Abstract : Many problems in science and engineering are described by high-dimensional PDEs. Over the years, different numerical techniques have been developed for these problems, including low rank method, sparse grid, tensor method, reduced order modeling, machine learning, optimization, and quantum algorithms, to name a few. In this minisymposium, we bring researchers from a wide spectrum of application areas, such as plasma physics, quantum dynamics, biology, etc. to address the common theme of solving high-dimensional PDEs and exchange ideas.
[04856] Sparse grid techniques for particle-in-cell simulation of kinetic plasmas
Format : Online Talk on Zoom
Author(s) :
Lee Forrest Ricketson (Lawrence Livermore National Laboratory)
Abstract : The Vlasov equation, which models collisionless plasma dynamics, is six-dimensional in general. To combat this high dimensionality, the most prevalent numerical method has long been particle-in-cell (PIC), in which the plasma is represented by particles which interact with electromagnetic fields specified on a spatial mesh. However, the inclusion of particles subjects the scheme to slow-converging sampling errors, while use of a spatial mesh admits only partial mitigation of the curse of dimensionality. We show that using sparse grids with PIC makes the algorithm’s complexity only logarithmically dependent on dimension and dramatically reduces the impact of sampling noise. We report recent progress combining sparse PIC advanced symplectic and implicit methods. Finally, we report on ongoing work toward adaptively choosing suitable coordinates for these sparse grid computations.
*Prepared by LLNL under Contract DE-AC52-07NA27344.
[03639] Nonlinear model reduction with adaptive bases and adaptive sampling
Format : Online Talk on Zoom
Author(s) :
Benjamin Peherstorfer (Courant Institute of Mathematical Sciences, New York University)
Abstract : We introduce an online-adaptive model reduction approach that can efficiently reduce convection-dominated problems. It exploits that manifolds are low dimensional in a local sense in time and iteratively learns and adapts reduced spaces from randomly sampled data of the full models to locally approximate the solution manifolds. Numerical experiments to predict pressure waves in combustion dynamics demonstrate that our approach achieves significant speedups in contrast to classical, static reduced models.
[04871] Quantum Algorithms for Accelerating the Solution of Partial Differential Equations
Format : Online Talk on Zoom
Author(s) :
Ilon Joseph (Lawrence Livermore National Laboratory)
Abstract : Quantum algorithms have been proposed to accelerate the solution of linear partial differential equations (PDEs) with polynomial to exponential speedup. Recent progress has significantly improved these algorithms with respect to numerical methods and error convergence. Due to the “no-cloning” theorem, solving nonlinear PDEs is more challenging. Newly proposed Koopman and Carleman approaches solve for the linear evolution of the probability distribution function of the solution and are closely related to simulating stochastic PDEs and quantum fields.
[03854] A Hybrid AMR Low-Rank Tensor Approach for Solving the Boltzmann Equation
Format : Talk at Waseda University
Author(s) :
William Tsubasa Taitano (Los Alamos National Laboratory)
Samuel Jun Araki (Air Force Research Laboratory)
Abstract : The Boltzmann equation describes the time evolution of a particle distribution function in a six-dimensional position-velocity phase space. The exponential growth in computational complexity often challenges a grid-based approach to modeling the Boltzmann equation as the dimensionality grows. Scalable low-rank tensor decomposition techniques have recently been developed with applications to high-dimension PDEs to address this issue. Despite the remarkable progress made in the community, low-rank structures in the phase-space are not evident in realistic engineering systems with complex geometries (e.g., electric propulsion systems and fusion reactors), where discontinuities, shocks, complex boundary conditions, and material-dependent physics (e.g., collisions, fusion reactions, ionization/excitation, charge-exchange processes) pose formidable challenges. In this talk, we propose a novel hybrid algorithm where quad-tree adaptive mesh refinement (AMR) is applied in real space while a low-rank approximation is applied in the velocity space. The AMR algorithm efficiently handles challenges pertaining to complex structures in real space, while the low-rank formulation targets dimensionality challenges in the velocity space. We present preliminary results on the new algorithm applied to challenging multi-dimensional gas kinetics problems.
[04128] Designing High-Dimensional Closed-Loop Optimal Control Using Deep Neural Networks
Format : Talk at Waseda University
Author(s) :
Jiequn Han (Flatiron Institute, Simons Foundation)
Abstract : Designing closed-loop optimal control for high-dimensional nonlinear systems remains a long-standing challenge. Traditional methods, such as solving the Hamilton-Jacobi-Bellman equation, suffer from the curse of dimensionality. Recent studies introduced a promising supervised learning approach, utilizing deep neural networks to learn from open-loop optimal control solutions. From a PDE standpoint, this method learns solutions along characteristic lines.
This talk will first overview this method and identify a limitation in its basic form, the distribution mismatch phenomenon, caused by controlled dynamics. We then propose the initial value problem enhanced sampling method to address this issue. The proposed method presents theoretical guarantees of improvement over the basic version in the classical linear-quadratic regulator and demonstrates significant improvement numerically on several high-dimensional nonlinear problems.
[04563] An Inverse Problem in Mean Field Games from Partial Boundary Measurement
Format : Talk at Waseda University
Author(s) :
Yat Tin Chow (University of California, Riverside)
Samy Wu Fung (Colorado School of Mines)
Siting Liu (University of California, Los Angeles)
Levon Nurbekyan (University of California, Los Angeles)
Stanley Osher (University of California, Los Angeles)
Abstract : In this talk, we consider a novel inverse problem in mean-field games (MFG). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture. Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult. Nonetheless, it is vital to recovering the model parameters stably and efficiently to uncover the underlying causes of population dynamics for practical needs.
Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a finite number of boundary measurements of population profile and boundary movement. To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method. Numerical experiments illustrate the effectiveness and robustness of the algorithm.
[04470] Automatic partitioning for Boolean CME Low-Rank integrator
Format : Talk at Waseda University
Author(s) :
Martina Prugger (University of Innsbruck)
Lukas Einkemmer (University of Innsbruck)
Abstract : Cell signaling processes are usually modeled by chemical reactions encoded in a system of ordinary differential equations. The resulting model is deterministic and omits the inherent stochasticity of cell reactions. This is mostly due do the fact, that solving the Chemical Master Equation (CME), which resolves the inherent probabilistic states of the chemical system suffers from the curse of dimensionality. This results in high computational and memory requirements, that prohibit the simulation of the CME for system sizes that are usually required for practical applications.
We developed a low-rank integrator for the CME for Boolean networks that enables us to simulate systems to a size of up to 41 different chemical species on a workstation. An integral part of this method is to partition the network into multiple sub-partition, on which the CME is solved exactly. Key to an efficient solver is the distribution of species, while keeping the approximation error between the networks to a minimum. While for small networks, this can still be done easily by a person, for a network of the size of e.g., 41 species, this is no longer feasible. We therefore introduce an automatic partitioner by using the Kernighan-Lin algorithm to select multiple networks that minimize the amount of connections between the sub-networks. We then use information enthropy that evaluates each of the chosen networks. This results in an automatic partitioning tool that reduces the rank that is required to faithfully resolve the biological dynamics.
