4:30 – 5:30 p.m. E.D.T. | Fri., May 20, 2022
Session F1: DG Methods
4:30 – 5:30 p.m. E.D.T. | Fri., May 20, 2022 | Room: 1060 East Hall
Yue Kang, Michigan Technological University
Bound-preserving discontinuous Galerkin methods with second-order implicit pressure explicit concentration time marching for compressible miscible displacements in porous media
In this talk, I will introduce the bound-preserving interior penalty discontinuous Galerkin (IPDG) methods with a second-order implicit pressure explicit concentration (SIPEC) time marching for the coupled system of two-component compressible miscible displacements. The SIPEC method is based on the traditional second-order strong-stability-preserving Runge Kutta (SSP-RK2) method. The main idea is to treat the pressure equation implicitly and the concentration equation explicitly. However, this treatment would result in a first-order accurate scheme. Therefore, we propose a correction stage to compensate for the second order accuracy in each time step. We can deal with the velocity in the diffusion term explicitly to avoid correction of the diffusion term. Also, we need to ensure that the bound preserving technique for the convection and source terms can be applied when the correction stage has been established. In fact, in the correction stage, the new approximation to the concentration can be chosen as the numerical solution in the previous stage, so the numerical cell averages are positivity-preserving. Numerical experiments will be given to demonstrate that the proposed scheme can reduce the computational cost significantly.
Fangyao Zhu, Michigan Technological University
Discontinuous Galerkin methods with Patankar time discretization for chemical reacting flow
I will talk about bound-preserving modified Patankar scheme with discontinuous Galerkin (DG) methods for chemical reactive flows. For this problem, the physical density and pressure must be kept positive. We start from production-destruction equations. Using the idea of the high order modified Patankar Runge-Kutta, we adapt it to multi-step method. Coupled with this time integration method, we use bound-preserving discontinuous Galerkin method in space to preserve those physical properties. Numerical examples will be shown.
Tamas Horvath, Oakland University
Sliding grid techniques for Space-Time Hybridizable Discontinuous Galerkin methods
The Space-time Discontinuous Galerkin (ST-DG) method is an excellent method to discretize problems on deforming domains. This method uses DG to discretize both the spatial and temporal directions, allowing for an arbitrarily high order approximation in space and time. We present a higher-order accurate Embedded-Hybridized Discontinuous Galerkin method for fluid-rigid body interactions. We apply a sliding grid technique or rotational movement that can handle arbitrary rotation. The numerical examples will include galloping and fluttering motions.
Session F2: Integral Equations
4:30 – 5:30 p.m. E.D.T. | Fri., May 20, 2022 | Room: 1068 East Hall
Yue Cao, Illinois Institute of Technology
Kernel free boundary integral method and its application
We developed a kernel-free boundary integral method (KFBIM) for solving variable coefficients partial differential equations (PDEs) in a doubly-connected domain. We focus our study on boundary value problems (BVP) and interface problems. A unique feature of the KFBIM is that the method does not require an analytical form of the Green’s function for designing quadratures, but rather computes boundary or volume integrals by solving an equivalent interface problem on Cartesian mesh. The method has second-order accuracy in space, and its complexity is linearly proportional to the number of mesh points. Numerical examples demonstrate that the method is robust for variable coefficients PDEs, even for cases when diffusion coefficients ratio is large and when two interfaces are close. We also develop two methods to compute moving interface problems whose coefficients in governing equations are spatial functions. Variable coefficients could be a non-homogeneous viscosity in Hele-Shaw problem or an uptake rate in tumor growth problems. The methods have second-order accuracy in both space and time. Machine learning techniques have achieved magnificent success in the past decade. We couple the KFBIM with supervised learning algorithms to improve efficiency. In the KFBIM, we apply a finite difference scheme to find dipole density of the boundary integral iteratively, which is quite costly. We train a linear model to replace the finite difference solver in GMRES iterations. The cost, measured in CPU time, is significantly reduced. We also developed an efficient data generator for training and derived an empirical rule for data set size.
Somveer Singh, Indian Institute of Technology BHU Varanasi, India
Numerical analysis of nonlinear weakly singular integro-partial differential equations arising from viscoelasticity
An efficient matrix method is presented for the solution of non-linear weakly singular partial integro-differential equation (SPIDE) arising from viscoelasticity subject to the given initial and boundary conditions. The method is based on the operational matrices of Legendre wavelets. By implementing the operational matrices of Legendre wavelets, the given intero PDE is reduced to the system of nonlinear equations. Some useful results concerning the convergence and error estimates associated to the suggested scheme are presented. Illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method.
