Session III.4 - Foundations of Numerical PDEs
Talks
No date set.
Swarm-Based Random Descent Methods for Non-Convex Optimization
Eitan Tadmor
University of Maryland, UDA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We introduce a new swarm-based descent (SBD) method for non-convex optimization. The swarm consists of agents, each identified with position x and mass m. There are three key aspects to the SBD dynamics: (i) persistent transition of mass from high to lower ground; (ii) a random choice of descent marching direction, which is aligned with the orientation of the steepest gradient descent; and (iii) a time stepping protocol, h(x,m), which decreases with m. The interplay between positions and masses leads to dynamic distinction between `leaders’ and `explorers’. Heavier agents lead the swarm near local minima with small time steps. Lighter agents explore the landscape in random directions with large time steps, and lead to improve position, i.e., reduce the ‘loss’ for the swarm. Convergence analysis and numerical simulations demonstrate the effectiveness of SBD method as a global optimizer.
Monday, June 19, 14:00 ~ 15:00
Asymptotically compatible discretization of parameterized nonlocal models
Qiang Du
Columbia University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
There has been much interest in nonlocal models associated with a horizon parameter $\delta$ which characterizes the effective range of nonlocal interactions. Asymptotically compatible discretization provides a framework to develop robust numerical discretization schemes that are insensitive to changes in the parameter $\delta$. We discuss the progress on the design and mathematical analysis of such discretization for nonlocal variational problems and nonlocal conservation laws.
Joint work with Xiaochuan Tian (UCSD), Kuang Huang (Columbia University), and other collaborators.
Monday, June 19, 15:00 ~ 15:30
Dissipative solutions of compressible fluid flows: What do we approximate by structure preserving numerical schemes?
Mária Lukáčová
University of Mainz, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we present an overview of our recent results on generalized dissipative solutions of compressible fluid flows. We will concentrate on the inviscid flows, the Euler equations, and mention also the relevant results obtained for viscous compressible flows, governed by the compressible Navier-Stokes equations.
The existence of dissipative solutions has been shown by the convergence analysis of suitable, invariant-domain preserving finite volume schemes [1,2,3] . In the case that the strong solution to the above equations exists, the dissipative solutions coincide with the strong solution on its lifespan. In this case we can also apply the relative entropy to derive rigorous error estimates between numerical solutions and the exact strong solution [4].
Otherwise, we apply a newly developed concept of $K$-convergence and prove the strong convergence of the empirical means of numerical solutions to a dissipative solution [5]. The latter is the expected value of the dissipative measure-valued solutions and satisfies a weak formulation of the Euler equations modulo the Reynolds stress tensor. In the class of dissipative solutions there exists a solution that is obtained as a vanishing viscosity limit of the Navier-Stokes system [6]. We will draw a connection to the Kolmogorov hypothesis and illustrated theoretical results by a series of numerical simulations.
If time permits, we present also our recent results on the error analysis of the Monte Carlo finite volume method for the approximation of statistical solutions of the compressible Navier-Stokes equations.
The present research has been partially supported by TRR 146 Multiscale simulation methods for soft matter systems, TRR 165 Waves to Weather funded by the German Science Foundation and by the Gutenberg Research College.
\textbf{References}
[1] E. Feireisl, M. Lukáčová-Medvid'ová, H. Mizerová, B. She, Numerical analysis of compressible fluid flows, Springer, 2021.
[2] E. Feireisl, M. Lukáčová-Medvid'ová, H. Mizerová, Convergence of finite volume schemes for the Euler equations via dissipative-measure valued solutions, {\em Found. Comput. Math.} \textbf{20} (2020), 923–966.
[3] E. Feireisl, M. Lukáčová-Medvid'ová, H. Mizerová, B. She, Convergence of a finite volume scheme for the compressible Navier-Stokes system, {\em ESAIM: Math. Model. Num.} \textbf{53} (2019), 1957–1979.
[4] M. Lukáčová-Medvid'ová, B. She, Y. Yuan, Error estimate of the Godunov method for multidimensional compressible Euler equations, {\em J. Sci. Comput.} \textbf{91} (2022), 71.
[5] E. Feireisl, M. Lukáčová-Medvid'ová, B. She, Y. Wang, Computing oscillatory solutions of the Euler system via $K$-convergence, {\em Math. Math. Models Methods Appl. Sci.} \textbf{31} (2021), 537–576.
[6] E. Feireisl, M. Lukáčová-Medvid'ová, S. Schneider, B. She, Approximating viscosity solutions of the Euler system, {\em Math. Comp.} \textbf{91} (2022), 2129-2164.
Joint work with Eduard Feireisl (Academy of Sciences, Prague, Czech Republic), Hana Mizerov\'a (Comenius University, Bratislava, Slovakia), Bangwei She (Capital Normal University, Beijing, China) and Yuhuan Yuan (University of Mainz, Germany).
Monday, June 19, 15:30 ~ 16:00
Going implicit: large time steps for hyperbolic problems
Gabriella Puppo
Sapienza Università di Roma, Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will discuss the construction of implicit schemes in hyperbolic problems, especially in the low Mach case. I will consider two different procedures. First, implicit schemes can be derived to target some specific speeds, neglecting faster phenomena. This is the realm of low Mach schemes, where the purpose is to trace convective flow, while preserving the incompressible limit. I will discuss the main issues and difficulties in the numerical treatment of these problems and present an approach based on relaxation for hyperbolic systems for elastic models and two-phase flow. A second approach is to construct fully implicit schemes. Here the difficulty lies in the high non linearity of non oscillatory high order schemes.
