the preferred optimization tool in the
mechanical, automotive, and aerospace industries. It has undergone tremendous development since its
introduction in 1988, and nowadays, it has spread to many other disciplines such as Acoustics, Optics, and
Material Design. The basic idea is to distribute material in a predefined domain by minimizing a selected
objective and fulfilling a set of constraints. The procedure consists of repeated system analyses, gradient
evaluation steps by adjoint sensitivity analysis, and design updates based on mathematical programming
methods. Regularization techniques ensure the existence of a solution.
The result of the topology optimization procedure is a bitmap image of the design. The ability of the method to
modify every pixel/voxel results in design freedom unavailable by any other alternative approach. However, this
freedom comes with the requirement of using the computational power of large parallel machines.
Incorporating a model accounting for exploitation and manufacturing variations in the optimization process and
the high contrast between the material phases increase further the computational cost. Thus, this talk focuses
on methods for reducing the computational complexity, ensuring manufacturability of the optimized design and
efficient handling of the high contrast of the material properties. The development will be demonstrated in
airplane wing design, compliant mechanisms, heat sinks, material microstructures for additive manufacturing,
and photonic devices.
Date: Friday, May 12, 2023
Speaker: Oleg Davydov (Giessen)
Title: Topics in Meshless Finite Difference Methods
Abstract:
Abstract: Meshless finite difference methods are the finite difference methods on irregular nodes, with stencils obtained by optimizing numerical differentiation using polynomials or radial basis
functions. These techniques are inherently meshless and isogeometric, and applicable to a large range of problems. I will report on several recent computational results for the elliptic equations on
manifolds, elliptic interface problems, Stokes equation and scalar conservation laws. In addition, a link to the collocation for the spaces of "overlap splines" will be discussed, that promises a
progress on the theoretical understanding of these methods.
TBA
Fall 2022
Date: Friday, Sept. 23, 2022 (CANCELED)
Speaker: Wasilij Barsukow, University of Bordeaux and CNRS
Title: Beyond Godunov’s method: A new structure preserving numerical method for conservation laws
Abstract:
Exact solutions to systems of conservation laws in multiple spatial dimensions often possess interesting additional properties which are a consequence of the equations. Examples are the evolution equations of vorticity or angular momentum, involutional constraints, stationary states or singular limits. A popular way to derive numerical methods for conservation laws (due to Godunov) approximates the solution by a piecewise constant function. This way, numerical diffusion is introduced which stabilizes the method, but generically prevents it from preserving any of those additional properties on grids with finite spacing (it is not “structure-preserving”). Excessive grid refinement, however, is expensive and impractical. Active Flux is a new kind of numerical method which uses a globally continuous approximation of the solution. The evolution of the averages is conservative, and the method is able to resolve shocks correctly. Its centerpiece is a short-time evolution of continuous data which provides the necessary upwinding and stability. The talk will describe this numerical method and show examples of it being naturally structure preserving.
Support provided from the C.V. Starr Foundation Lectureship Fund.
Date: Friday, Nov. 18, 2022
Speaker: Jay Gopalakrishnan, Portland State University
Title: A diagram chase to get a discrete elasticity complex
Abstract:
Differential complexes have shed new insight into the finite elements in recent years. This talk is devoted to the elasticity complex, which provides an example of how complicated exact sequence of spaces can be built from simple ones. Lining up two simpler complexes, we start by performing a "diagram chase", which often goes by the name of Bernstein-Gelfand-Gelfand resolution. In the remainder of the talk, we show how this process can be perfectly mimicked at the discrete level on a three-dimensional mesh of macroelements of Alfeld's type. It results in a discrete elasticity complex, complete unisolvent degrees of freedom and cochain projections. This is joint work with S. Christiansen, J. Guzman and K. Hu.
Date: Friday, Dec. 9, 2022
Speaker: Nisha Chandramoorthy (MIT)
Title: Learning from dynamics, learning dynamics and the dynamics of learning
Abstract:
In this talk, we take a dynamical systems approach toward three algorithmic questions that arise from complex systems in scientific and machine learning applications.
In the first part, we discuss the computation of linear response: the derivative of statistics or long-time averages of a dynamical system with respect to its input parameters. In many ergodic chaotic systems, such as certain turbulent fluid flows, detailed climate models, etc., linear response exists but has been notoriously difficult to compute. Apart from the curse of dimensionality, this difficulty can be attributed to a defining aspect of chaos: infinitesimal perturbations along a given orbit grow in norm exponentially. In this talk, we present a new alternative for linear response computation called the space-split sensitivity (S3) algorithm. One key component of S3 is a fast computation of conditional scores – log gradients of probability measures conditioned on the unstable manifold.
In the second third, we discuss the problem of Bayesian filtering in chaotic systems, where the goal is to sample from a sequence of filtering distributions, which are probability distributions of the state conditioned on past observations. Filtering algorithms based on measure transport do not yet exploit the structure in the target (filtering) distributions that arises from the underlying chaotic dynamics. Here we propose an ansatz, based on the concept of conjugacies, for the transport map and compute it by exploiting our fast computation of scores.
Finally, we discuss a problem where taking the dynamical systems approach is insightful for generalization in machine learning: the performance of a learning algorithm on unseen data. We consider local descent training algorithms that do not converge to a fixed point but whose long-time averages converge. We redefine generalization and training errors, which traditionally use loss values at fixed parameters, in terms of loss statistics. We then extend classical generalization analyses to such non-converging regimes. Further, we show how training dynamics can provide clues for generalization.
Bio: Nisha Chandramoorthy is a postdoctoral researcher at the Institute for Data, Systems and Society at MIT. She received her Ph.D. in computational science from MIT. Her work lies at the intersection of dynamical systems and ergodic theory with other areas of applied math, including statistical learning theory and Bayesian statistics. She is interested in computational methods development and analyses for complex systems that can have a positive societal impact, including those that arise in systems biology and climate studies.
Support provided from the C.V. Starr Foundation Lectureship Fund.
Fall 2023