Lei Wang, University of Wisconsin, Milwaukee
Application of the Barycentric Lagrange Treecode for Computing Correlated Random Displacements Using the Spectral Lanczos Decomposition Method
In Brownian Dynamics simulations, correlated random displacements g =√Dz can be computed using the Spectral Lanczos Decomposition Method (SLDM), where D is the diffusion matrix based on the Rotne-Prager-Yamakawa tensor and z is a normal random vector. Each step of the SLDM requires computing a matrix-vector product Dqk, where qk is the kth Lanczos vector. The present work applies the barycentric Lagrange treecode (BLTC) to accelerate the matrix-vector product in the SLDM. Numerical results show the performance of the SLDM-BLTC in serial and parallel simulations.
Session F3: DG Wave Propagation
4:30 – 5:30 p.m. E.D.T. | Fri., May 20, 2022 | Room: 1084 East Hall
Zhichao Peng, Michigan State University
EM-WaveHoltz: a flexible frequency-domain Maxwell solver built from time-domain solvers
Two main challenges to design efficient iterative solvers for the frequency-domain Maxwell equations are the indefinite nature of the underlying system and the high resolution requirements. Scalable parallel frequency-domain Maxwell solvers are highly desired. This talk will introduce the EM-WaveHoltz method which is an extension of the recently developed Wave Holtz method for the Helmholtz equation to the time-harmonic Maxwell equations. Three main advantages of the proposed method are as follows. (1) It always results in a positive definite linear system. (2) Based on the framework of EM-WaveHoltz, it is flexible and simple to build efficient frequency-domain solvers from current scalable time-domain solvers. (3) It is possible to obtain solutions for multiple frequencies in one solve. The formulation of the EM-WaveHoltz and analysis in the continuous setting for the energy conserving case will be discussed. The performance of the proposed method will be demonstrated through numerical experiments.
Yann-Meing Law, Michigan State University
The Hermite-Taylor Correction Function Method for Maxwell’s Equations
Hermite-Taylor methods are high-order methods for hyperbolic problems that rely on a Hermite interpolation procedure in space and a Taylor method in time. The stability condition of these methods depends only on the largest wave-speed, independent of the order. However, the treatment of general boundary conditions is still a challenge since these methods require not only to know the electromagnetic fields on the boundary but also their m first derivatives in space to achieve a (2m+1)-order method. In this talk, we investigate the Correction Function Method (CFM) to seek all the needed information on the boundary in the Hermite-Taylor setting. The CFM relies on the minimization of a functional based on Maxwell’s equations. This minimization problem has to be solved for only some nodes in the vicinity of the boundary at each time step. Numerical examples in 1-D and 2-D are performed and the expected convergence order is observed for reasonable values of m.
Yang Yang, Michigan Technological University
Stability analysis of the Eulerian-Lagrangian finite volume methods for nonlinear hyperbolic equations in one space dimension
In this talk, we construct a novel Eulerian-Lagrangian finite volume (ELFV) methods for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose the ELFV method, where the space-time domain was partitioned based on the characteristics that originated from the cell interfaces in the spatial domain. Unfortunately, to avoid the intersection of the characteristics, the time step sizes are still limited. To fix this gap, we detect ETCs and carefully design the influence region of each ETC within which the cells are merged together to form a new one. Then with the new partition of the space-time domain, it is possible to gain larger time step sizes. Moreover, we theoretically prove that the proposed first-order scheme with Euler forward time discretization is total variation-diminishing and maximum-principle-preserving. Numerical experiments verify the optimality of the designed time step sizes.
Session F4: Multiscale, Geometric, and Exponential Methods
4:30 – 5:30 p.m. E.D.T. | Fri., May 20, 2022 | Room: 1096 East Hall
Siu Wun Cheung, Lawrence Livermore National Laboratory
Multiscale and numerical upscaling methods for physical processes in high contrast heterogeneous porous media
Many applications related to subsurface formations requires accurate numerical simulations in high contrast heterogeneous porous media, in which the material properties within fractures can have a large difference from the material properties in the background media, which can also contain highly heterogeneous and high contrast regions. These large contrasts in material properties and the complex geometries of the fractures lead to difficulties in traditional numerical simulations due to the fact that solutions contain various scales and resolving these scales requires huge computational costs. In this talk, we will discuss the development of several multiscale and upscaling methods for physical processes in high contrast heterogeneous porous media, which aims at efficiently compute numerical approximations by formulating coarse-scale equations for dominant components of the solution representation. Using the concepts of model order reduction and localized energy minimization, we develop multiscale and numerical upscaling methods which significantly reduces the problem size and exhibits convergence with respect to the coarse mesh size. We will discuss recent developments in multiscale methods, such as online adaptivity and iterative oversampling techniques in basis constructions.