Monday, June 19, 16:30 ~ 17:30
Adaptive Approximations for PDE-Constrained Parabolic Control Problems
Angela Kunoth
University of Cologne, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Numerical solvers for PDEs have matured over the past decades in efficiency, largely due to the development of sophisticated algorithms based on closely intertwining theory with numerical analysis. Consequently, systems of PDEs as they arise from optimization problems with PDE constraints also have become more and more tractable.
Optimization problems constrained by a parabolic evolution PDE are challenging from a computational point of view, as they require to solve a system of PDEs coupled globally in time and space. For their solution, time-stepping methods quickly reach their limitations due to the enormous demand for storage. An alternative approach is a full space-time weak formulation of the parabolic PDE which allows one to treat the constraining PDE as an operator equation without distinction of the time and space variables. An optimization problem constrained by a parabolic PDE in full space-time weak form leads to a coupled system of corresponding operator equations which is, of course, still coupled globally in space and time.
For the numerical solution of such coupled PDE systems, adaptive methods appear to be most promising, as they aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities in the data or domain. Employing wavelet schemes, we can prove convergence and optimal complexity. I will also address results on corresponding finite element approximations.
The theoretical basis for proving convergence and optimality of wavelet-based algorithms for such type of coupled PDEs is nonlinear approximation theory and the characterization of solutions of PDEs in Besov spaces.
Finally, I would like to address the possibility of solving such coupled PDEs by neural networks, combined with the characterization of solutions of PDEs in Barron spaces.
Monday, June 19, 17:30 ~ 18:00
Swarm-Based Random Descent Methods for Non-Convex Optimization
Eitan Tadmor
University of Maryland, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We introduce a new swarm-based descent (SBD) method for non-convex optimization. The swarm consists of agents, each identified with position x and mass m. There are three key aspects to the SBD dynamics: (i) persistent transition of mass from high to lower ground; (ii) a random choice of descent marching direction, which is aligned with the orientation of the steepest gradient descent; and (iii) a time stepping protocol, h(x,m), which decreases with m. The interplay between positions and masses leads to dynamic distinction between `leaders’ and `explorers’. Heavier agents lead the swarm near local minima with small time steps. Lighter agents explore the landscape in random directions with large time steps, and lead to improve position, i.e., reduce the ‘loss’ for the swarm. Convergence analysis and numerical simulations demonstrate the effectiveness of SBD method as a global optimizer.
Monday, June 19, 18:00 ~ 18:30
Mean field theory in Inverse Problems: from Bayesian sampling to overparameterization of networks
Qin Li
University of Wisconsin-Madison, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Bayesian sampling and neural networks are seemingly two different machine learning areas, but they both deal with systems with many particles. In sampling, one evolves a large number of samples (particles) to match a target distribution function, and in optimizing over-parameterized neural networks, one can view neurons particles that feed each other information in the DNN flow. These perspectives allow us to employ mean-field theory, a powerful tool that translates dynamics of many particle system into a partial differential equation (PDE), so rich PDE analysis techniques can be used to understand both the convergence of sampling methods and the zero-loss property of over-parameterization of ResNets. I would like to showcase the use of mean-field theory in these two machine learning areas, and I'd also love to hear feedbacks from the audience on other possible applications.
Joint work with Shi Chen (University of Wisconsin-Madison), Zhiyan Ding (University of California-Berkeley) and Steve Wright (University of Wisconsin-Madison).
Tuesday, June 20, 14:00 ~ 15:00
Laplacian Eigenfunctions that Do Not Feel Boundary: Theory, Computation, and Applications
Naoki Saito
University of California, Davis, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will discuss Laplacian eigenfunctions defined on a Euclidean domain of general shape, which "do not feel the boundary." These Laplacian eigenfunctions satisfy the Helmholtz equation inside the domain and can be extended smoothly and harmonically outside of the domain. These eigenfunctions satisfy the "nonlocal" boundary conditions, not the usual Dirichlet or Neumann boundary conditions. However, they can be computed via the eigenanalysis of the integral operator with rather simple potential kernel. The key here is the use of integral kernel that is a function of "distance" between a pair of points in a given domain. Compared to directly solving the Helmholtz equations on such domains, the eigenanalysis of this integral operator has several advantages including the numerical stability and amenability to modern fast numerical algorithms (e.g., the Fast Multipole Method). I will also discuss its relationship with the Laplacians satisfying various standard boundary conditions (e.g., Dirichlet, Neumann, Robin, periodic, anti-periodic, etc.) as well as the Krein-von Neumann Laplacian. Finally, I will discuss the extension of this integral operator approach to combinatorial graphs and simplicial complexes as its domain, and certain applications, e.g., image classification, heat equations, and characterization of biological shapes.
Tuesday, June 20, 15:00 ~ 15:30
Least-Squares Neural Network (LSNN) Method for Hyperbolic Conservation Laws
Zhiqiang Cai
Purdue University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Solutions of nonlinear hyperbolic conservation laws (HCLs) are often discontinuous due to shock formation; moreover, locations of shocks are a priori unknown. This presents a great challenge for traditional numerical methods because most of them are based on continuous or discontinuous piecewise polynomials on fixed meshes.
As an alternative, by employing a new class of approximating functions, neural network (NN), recently we proposed the least-squares neural network (LSNN) method for solving HCLs. The LSNN method shows a great potential to sharply capture shock without oscillation or smearing; moreover, its degrees of freedom are much less than those of mesh-based methods. Nevertheless, current iterative solvers for the LSNN discretization are computationally intensive and complicated.
In this talk, I will present our recent work on the LSNN for solving linear and nonlinear scalar HCLs.
[1] Cai, Z., Chen, J., and Liu, M., Least-squares ReLU} neural network (LSNN)} method for linear advection-reaction equation, J. Comput. Phys., 443 (2021) 110514.
[2] Cai, Z., Chen, J., and Liu, M., Least-squares {ReLU} neural network (LSNN) method for scalar nonlinear hyperbolic conservation law, Appl. Numer. Math., 174 (2022), 163-176.
[3] Cai, Z., Chen, J., and Liu, M., LSNN method for scalar nonlinear HCLs: discrete divergence operator, arXiv:2110.10895v2 [math.NA].
Joint work with Jingshuang Chen (Microsoft, USA) and Min Liu (Purdue University, USA).
Tuesday, June 20, 15:30 ~ 16:00
Quantitative convergence rates for Poisson learning
Jeff Calder
University of Minnesota, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Graph-based learning is a field within machine learning that uses similarities between datapoints to create efficient representations of high-dimensional data for tasks like semi-supervised classification, clustering and dimension reduction. Poisson learning was recently proposed for graph-based semi-supervised learning problems with very few labeled examples, where the widely used Laplacian regularization performs poorly. In contrast to Laplacian regularized learning, where labels are represented as Dirichlet boundary conditions, Poisson learning encodes the labels as point sources and sinks in a graph Poisson equation. In this work, we prove quantitative convergence rates for discrete to continuum convergence for Poisson learning. The problem is challenging since the source term is measure-valued in the continuum, and the continuum Poisson equation does not admit a variational interpretation. This work gives a rigorous mathematical justification for using Poisson learning for semi-supervised learning at low label rates.
Joint work with Leon Bungert (University of Bonn), Franca Hoffmann (Caltech), Kodjo Houssou (University of Minnesota), Max Mihailescu (University of Bonn) and Amber Yuan (Spotify).
Tuesday, June 20, 16:30 ~ 17:30
Time-fractional operators: From theory to simulation
Barbara Wohlmuth
TU München, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we discuss several aspects of time-fractional differential operators. The non-local nature of the operator is challenging from a theoretical and algorithmic point of view but allows to account for memory effects in the mathematical model. We consider a wide class of time-fractional PDE models, including gradient flow, viscoelasticity, random field generators and polymeric fluids. Of special interest is the equivalence of the time-fractional gradient flow model with an integer order system in a higher dimension. Exploiting this equivalence allows us to construct efficient algorithms for time fractional PDEs. Replacing the kernel by a finite sum of exponentials transforms the non-local PDE into a integer-order system for the modes. To balance the discretization error in time and space with the kernel approximation error, the number of modes has to grow logarithmically. However, we observe numerically that the number of modes is very moderate if the finite sum of exponentials is based on a rational approximation of the kernel function. In this talk we present theoretical existence results, algorithmic aspects and show several applications in which fractional operators play an important role. Numerical examples illustrating the large flexibility of the proposed techniques.
Joint work with Jonas Beddrich (TU München, Germany), Marvin Fritz (RICAM, Austria), Brendan Keith (Brown University, USA), Ustim Khristenko (Safran, France) and Endre Süli (University of Oxford, UK).
Tuesday, June 20, 17:30 ~ 18:00
Diffusion Maps for solving the Backward Kolmogorov PDEs in moderately high dimensions
Maria Cameron
University of Maryland, College Park, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
The diffusion map algorithm introduced by Coifman and Lafon in 2006 as a nonlinear dimensional reduction tool with proven theoretical guarantees has an important ability to approximate differential operators on point clouds. We show that by changing the kernel function inherent in diffusion maps and using renormalizations one can approximate the Backward Kolmogorov Operator for the stochastic differential equation governing the dynamics of biomolecules or atomic clusters described in collective variables: time-reversible dynamics with position-dependent and anisotropic diffusion. Moreover, the point cloud used as an input does not need to be sampled from the invariant density but can be generated by any standard enhanced sampling algorithm. Using the solution to the Backward Kolmogorov PDE on a point cloud with appropriate boundary conditions one can identify reaction channels and calculate the transition rate between metastable states of interest. An application to alanine dipeptide in four collective variables whose configurational space is the four-dimensional torus will be discussed.
Joint work with Luke Evans (University of Maryland, USA), Pratyush Tiwary (University of Maryland, USA).
Tuesday, June 20, 18:00 ~ 18:30
Greedy algorithms and tensor methods for the approximation of high-dimensional PDEs
Virginie Ehrlacher
Ecole des Ponts ParisTech, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
High-dimensional problems are ubiquitous in a large variety of applications: materials science, finance, uncertainty quantification, data science, stochastic game theory etc.
However, standard numerical methods cannot be used in practice for the resolution of Partial Differential Equations the solutions of which depend on a large number of variables because of the so-called curse of dimensionality. The most direct manifestation of this curse lies in the fact that the complexity of the representation of a function depending on d variables with a fixed number of degrees of freedom per variable grows exponentially with d.
In the last decades, several dedicated numerical strategies have been developped by applied mathematicians in order to circumvent this curse for the resolution of high-dimensional Partial Differential Equations. Among these, tensor methods have been a very active field of research in the past few years and are nowadays one of the most successfull family of approaches for the resolution of such problems. The bottom line of these methods is to use the well-known principle of separation of variables to define appropriate subsets of functions, called tensor formats, depending on a large number of variables and which can be represented with low complexity. There exists a wide variety of tensor formats, the most widely used of those being for instance the so-called Canonical Polyadic, Tucker, Tensor Train or Hierarchical Tucker formats.