Jiahui Chen, Michigan State University
Evolutionary de Rham-Hodge method
The de Rham-Hodge theory is a landmark of the 20th Century’s mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of unique evolutionary Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0, 1, and 2. Additionally, three sets of non-zero eigenvalues further reveal both topological persistence and geometric progression during the manifold evolution. Extensive numerical experiments are carried out via the discrete exterior calculus to demonstrate the potential of the proposed paradigm for data representation and shape analysis. To demonstrate the utility of the proposed method, application is considered to the protein B-factor predictions of a few challenging cases for which other existing models do not work well.
Van Hoang Nguyen, Mississippi State University
Efficient exponential methods for genetic regulatory systems
In recent years, gene regulatory networks (GRNs) have attracted a lot of interest as they play a crucial role in modeling the protein regulation processes. Among many numerical schemes which were introduced for simulating these systems, the classical fourth-order Runge-Kutta method (RK4) has been often used. When simulating GRNs, it is difficult to characterize some important properties of the system, such as stability and phase properties. Very recently, splitting methods, exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods and classical exponential Runge-Kutta (ExpRK)-type methods have been also considered. In this work, we propose a class of partitioned ExpRK methods and exponential Rosenbrock methods for the simulation of genetic regulatory systems. Moreover, we also investigate the stability and phase properties of the proposed schemes. Finally, based on a set of benchmark test problems including two- and three-gene systems, we demonstrate the accuracy and efficiency of the newly derived methods in comparison with several popular schemes.
11:30 – 12:30 p.m. E.D.T. | Sat., May 21, 2022
Session S1: DG Methods
11:30 a.m. – 12:30 p.m. E.D.T. | Sat., May 21, 2022 | Room: 1060 East Hall
Mahboub Baccouch, University of Nebraska at Omaha
A local discontinuous Galerkin method for elliptic problems on Cartesian grids: Supercon vergence, a posteriori error estimation, and adaptivity
In this talk, we present a local discontinuous Galerkin (LDG) methods for two-dimensional second-order elliptic problems. Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. We prove that the LDG solution is superconvergent towards a particular projection of the exact solution. The order of convergence is proved to be p+2, when tensor product polynomials of degree at most p are used. Then, we show that the actual error can be split into two parts. The components of the significant part can be given in terms of (p +1)-degree Radau polynomials. We use these results to construct a reliable and efficient residual-type a posteriori error estimates. We prove that the proposed residual-type a posteriori error estimates converge to the true errors in the L2-norm under mesh refinement. The order of convergence is proved to be p+2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. We provide several numerical examples illustrating the effectiveness of our procedures.
Andrés Galindo Olarte, Michigan State University
Accuracy enhancement of discontinous Galerkin solutions for Vlasov-Maxwell equations
In this talk, we will establish negative-order estimates for the accuracy of discontinuous Galerkin (DG) approximations to smooth solutions for the Vlasov-Maxwell (VM) system of equations. For the approximated solutions, we are able to extract this “hidden accuracy” through the use of a Smooth-Increasing Accuracy-Conserving (SIAC) filter which is a convolution kernel that is composed of a linear combination of B-splines. We provide rigorous error estimates for the DG solutions that show improvement to (2k + 1/2)-th order in the negative-order norm.
Devina Sanjaya, University of Tennessee, Knoxville
Global mesh optimization: Is it worth the cost?
Global mesh optimization is a natural approach when we view meshing as a method to connect geometry nodes, construct valid mesh elements of certain types, and arrange these elements to cover the entire computational domain. From this viewpoint, we can easily see that the effect of one node movement will propagate not only to all of the neighboring elements, but also to the entire domain. Hence, one may argue that global mesh optimization is simply too expensive and impractical. However, if we were to overcome the cost barrier, global mesh optimization could significantly improve the accuracy and robustness of computational simulations, closely couple multiple mesh adaptation techniques, eliminate heuristics in meshing, and fully automate meshing procedures. In this talk, I will describe several ongoing efforts in developing effective optimization problem formulations and architectures as well as identify the main challenges. The talk will be presented in the context of high-order, finite-element meshes, and in this context, one may view global mesh optimization as a way to capture the global coupling between the optimal placement of element’s vertices and high-order geometry nodes within the computational domain. The ultimate goal is to build an automated, high-order meshing suite for maximizing the accuracy and robustness of computational fluid dynamics (CFD) simulations while efficiently using the high-performance computing (HPC) systems to reduce human workload.