A first objective of this talk is to give a comprehensive introduction to these tensor formats, and explain how these methods can be used for the resolution of high-dimensional Partial Differential Equations. A particular emphasis will be put on theoretical results concerning the analysis of numerical methods which consists in combining these tensor formats with so-called greedy algorithms from nonlinear approximation theory.
A second objective of this talk is to illustrate the efficiency of such approaches for the resolution various high-dimensional problems stemming from materials science applications. Indeed, interacting particle systems are ubiquitous in materials science applications in order to understand the macroscopic properties of materials from its microscopic or mesoscopic features. Several mathematical models exist to account for the evolution of such systems at different scales. Among those, let us mention for instance kinetic models, Fokker-Planck equations for molecular dynamics or quantum models for electronic structure calculations. All these models are defined on high-dimensional spaces, the high dimension stemming either from the large number of particles in the system of interest or the high number of features characterizing the state of each particle.
A specific focus will be put in this talk on kinetic equations, which are mesoscopic models (used for instance for the study of plasmas, neutronics or electron transport) which describe the state of large particle systems at the statistical level by a time-dependent probability distribution function, which encodes the probability of finding a particle at a certain position in space and with a certain speed. This distribution function is thus defined on a high-dimensional phase space and its evolution is typically modeled via a Boltzmann Partial Differential Equation. Numerical results will be presented which illustrates the successful use of tensor methods and greedy algorithms for the resolution of such models, in particular the resolution of the so-called Vlasov-Poisson system in some 3d-3d test cases.
Wednesday, June 21, 14:00 ~ 14:30
Finite element methods for ill-posed partial differential equations with conditional stability
Erik Burman
UCL, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.
Ill-posed partial differential equation are omnipresent in computational science, however the numerical analysis of their approximation methods is not so well developed. In this talk we will discuss recent results on stabilized finite element methods for the approximation of ill-posed problems that satisfy a conditional stability estimate. This means that the problem is stable under some additional a priori assumption on the solution. Using the classical linear unique continuation problem with smooth solution as model problem we will show how to design finite element methods whose accuracy reflects the order of the approximation space, the strength of perturbations in data and the stability of the continuous problem. We will then discuss the optimality of the results and how the introduction of further a priori assumptions can be used to improve the accuracy of the method.
Joint work with Lauri Oksanen.
Wednesday, June 21, 14:30 ~ 15:00
An element based spectral collocation technique
Adrianna Gillman
University of Colorado at Boulder, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
In many applications, it is ideal to have high order approximations of the solution. For example, problems involving high oscillatory solutions. A high order approximation is able to efficiently capture the solution. This talk presents a high order discretization technique based on spectral collocation on each of the elements. Continuity of the solution and the flux on the interfaces between the elements is done strongly and in a high order manner. This discretization technique is called the Hierarchical Poincar\'e-Steklov (HPS) method. Numerical results will illustrate the potential of the method including its ability to accurately handle mid-to-high frequency Helmholtz problems. Efficient direct and iterative solution techniques for the sparse linear system that results from the HPS discretization will also be presented.
Wednesday, June 21, 15:00 ~ 15:30
Some recent advances in parametric finite element approximation of surface evolution under geometric flows
Buyang Li
The Hong Kong Polytechnic University, Hong Kong - This email address is being protected from spambots. You need JavaScript enabled to view it.
We report some recent advances in the numerical analysis of parametric finite element approximation to surface evolution under geometric flows, and then introduce a new approach for proving convergence of parametric finite element approximation to surface evolution, which allows the algorithm to include certain artificial tangential motions to improve the mesh quality of the triangulation on the approximate surface.
Wednesday, June 21, 15:30 ~ 16:00
Trefftz and quasi-Trefftz methods
Lise-Marie Imbert-G\'erard
University of Arizona, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
In the field of numerical partial differential equations (PDEs), many methods rely on polynomial bases. These are very versatile tools, for instance in terms of their approximating properties: it is well-known that high order properties can be achieved to approximate any function with high enough regularity by choosing a sufficiently large space of polynomials.
Alternatively, various methods rely on problem-dependent bases, hence incorporating knowledge from the governing PDE into the basis functions. Such bases can achieve high order properties to approximate any sufficiently regular function {\bf{satisfying the governing PDE}}, at a lower cost in terms of degrees of freedom compared to polynomial bases. However they cannot do so to approximate any function with high enough regularity.
Several methods leveraging PDE-dependent functions have been developed under the name of Trefftz methods. This talk will focus on Discontinuous Galerkin types of Trefftz methods (TDG methods), in the particular context of time-harmonic wave propagation problems. Here information about the ambient medium is incorporated into oscillating basis functions via the appropriate wave-number. Some properties and drawbacks of basis of oscillating functions will be presented.
One limitation of TDG methods is that explicit basis functions that solve the PDE are needed. But wave propagation in inhomogeneous media is modeled by PDEs with variable coefficients, and in general no exact solutions are available in this case. Quasi-Trefftz methods have been introduced, in the case of the Helmholtz equation with variable coefficients, to address this problem: they do not rely on exact solutions to the PDE but instead on approximate solutions constructed locally. We will discuss the origin, the construction, and the properties of these so-called quasi-Trefftz functions. We will also discuss the consistency error introduced in the TDG method by this construction process.