Session S2: Biological Modeling
11:30 a.m. – 12:30 p.m. E.D.T. | Sat., May 21, 2022 | Room: 1068 East Hall
Dexuan Xie, University of Wisconsin-Milwaukee
A new nonlinear iterative scheme for solving a nonuniform size modified Poisson-Boltzmann ion channel model
The nonuniform size modified Poisson-Boltzmann ion channel (nuSMPBIC) model is difficult to solve numerically due to that it is not only an exponentially nonlinear system with strong solution singularity but also involves an interface boundary value problem defined in a box domain with three complex regions (an ion channel protein region, a solvent region, and a membrane region) and multiple nonlinear algebraic equations defined on the solvent region only. To overcome these numerical difficulties, we recently developed a novel iterative technique and an effective nonlinear iterative scheme for solving the nuSMPBIC model. Moreover, this new scheme has been implemented as a finite element program package, which works for an ion channel protein with a three-dimensional molecular structure and a mixture solution of multiple ionic species with distinct ion sizes. In this talk, I will describe the construction of this new scheme. I then will report numerical results to demonstrate the fast convergence of the new nonlinear iterative scheme and the high performance of the nuSMPBIC finite element software. This work is partially supported by the Simons Foundation, USA, through research award 711776.
Zhen Chao, University of Michigan
Integral method for solving one-dimensional Poisson-Nernst-Planck ion channel model
A system of Poisson-Nernst-Planck (PNP) dielectric continuum model has been used to describe the dynamics of ion transport through biological ion channels besides being widely employed in semiconductor industry. Most of the existing literature focus on the finite difference method and finite element method for solving PNP system. In this talk, we employ the integral method with the Gummel iteration method to solve the one-dimensional (1D) steady state PNP system. We describe the 1D steady state PNP model and the implementation of our numerical method. Several numerical examples are presented to show the stability and usefulness of the numerical method, we notice that the integral method with Gummel iteration still works for high electrostatic potential and nonelectroneutral cases without using Slotboom transformation.
Will Zhang, University of Michigan
Fractional Viscoelastic Modeling of Cardiovascular Soft Tissues
While experimental evidence shows that the mechanical response of most tissues is viscoelasticity, current biomechanical models used in computational studies often assume only hyperelasticity. Fractional viscoelastic constitutive models have been successfully used in literature to capture the material response. However, the translation of these models into computational platforms remains limited due to a computational cost that is O(N2T) and a storage cost that is O(NT ) in order to store and integrate historical data. We developed a novel numerical approximation to the Caputo derivative which exploits a recurrence relation to give a computational cost that is O(N) and a storage cost that is fixed over time. We used this approach to extend the conventional analysis of residual stress in aortic soft tissues by considering the effects of viscoelasticity. A substantial portion viscoelastic tissue is necessary to capture the long-term dynamics of the residual stress experiment. In addition, the results show that conventional estimates of residual stress are overestimated by a factor of 2 and that viscoelasticity is necessary to accurately estimate residual strain. Using this approach, we developed a fractional viscoelastic model for myocardium that improves the quality of fit compared to current models in the literature. We analyzed the numerical properties of this approach using simplified examples and application in simulations of the right ventricle. These results demonstrates that fractional viscoelasticity has enormous potential for facilitating the analysis of viscoelastic properties of soft biological tissues in computational problems.
Session S3: High-order Methods
11:30 a.m. – 12:30 p.m. E.D.T. | Sat., May 21, 2022 | Room: 1084 East Hall
Luan Vu Thai, Mississippi State University
High-order efficient multirate infinitesimal step-type methods
Multirate time integrators use different time steps for integrating different components of multiphysics systems, thereby increasing computational efficiency over classical time integration methods. So far, multirate schemes of orders up to four have been proposed in the literature such as MIS or MRI-GARK methods. Their derivations, however, require solving a complicated set of coupling and decoupling order conditions, which pose difficulties to construct higher orders. In this talk, we propose a new family of high-order multirate time integrators based on recent developed exponential Runge-Kutta/Rosenbrock methods, thereby named MERK/MERB methods. They can be classified as MIS-type methods. Their rigorous convergence analysis was carried out. Schemes up to order 6 have been constructed in an elegant way. Moreover, these new schemes allow to use a smaller number of modified differential equations in each step when compared to existing MIS or MRI-GARK of the same orders. Numerical experiments are given to demonstrate the accuracy and efficiency of the newly derived MERB schemes.
Trky Alhsmy, Mississippi State University
High-order adaptive exponential Runge-Kutta methods
Exponential Runge-Kutta (ExpRK) methods have shown to be well-suited for the time discretization of stiff semilinear parabolic PDEs. The construction of stiffly-accurate ExpRK schemes requires solving a system of stiff order conditions which involve matrix functions. So far, methods up to order 5 have been derived by relaxing one or more order conditions (depending on a given order of accuracy). These schemes, however, allow using with constant step sizes only. In this talk, we will derive new and efficient ExpRK schemes of high orders which not only fulfill the stiff order conditions in the strong sense and but also support variable step sizes implementation. Numerical examples are given to verify the accuracy and to illustrate the efficiency of the newly constructed ExpRK schemes.