Wednesday, June 21, 16:30 ~ 17:00
** TALK CANCELLED ** (Well-balanced high-order methods for systems of balance laws: recent results and applications)
Carlos Parés
University of Málaga, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
This work summarizes some recent advances in the design of high-order well-balanced methods for systems of conservation laws by the authors and collaborators. The starting point is the family of high-order finite-volume methods proposed in [1] for hyperbolic 1d-systems of balance laws of the form \[ U_t + F(U)_x = S(U)H_x, \] where $H$ is a known possibly discontinuous function. The key ingredient in [1] was the reconstruction of fluctuations with respect to local equilibria. More precisely, once the numerical approximations at time $t^n$, $U_i^n$, have been obtained, the steps to compute the high-order reconstruction at the $i$th cell are the following: first, a stationary solution $U^*_i(x)$ of the system whose average at the $i$th-cell, $[x_{i-1/2}, x_{i + 1/2}]$, is equal to $U_i^n$ is obtained, i.e. $U_i^*$ solves the problem \[ F(U_i^*)_x = S(U^*_i)H_x, \quad \frac{1}{\Delta x}\int_{x_{i - 1/2}}^{x_{i + 1/2}} U^*_i (x) \,dx = U_i^n.\] Next, the fluctuations \[ V^n_j = U_j^n - \frac{1}{\Delta x}\int_{x_{j- 1/2}}^{x_{j + 1/2}} U^*_i (x) \,dx, \quad j \in \mathcal{S}_i \] are computed, where $\mathcal{S}_i$ represents the stencil of the $i$th cell. Then, a standard high-order reconstruction operator (like MUSCL, ENO; WENO, CWENO, etc.) is applied to these fluctuations to obtain \[ Q_i(x) = Q_i(x; \{ V_j^n\}_{j \in \mathcal{S}_i} ). \] Finally, the reconstruction operator is defined as follows: \[ P_i(x) = U_i^*(x) + Q_i(x). \]
This strategy has been since then extended to systems for which the analytic expression of the stationary solutions is not available neither in explict or implicit form. Moreover it has been extended to the design of implicit and assymptotic preserving well-balanced finite volume methods and it has been also succesfully extended to the design of well-balanced finite-difference, Discontinuous-Galerkin, or Taylor methods.
Well-balanced methods designed following this strategy have been applied to fluid models in different contexts: shallow water models, gas dynamic, relativistic fluid models, blood flow models, etc.
An overview of the different extensions and applications of this methodology will be presented, including a brief discussion about the extension to multidimensional problems.
REFERENCES:
[1] M.J. Castro, C. Parés. Well-balanced high-order finite volume methods for systems of balance laws. Journal of Scientific Computing 82 (2), 1-48, 2021.
Joint work with Manuel J. Castro (University of Málaga, Spain), Irene Gómez-Bueno (University of Málaga, Spain), Sebastiano Boscarino (University of Catania, Italy) and Giovanni Russo (University of Catania, Italy).
Wednesday, June 21, 16:30 ~ 17:00
Well-balanced high-order methods for systems of balance laws: recent results and applications
CANCELLED (Carlos Parés) CANCELLED
University of Málaga, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
This work summarizes some recent advances in the design of high-order well-balanced methods for systems of conservation laws by the authors and collaborators. The starting point is the family of high-order finite-volume methods proposed in [1] for hyperbolic 1d-systems of balance laws of the form \[ U_t + F(U)_x = S(U)H_x, \] where $H$ is a known possibly discontinuous function. The key ingredient in [1] was the reconstruction of fluctuations with respect to local equilibria. More precisely, once the numerical approximations at time $t^n$, $U_i^n$, have been obtained, the steps to compute the high-order reconstruction at the $i$th cell are the following: first, a stationary solution $U^*_i(x)$ of the system whose average at the $i$th-cell, $[x_{i-1/2}, x_{i + 1/2}]$, is equal to $U_i^n$ is obtained, i.e. $U_i^*$ solves the problem \[ F(U_i^*)_x = S(U^*_i)H_x, \quad \frac{1}{\Delta x}\int_{x_{i - 1/2}}^{x_{i + 1/2}} U^*_i (x) \,dx = U_i^n.\] Next, the fluctuations \[ V^n_j = U_j^n - \frac{1}{\Delta x}\int_{x_{j- 1/2}}^{x_{j + 1/2}} U^*_i (x) \,dx, \quad j \in \mathcal{S}_i \] are computed, where $\mathcal{S}_i$ represents the stencil of the $i$th cell. Then, a standard high-order reconstruction operator (like MUSCL, ENO; WENO, CWENO, etc.) is applied to these fluctuations to obtain \[ Q_i(x) = Q_i(x; \{ V_j^n\}_{j \in \mathcal{S}_i} ). \] Finally, the reconstruction operator is defined as follows: \[ P_i(x) = U_i^*(x) + Q_i(x). \]
This strategy has been since then extended to systems for which the analytic expression of the stationary solutions is not available neither in explict or implicit form. Moreover it has been extended to the design of implicit and assymptotic preserving well-balanced finite volume methods and it has been also succesfully extended to the design of well-balanced finite-difference, Discontinuous-Galerkin, or Taylor methods.
Well-balanced methods designed following this strategy have been applied to fluid models in different contexts: shallow water models, gas dynamic, relativistic fluid models, blood flow models, etc.
An overview of the different extensions and applications of this methodology will be presented, including a brief discussion about the extension to multidimensional problems.
REFERENCES:
[1] M.J. Castro, C. Parés. Well-balanced high-order finite volume methods for systems of balance laws. Journal of Scientific Computing 82 (2), 1-48, 2021.
Joint work with Manuel J. Castro (University of Málaga, Spain), Irene Gómez-Bueno (University of Málaga, Spain), Sebastiano Boscarino (University of Catania, Italy) and Giovanni Russo (University of Catania, Italy).