Qicang Shen, University of Michigan [CANCELLED]
High-order accurate solution of the spatial kinetics problem for nuclear reactor simulation using the spectral deferred correction method
Solving initial value problems with high-order methods receives considerable attention in many fields because these methods can potentially improve the accuracy of the simulation results with lower computational cost than low-order methods. Most methods, however, are either complicated to implement or unstable when the order of accuracy is high. The Spectral Deferred Correction (SDC) method is a stable, robust, and efficient high-order time integration scheme capable of an arbitrary order of accuracy. We apply the SDC method to solve the time-dependent diffusion equation of the spatial kinetics problem of the nuclear reactor. We will present how we combine the SDC with the common-used low-order methods for modeling the spatial kinetics models. Numerical results will be presented. The investigations made here can provide the foundation for future investigations on simulating the neutron transport problem using the high-order methods both for the spatial discretization and time integration.
Session S4: Inverse Problems, Statistical Methods
11:30 a.m. – 12:30 p.m. E.D.T. | Sat., May 21, 2022 | Room: 1096 East Hall
Ying Liang, Purdue University
Least-squares method for recovering multiple medium parameters
We present a two-stage least-squares method for inverse medium problems of reconstructing multiple unknown coefficients simultaneously from noisy data. A direct sampling method is applied to detect the location of the inhomogeneity in the first stage, while a total least squares method with a mixed regularization is used to recover the medium profile in the second stage. The total least-squares method is designed to minimize the residual of the model equation and the data fitting, along with an appropriate regularization, in an attempt to significantly improve the accuracy of the approximation obtained from the first stage. We shall also present an analysis on the well-posedness and convergence of this algorithm. Numerical experiments are carried out to verify the accuracies and robustness of this novel two-stage least-squares algorithm, with high tolerance of noise in the data.
Neophytos Charalambides, University of Michigan
Orthonormal Sketches for Secure Coded Regression
We propose a method for speeding up linear regression distributively, while ensuring security. We leverage randomized sketching techniques, and improve straggler resilience in asynchronous systems. Specifically, we apply a random orthonormal matrix and then sub sample in blocks, to simultaneously secure the information and reduce the dimension of the regression problem. In our setup, the transformation corresponds to an encoded encryption in an approximate gradient coding scheme, and the subsampling corresponds to the responses of the non-straggling workers; in a centralized coded computing network. We focus on the special case of the Subsampled Randomized Hadamard Transform, which we generalize to block sampling; and discuss how it can be used to secure the data. We illustrate the performance through numerical experiments.
Session S5: Remote Talks
11:30 a.m. – 12:30 p.m. E.D.T. | Sat., May 21, 2022 | Room: 1372 East Hall
Songting Luo, Iowa State University
A Fixed-point Iteration Method for High Frequency Helmholtz Equations
In order to obtain globally valid solutions for high frequency Helmholtz equations efficiently without “pollution effect”, we will transfer the problem into a fixed-point problem related to an exponential operator, and the associated functional evaluations are achieved by unconditionally stable operator-splitting based pseudospectral schemes such that large step sizes are allowed to reach the approximated fixed point efficiently for prescribed accuracy requirement. Both two-dimensional and three-dimensional numerical experiments are presented to demonstrate the method.
Xiaoming He, Missouri University of Science and Technology
PIFE-PIC: Parallel Immersed-Finite-Element Particle-in-Cell for 3-D kinetic simulations of plasma-material interactions
In this presentation, we present a recently developed particle simulation too PIFE-PIC, which is a novel three-dimensional (3-D) Parallel Immersed-Finite-Element (IFE) Particle-in-Cell (PIC) simulation model for particle simulations of plasma-material interactions. This framework is based on the recently developed non-homogeneous electrostatic IFE-PIC method, which is designed to handle complex plasma-material interface conditions associated with irregular geometries using a Cartesian-mesh-based PIC. Three-dimensional domain decomposition is utilized for both the electrostatic field solver with IFE and the particle operations in PIC to distribute the computation among multiple processors. A simulation of the orbital motion-limited (OML) sheath of a dielectric sphere immersed in a stationary plasma is carried out to validate PIFE-PIC and profile the parallel performance of the code package. Parallel efficiency up to approximately 110 superlinear speedup was achieved for strong scaling test. Furthermore, a large-scale simulation of plasma charging at a lunar crater containing 2 million PIC cells (10 million FE/IFE cells) and about 1 billion particles, running for 20,000 PIC steps in about 154 wall-clock hours, is presented to demonstrate the high-performance computing capability of PIFE-PIC.