Wednesday, June 21, 17:00 ~ 17:30
Numerical methods for models of liquid crystal dynamics
Franziska Weber
UC Berkeley, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Liquid crystal is an intermediate state of matter between the liquid and the solid phase, where the elongated molecules of the material are in partial alignment. Due to this, liquid crystals have unique physical properties that are used in various real-life applications, such as monitors (LCDs), smart glasses, navigation systems, shampoos, and others. Various mathematical models are available to describe their dynamics, among the most commonly used ones are the Oseen-Frank model, and the Q-tensor model by Landau and de Gennes, in which the alignment and its variation over time are described through systems of nonlinear PDEs. In this talk, I will describe these models and present results on the numerical approximation of PDEs that arise in these two modeling frameworks. These schemes are energy-stable, preserve the physical constraints in the discrete setting, and can be shown to converge within a particular range of material parameters.
Joint work with Yukun Yue (University of Wisconsin - Madison, USA), Varun Gudibanda (University of Wisconsin - Madison, USA) and Max Hirsch (Carnegie Mellon University, USA).
Wednesday, June 21, 17:30 ~ 18:00
The periodic KdV equation: Computing with nonlinear Fourier series
Thomas Trogdon
University of Washington, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
The integrability of the Korteweg-de Vries (KdV) equation affords an inverse scattering transform (IST) that "solves" the associated initial-value problem. First, historically, this fact allowed one to determine a class of explicit solutions, including both soliton solutions and (almost) periodic so-called finite-genus solutions. Then the IST allowed one to compute the long-time behavior of the solution to the Cauchy problem on the line using the Deift-Zhou method of nonlinear steepest descent. More recently, the IST has been used to compute solutions of the Cauchy problem on the line in the entire space-time plane.
In this talk, I will discuss a recent development that allows one to compute finite-genus solutions when the genus is large and to then effectively approximate the solution of the KdV equation with periodic initial data. This new approach uses a Riemann-Hilbert problem posed on (possibly thousands of) intervals and gives a natural interpretation as computing a nonlinear Fourier series approximation of the solution under consideration. Implications for dispersive quantization will be discussed.
Joint work with Deniz Bilman (University of Cincinnati), Ken McLaughlin (Tulane University) and Patrik Nabelek (Oregon State University).
Wednesday, June 21, 18:00 ~ 18:30
The Active Flux Method for Hyperbolic Conservation Laws
Christiane Helzel
Heinrich-Heine-University Düsseldorf, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
In 2011, Eymann and Roe introduced a new class of truly multidimensional finite volume methods. These so-called Active Flux methods uses some concepts which are quite different from concepts that are typically used in numerical schemes for hyperbolic problems. In particular the method is based on the use of point values as well es cell average values, it uses a continuous reconstruction and does not rely on the use of Riemann solvers.
I will review the current state of the art of Active Flux methods and present recent results of our group. These include a discussion of different evolution formulas for the update of the point values, results on the linear stability of the resulting methods as well as a discussion of limiters. Furthermore, we explored the use of Active Flux methods on Cartesian grids with adaptive mesh refinement as well as on cut cell grids.
Posters
Low-rank tensor product approximations for radiative transfer in plane-parallel geometry
Riccardo Bardin
University of Twente, The Netherlands - This email address is being protected from spambots. You need JavaScript enabled to view it.
The radiative transfer equation (RTE) has been established as a fundamental tool for the description of energy transport, absorption and scattering in many relevant scientific and societal applications, and requires numerical approximations. Classical numerical schemes in radiative transfer, such as the PN−and SN− approximations, exploit the tensor product structure of the underlying phase space, but they are affected by the so-called curse of dimensionality, which describes the exponential scaling of computational complexity with the physical dimension. For significant models like the RTE, this matter is particularly limiting.
The aim of our work is to introduce a low-rank tensor product framework for the approximation of the RTE. In the context of plane-parallel geometry, in order to tackle the dimensionality issue, we propose to construct tensor product solutions with low rank. An appropriate variational formulation, the even-parity formulation [1], allows us to recast the hyperbolic radiative transfer problem as a degenerate elliptic equation. Galerkin projection of this degenerate elliptic equation yields an equation for an operator that has a low-rank decomposition, which allows efficient application to objects which also exhibit a low-rank decomposition. To solve the projected equation, we consider a preconditioned Richardson iteration with rank control, which has been used in the context of high-dimensional elliptic equations in [2]. Due to the degeneracy of the elliptic equation, the construction of a preconditioner with a low-rank decomposition is challenging: we propose a suitable change of basis to highlight a Kronecker-sum structure of the new preconditioner, and we describe its application to low-rank object through exponential sums approximations and a particular summation procedure based on inexact evaluation of residuals.
The construction of the low-rank framework highlights the link between Hilbert space iterations and Linear Algebra, providing a solid motivation for the use of low-rank structure in the approximation of radiative transfer.
[1] - Egger, H. and Schlottbom, M., A mixed variational framework for the radiative transfer equation, Math. Models Methods Appl. Sci. 22 (2012)
[2] - Bachmayr, M. and Schneider, R., Iterative methods based on soft thresholding of hierarchical tensors, FoCM 17, 1037--1083 (2017)
Joint work with Matthias Schlottbom (University of Twente, The Netherlands) and Markus Bachmayr (RWTH Aachen University, Germany)..
Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non- Conservative Products
Deepak Bhoriya
University of Notre Dame, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Higher order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes have been constructed for conservation laws. For multidimensional problems, they offer high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of comparable accuracy. This makes them quite attractive for several science and engineering applications. But, to the best of our knowledge, such schemes have not been extended to non-linear hyperbolic systems with non-conservative products. In this paper, we perform such an extension which improves the domain of applicability of such schemes. The extension is carried out by writing the scheme in fluctuation form. We use the HLLI Riemann solver of Dumbser and Balsara (2016) as a building block for carrying out this extension. Because of the use of an HLL building block, the resulting scheme has a proper supersonic limit. The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme, thus expanding its domain of applicability. Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions, making it very easy for users to transition over to the present formulation.
For conservation laws, the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO, with two major advantages:- 1) It can capture jumps in stationary linearly degenerate wave families exactly. 2) It only requires the reconstruction to be applied once. Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of accuracy for smooth multidimensional flows. Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow, multiphase debris flow and two-layer shallow water equations are also shown to document the robustness of the method. For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results. Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.
Joint work with Dinshaw S. Balsara (University of Notre Dame, USA), Chi-Wang Shu (Brown University, USA) and Harish Kumar (Indian Institute of Technology, Delhi, India).
Model order reduction for parametric generalised EVPs
Joanna Bisch
Aalto University, Finland - This email address is being protected from spambots. You need JavaScript enabled to view it.
We investigate the resolution of parametric generalized eigenvalue problems in the form $ A(\sigma)x(\sigma)=\lambda(\sigma)Mx(\sigma) $ for given $ \sigma\in S $ with $ (\lambda(\sigma),x(\sigma))\in (0,\Lambda)\times \mathbb{R}^n $, $ A(S)\subset \mathbb{S}^{n\times n}_{++} $ and $ M\in \mathbb{S}^{n\times n}_{++} $. We look for a rapid solution to this GEVP using the Ritz method. We first find a basis for the Ritz subspace $ V\subset \mathbb{R}^{n} $ and solve the GEVP in this subspace. We built $ V $ using an average matrix $ \overline{A} $ related to $ (0,\rho \Lambda) $ with the help of a correction formula. For this, we use a bivariate sparse collocation operator of the correction function to built the Lagrange interpolants of the correction. We finally use the average eigenbasis to solve the GEVP to estimate the approximation error and prove the convergence rate.
Joint work with Antti Hannukainen (Aalto University, Finland).
Space-time pseudospectral method for the variable-order space-time fractional diffusion equation
Rupali Gupta
Sardar Vallabhbhai National Institute of Technology Surat, India - This email address is being protected from spambots. You need JavaScript enabled to view it.
Even though constant order fractional calculus is sufficient to cover most physical problems, it lacks when the fractional order behavior of the system changes with time or space or both. Variable order fractional calculus is the advancement of constant order fractional calculus. Finding the analytical solution is still tricky despite its significant importance in real-life modeling problems. Hence several computational and approximation methods are proposed to achieve the approximate solution. We discuss a numerical method for solving the space-time variable order fractional diffusion equations with various boundary conditions.
We used the pseudospectral method with Chebyshev polynomial as an orthogonal basis function which converts the considered problem into a set of linear algebraic equations. Then it can be solved for the unknowns to get the numerical solution. We solved a few examples to show that the proposed method is reliable and efficient. The convergence of the method is also presented. Error analysis is done theoretically and then verified through graphs of the numerical solutions. We also observed that using variable order derivatives requires fewer basis functions for a more accurate solution.
Joint work with Sushil Kumar(Sardar Vallabhbhai National Institute of Technology Surat, Surat, India).
Numerical solution of the isotropic Landau–Lifshitz equation using the Hasimoto transform
Georg Maierhofer
Sorbonne Université, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
The isotropic Landau–Lifshitz (LL) equation provides a model for a wide range of physical phenomena describing inter alia magnetization dynamics in ferromagnetism and the evolution of vortex filaments in ideal fluids. The fully nonlinear structure of this equation makes computations exceedingly difficult and prior approaches had to resort to comparatively expensive implicit formulations to achieve stable approximations to this equation.
In this work, we introduce a novel numerical approach to computing solutions to the LL equation based on the Hasimoto transform which relates the LL flow to a cubic nonlinear Schrödinger (NLS) equation. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators for the LL equation. Our approach consists of two parts: an integration of the NLS equation followed by the numerical evaluation of the Hasimoto transform. Motivated by the desire to study rough solutions to the LL equation, we also introduce a new symmetric low-regularity integrator for the NLS equation. This is combined with our novel fast low-regularity Hasimoto (FLowRH) transform, based on a tailored analysis of the resonance structures in the Magnus expansion and a fast realisation based on block-Toeplitz partitions, to yield an efficient low-regularity integrator for the LL equation.
In this poster we will present an overview of the methodology of this novel approach whose favorable properties are exhibited both in theoretical convergence analysis and in numerical experiments.
Joint work with Valeria Banica (Sorbonne Université, France) and Katharina Schratz (Sorbonne Université, France).
Wavelet Galerkin Method for an Electromagnetic Scattering Problem
Michelle Michelle
Purdue University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
The Helmholtz equation is challenging to solve numerically due to the pollution effect, which often results in a huge ill-conditioned linear system. In this paper, we present a high order wavelet Galerkin method to numerically solve an electromagnetic scattering from a large cavity problem modeled by the 2D Helmholtz equation. The high approximation order and the sparse stable linear system offered by wavelets are useful in dealing with the pollution effect. By using the direct approach presented in our past work [B.~Han and M.~Michelle, Appl. Comp. Harmon. Anal., 53 (2021), 270-331], we present various optimized spline biorthogonal wavelets on a bounded interval. We provide a self-contained proof to show that the tensor product of such wavelets form a 2D Riesz wavelet in the appropriate Sobolev space. Compared to the coefficient matrix of a standard Galerkin method, when an iterative scheme is applied to the coefficient matrix of our wavelet Galerkin method, much fewer iterations are needed for the relative residuals to be within a tolerance level. Furthermore, for a fixed wavenumber, the number of required iterations is practically independent of the size of the wavelet coefficient matrix. In contrast, when an iterative scheme is applied to the coefficient matrix of a standard Galerkin method, the number of required iterations doubles as the mesh size for each axis is halved. The implementation can also be done conveniently thanks to the simple structure, the refinability property, and the analytic expression of our wavelet bases.