Ahmed Al-Taweel, University of Arkansas at Little Rock
A stabilizer free weak Galerkin finite element method with supercloseness of order two
The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. A simple WG finite element method is introduced for second-order elliptic problems. First we have proved that stabilizers are no longer needed for this WG element. Then we have proved the supercloseness of order two for the WG finite element solution. The numerical results confirm the theory.
4:30 – 5:30 p.m. E.D.T. | Sat., May 21, 2022
Session S6: Reduced Order Modeling
4:30 – 5:30 p.m. E.D.T. | Sat., May 21, 2022 | Room: 1060 East Hall
Jörn Zimmerling, University of Michigan
Imaging with Reduced-order models
TBD
Siddhartha Srivastava, University of Michigan
Non-local calculus on graphs with application to reduced order modeling
Due to the expense and complexity of numerically solving Partial Differential Equations (PDE), high fidelity simulations do not lend themselves to typical engineering design and decision-making, which must instead rely on reduced-order models. We present an approach to reduced-order modeling that builds off recent graph-theoretic work for representation, exploration, and analysis on computed states of physical systems [1]. This graph theoretic framework consists of representing a physical system with a discrete manifold, where high fidelity and high dimensional states are mapped to vertices on a graph, with a lower-dimensional state vector associated with each vertex. Weighted edges between vertices can then be assigned, or found through graph-theoretic principles, indicating relationships or transitions between states and the magnitude of such correlations. We extend a non-local calculus on finite weighted graphs [2] to build models by exploiting polynomial expansions and the Taylor series, making graph-based representation amenable to modeling the PDE system. We analyze the consistency of the non-local derivatives in this setting, a crucial requirement for numerical applications. We show that the weight of the graph can be fine-tuned to achieve arbitrary order of accuracy in any number of dimensions without any assumptions of symmetry in the underlying data. Finally, applications of the method are demonstrated by extracting reduced-order models in the form of ordinary differential equations from parabolic partial differential equations of progressive complexity.
[1] Banerjee, R., et al. “A graph theoretic framework for representation, exploration and analysis on computed states of physical systems.” Computer Methods in Applied Mechanics and Engineering 351 (2019): 501-530.
[2] Gilboa, Guy, and Stanley Osher. “Nonlocal operators with applications to image processing.” Multiscale Modeling & Simulation 7.3 (2009): 1005-1028.
Jovan Zigic, Virginia Polytechnic Institute and State University
A Splitting Approach to Dynamic Mode Decomposition of Nonlinear Systems
Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This talk focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides a study of a newly proposed optimization framework for the DMD method called the Split DMD.
Session S7: Phase-field Modeling
4:30 – 5:30 p.m. E.D.T. | Sat., May 21, 2022 | Room: 1068 East Hall
Zachary Candelaria, Virginia Polytechnic Institute and State University
Phase-Field Modeling of Oscillatory Wetting Phenomena with Contact Angle Hysteresis
A recently published experimental study by Xia and Steen examines the connection between the contact line velocity and contact angle of liquid drops on an oscillating substrate that display contact angle hysteresis. Of particular interest in this experimental study is the analysis of the dependence of contact angle deviation on contact line velocity. Indeed it is found that for small angular displacements, there is a linear relationship between the two. Moreover, the oscillating drop exhibits contact angle hysteresis that is much greater than that measured from quasi-static experiments. Here we use a phase-field model of dynamic wetting which directly includes the contact angle hysteresis to simulate the results of the aforementioned authors. A thorough derivation of the governing equations is presented, starting from the pioneering work of Cahn and Hilliard. Our model is unique due to the explicit inclusion of contact angle hysteresis, a notoriously difficult physical phenomenon to model mathematically. By choosing appropriate parameters, our model can achieve very good agreement with experimental data. Further, we investigate the effects of contact line friction and the hysteresis window, which are otherwise very difficult to explore experimentally.
Min-Jhe Lu, Illinois Institute of Technology
Mechano-chemical Tumor Models: An Energetic Variational Approach
The building of the mechano-chemical tumor models aims to understand how the two key factors of (1) the mechanical interaction between the tumor cells and their surroundings, and (2) the biochemical reactions in the microenvironment of tumor cells can influence the dynamics of tumor growth. The mechanical interaction, realized by pressure or force between cells, is in view of the experimental discovery of microskeletons within cells, which give them mechanical strength. The biochemical reactions are involved with chemical species which either supply tumor with nutrients or change the cell cycle events of a tumor cell population. In this talk I will demonstrate how we apply the Energetic Variational Approach to build such models. The numerical simulation results in both sharp interface and diffuse interface (phase-field) formulation will also be given. This is a joint work with Prof. Chun Liu, Prof. John Lowengrub, Prof. Shuwang Li, Dr. Yiwei Wang and Dr. Huaming Yen.
Ziyang Huang, University of Michigan
Modeling multiphase problems with the consistent and conservative Phase-Field method
In the presentation, I will introduce the consistent and conservative Phase-Field method that addresses issues of coupling the Phase-Field modeling techniques to hydrodynamics, mass or heat transfer, and phase changes. Thanks to the proposed consistency conditions, the method results in multiphase models that enjoy not only the conservation of mass and momentum, but also the kinetic energy conservation and Galilean invariance, independent of the choice of the Phase-Field models to capture the interfaces or of the number and material properties of the phases. As a result, the mechanical and thermal equilibrium is guaranteed for arbitrarily large ratios of density and thermal capability. In the method, a volume distribution algorithm is developed to eliminate the production of fictitious phases, local voids, or overfilling, but simultaneously ensure the conservation and boundedness of the Phase-Field function.
Session S8: Parallel Computing, Newton’s Method
4:30 – 5:30 p.m. E.D.T. | Sat., May 21, 2022 | Room: 1084 East Hall
Zachary Miksis, University of Notre Dame
Parallel implementation of a sparse grid fast sweeping WENO method for Eikonal equations
Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady state solutions of hyperbolic PDEs. They are explicit and do not involve inverse operation of any nonlinear local system. Hence they are robust and flexible, and have been combined with high order WENO schemes to solve various hyperbolic PDEs in the literature. For multidimensional nonlinear problems, high order fixed-point fast sweeping WENO methods still require quite large amount of computational costs. We apply sparse-grid techniques, a naturally parallelizable and effective approximation tool for multidimensional problems, to reduce the computational costs of fixed-point fast sweeping WENO methods. Here we focus on a robust Runge-Kutta (RK) type fixed-point fast sweeping WENO scheme with third order accuracy for solving Eikonal equations, an important class of static Hamilton-Jacobi (H-J) equations. Numerical experiments on solving multidimensional Eikonal equations in parallel are performed to show that the sparse grid computations of the scheme achieve large savings of CPU times, and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids. This work is by Zachary Miksis and Yong-Tao Zhang.
Zhonggang Zeng, Northeastern Illinois University
Newton’s iteration at nonisolated solutions
The textbook Newton’s iteration has been unnecessarily restricted to isolated solutions of square systems with nonsingular Jacobian. As a recent discovery, Newton’s iteration can easily be applied to nonsquare systems with nonisolated solutions and rank-deficient Jacobians while maintaining quadratic convergence. This talk will elaborate on the extension of Newton’s iteration, its quadratic convergence on nonisolated solutions, and a wide range of applications in solving of nonlinear systems of equations.
Emmanuel Djegou, Missouri University of Science and Technology
To be determined.
TBD
Session S9: Mix and Match In-person and Remote
4:30 – 5:30 p.m. E.D.T. | Sat., May 21, 2022 | Room: 1372 East Hall
Chunmei Wang, University of Florida
A New Numerical Method for Div-Curl Systems with Low Regularity Assumptions
The speaker will present a numerical method for div-curl systems with normal boundary conditions by using a finite element technique known as primal-dual weak Galerkin (PDWG). The PDWG finite element scheme for the div-curl system has two prominent features in that it offers not only an accurate and reliable numerical solution to the div-curl system under the low Hα-regularity (α > 0) assumption for the true solution, but also an effective approximation of normal harmonic vector fields regardless the topology of the domain.
Chenyang Cao, Purdue University
A fast stable matrix form to some transforms
In this talk, we present a fast stable matrix-version method for evaluating the discretized transforms in one dimension, such as Gaussian transform, Hilbert transform. With the help of exponential expansions and diagonal form of translation, the kernel matrices from these transforms can be quickly approximated into structured matrices called sequentially semiseparable (SSS) matrices. Further, we can transfer SSS matrix representation to hierarchical semiseparable (HSS) matrix representation, which would be a more stable form for evaluating. Numerical experiments will be demonstrated on the performance and accuracy of evaluations.
Jiguang Sun, Michigan Technological University
Local estimators and Bayesian inverse problems with non-unique solutions
The Bayesian approach is effective for inverse problems. The posterior density distribution provides useful information of the unknowns. However, for problems with non-unique solutions, the classical estimators such as the maximum a posterior (MAP) and conditional mean (CM) are not suitable. We introduce two new estimators, the local maximum a posterior (LMAP) and local conditional mean (LCM). A simple algorithm based on clustering to compute LMAP and LCM is proposed. Their applications are demonstrated by three inverse problems: an inverse spectral problem, an inverse source problem, and an inverse medium problem.