Joint work with Bin Han (University of Alberta, Canada).
Central-Upwind Schemes for Weakly Compressible Two-layer Shallow-Water Flows
Sarswati Shah
Universidad Nacional Autónoma de México, México - This email address is being protected from spambots. You need JavaScript enabled to view it.
We present a weakly compressible approach to describe two-layer shallow water flows in channels with arbitrary cross sections \cite{1, 3}. The standard approach for those flows results in a conditionally hyperbolic balance law with non-conservative products while the current model is unconditionally hyperbolic. A detailed description of the properties of the model is provided, including entropy inequalities and entropy stability. Furthermore, a high-resolution, non-oscillatory semi-discrete central-upwind scheme is presented. The scheme extends existing central-upwind semi-discrete numerical methods for hyperbolic balance laws. Properties of the model such as positivity and well balance will be discussed. Along with the description of the scheme and proofs of these properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.
Joint work with Gerardo Hernández Dueñas (Universidad Nacional Autónoma de México, México).
An algorithm for solving a nonlocal problem for hyperbolic equations with piecewise constant argument of generalized type
Roza Uteshova
Institute of Mathematics and Mathematical Modeling, Kazakhstan - This email address is being protected from spambots. You need JavaScript enabled to view it.
The theory of differential equations with piecewise constant argument has been developed in the context of mathematical modeling of various processes in biology, chemistry, mechanics, electronics, etc. A new class of differential equations with piecewise constant argument was proposed by M. Akhmet. He suggested the delay argument to be an arbitrary piecewise constant function as opposed to the greatest integer function. Differential equations with piecewise constant argument of generalized type have proven to be more suitable models for studying and solving various application problems, including neural networks, discontinuous dynamical systems, hybrid systems, etc.
We present an algorithm for solving a nonlocal problem for a system of second-order hyperbolic equations with piecewise constant time argument of generalized type. The method we use is based on the introduction of functional parameters that are set as the values of the desired solution along the lines of the domain partition with respect to the time variable. With the aid of the functional parameters and new unknown functions, the considered problem is reduced to an equivalent problem for a system of hyperbolic equations with data on the interior partition lines and functional relations with respect to the introduced parameters. We developed a two-stage procedure to approximately solve the latter problem: firstly, we solve an initial-value problem for a system of differential equations in functional parameters; then, we solve a problem for a system of hyperbolic equations in new unknown functions with data on the interior partition lines. We derived some conditions for the convergence of approximate solutions to the exact solution of the problem under study in terms of input data and proved that these conditions guarantee the existence of a unique solution of the equivalent problem. Finally, we established coefficient conditions for the unique solvability of the nonlocal problem.
Joint work with Anar Assanova (Institute of Mathematics and Mathematical Modeling, Kazakhstan).
Error estimates of numerical methods for the nonlinear Schr\"{o}dinger equation with low regularity potential and nonlinearity
Chushan Wang
National University of Singapore, Singapore - This email address is being protected from spambots. You need JavaScript enabled to view it.
We prove optimal error bounds of time-splitting methods and the exponential wave integrator for the nonlinear Schr\"{o}dinger equation (NLSE) with low regularity potential and nonlinearity, including purely bounded potential and locally Lipschitz nonlinearity. Arising from different physical applications, low regularity potential and nonlinearity are introduced into the NLSE such as some discontinuous or disorder potential widely used in the physics literature or the non-integer power nonlinearity in the Lee-Huang-Yang correction which is adopted in modelling and simulation of quantum droplets. Most of the classical numerical methods can be directly extended to solve the NLSE with the aforementioned potential and nonlinearity. However, the performance of these methods becomes very different from the smooth case and the error estimates are subtle and challenging.
Joint work with Weizhu Bao (National University of Singapore, Singapore).
A posteriori error estimates and their use for a least-cost strategy to achieve target accuracy
Yipeng Wang
Sorbonne Université , France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Our work aims at providing an optimal cost strategy to achieve the targeted accuracy when approximating the solution of a nonlinear PDE. The numerical error comes from two sources: the number of iterations and the finite dimensional approximate space. We first apply a probabilistic method to explore an optimal path. Based on the analysis of this optimal path, we propose a near-optimal strategy to achieve a given accuracy based on a posteriori estimates.
Joint work with Yvon MADAY (Sorbonne Université, France) and Muhammad Hassan (Sorbonne Université, France).
Limits of non-local approximation to the Eikonal equation on manifolds
Rita Zantout
INSA de Rouen, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider a non-local approximation of the time-dependent Eikonal equation defined on a Riemannian manifold. We show that the local and the non-local problems are well-posed in the sense of viscosity solutions and we prove regularity properties of these solutions in time and space. If the kernel is properly scaled, we then derive error bounds between the solution of the non-local problem and the one of the local problem, both in continuous-time and Forward Euler discretization. Finally, we apply these results to a sequence of random weighted graphs with $n$ vertices. In particular, we establish that the solution of the problem on graphs converges almost surely uniformly to the viscosity solution of the local problem as the kernel scale parameter decreases at an appropriate rate as the number of vertices grows and the time step vanishes.
Joint work with Nicolas Forcadel (INSA de Rouen,France) and Jalal Fadili (École nationale supérieure d'ingénieurs de Caen (ENSICAEN),France).