Posters’ Titles & Abstracts
4 – 4:30 p.m. EDT | Fri., May 20, 2022
Ali Akhtari, University of Michigan – Dearborn
Magnetoconvection in a long vertical enclosure with walls with finite electrical conductivity
First results of implementation of a new solver for magnetohydrodynamic flows of liquid metals with natural convection are presented. A nearly fully conservative finite-difference scheme, which is known to improve accuracy and efficiency of simulations in the case of strong magnetic field effects, is combined with a tensor-product-Thomas solution of elliptic problems. The method is adapted to flows in domains with thin walls of finite electric conductivity and validated for a magnetoconvection flow in a long vertical box. The effects of wall electrical conductivity and the strength of the magnetic field are explored.
Vishwas Goel, University of Michigan
Simulation-based Optimization of the HOLE Architecture to Enable Fast Charging in Energy-Dense Li-ion Batteries
Li-ion batteries that are used in electric vehicles (EVs) primarily suffer from three main issues – limited capacity, slow charging, and fire outbreaks. The issue of limited capacity can be solved by designing energy-dense batteries, which entails the use of thick electrodes with low porosity. However, such an electrode design proves to be too tortuous for the transport of Li-ions with the electrode, which limits the charging rate. Furthermore, a graphite anode (most common anode material) based on this design is highly susceptible to Li-plating under fast charging conditions. Thus, it can be said that all three issues cannot be overcome by employing the conventional electrode design. To overcome these challenges, we have developed the Highly Ordered Laser-patterned Electrode (HOLE) architecture, which enables fast charging (6C) in energy dense electrodes (∼ 3 mAh/cm2) while eliminating the Li plating susceptibility, and thereby, solving the Li-plating caused fire outbreak issues in conventional Li-ion batteries.
Spencer Lee, Michigan State University
Numerical Methods for Highly Oscillatory Problems in Quantum Computing
We present a numerical methods tailored for Schr¨odinger equations with time dependent Hamiltonians and rapidly oscillating solutions, such as those arising in the modeling of controlled qudits. Our method discretizes the Picard form of the ordinary differential equation (ODE) by Filon quadrature. The method is implicit but the size of the linear system is always the dimension of the ODE independent of order. We illustrate that the new method is superior to the classic RK4 method.
Robert Malinas, University of Michigan
Change Detection for High-Dimensional Covariance Using Random Matrix Theory
Time series data often violates stationarity assumptions that are crucial to the success of most analysis methods. We introduce a novel test of stationarity based on random matrix theory. The test uses the spectral distribution of the sample covariance to detect deviations from second-order stationarity under the single change point model: the covariance matrix changes once over the observation interval. Using random matrix theory and free probability, we develop a statistic that is asymptotically zero for all indices under the null hypothesis under which no change is present. Furthermore, we show that the statistic is asymptotically maximized at the true change point with high probability under the alternative hypothesis. This yields a procedure by which we can detect a change in covariance by thresholding and subsequently estimate the change point by maximizing the statistic over the observed indices.
Som Phene, University of Michigan
Perturbed Steklov Eigenvalues via Asymptotic Analysis
We detect presence of impurities in a material by calculating Asymptotic series for Steklov Eigenvalue expansion.
Amit Rotem, Michigan State University
The WaveHoltz Heterogeneous Multiscale Method
We present the WaveHoltz Heterogeneous Multiscale Method (HMM), an iterative method for efficiently computing the homogenized solution to the Helmholtz equation with highly oscillatory variable coefficients that would make a direct numerical simulation prohibitively expensive. Unlike other numerical methods for the Helmholtz equation which directly discretize in the frequency domain, the WaveHoltz method discretizes in the time domain and iterates on the wave-equation. The WaveHoltz operator has many benefits: it is positive definite, has a bounded condition number, and is highly parallelizable. We solve the wave equation within the WaveHoltz iteration via the HMM which finds the homogenized solution to the wave-equation with highly oscillatory coefficients without having to resolve the high frequency information on the entire domain. When combined the WaveHoltz HMM is a highly efficient iterative method for computing the homogenized solution to the Helmholtz equation and has the additional benefit that, when compared to other multiscale methods which directly solve the elliptic problem, WaveHoltz HMM does not introduce additional error from artificial numerical boundary conditions.
Marc Tunnell, Grand Valley State University
bf Fast Gaussian Process Emulation of Mars Global Climate Model
The NASA Ames Mars Global Climate Model (MGCM) software has been in steady use at NASA for decades and was recently released to the public. This model simulates the complex interactions of various weather cycles that exist on Mars, namely the Dust Cycle, the CO2 Cycle, and the Water cycle. Utilized by NASA, the MGCM is used to help understand their empirically observed data through the use of sensitivity studies. However, these sensitivity studies are computationally taxing, requiring weeks to run. To address this issue, we have developed a surrogate model using Gaussian processes (GP) that can emulate the output of this model with relatively small amounts of data in a reduced amount of time (on the order of minutes). We demonstrate the effectiveness of our emulator using backward error analysis.