Applied Mathematics and Machine Learning
Department of Mathematics and Statistics
Texas Tech University
Abstract. Abstract. We consider sewing machinery between finite difference and analytical solutions defined at different scales: far away and near the source of the perturbation of the flow. One of the essences of the approach is that coarse problem and boundary value problem in the proxy of the source model two different flows. In his remarkable paper Peaceman propose a framework how to deal with solutions defined on different scale for linear time independent problem by introducing famous, Peaceman well block radius. In this article we consider novel problem how to solve this issue for transient flow generated by compressiblity of the fluid. We are proposing method to glue solution via total fluxes, which is predefined on coarse grid and
changes in the pressure, due to compressibility, in the block containing production(injection) well. It is important to mention that the coarse solution "does not see" boundary.
From industrial point of view our report provide mathematical tool for analytical interpretation of simulated data for compressible fluid flow around a well in a porous medium. It can be considered as a mathematical "shirt" on famous Peaceman well-block radius formula for linear (Darcy) transient flow but can be applied in much more general scenario. In the article we use Einstein approach to derive Material Balance equation, a key instrument to define R0. We will enlarge Einstein approach for three regimes of the Darcy and non-Darcy flows
for compressible fluid (time dependent): I.Stationary; II.P seudo Stationary(P SS); III.Boundary Dominated(BD).
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Abstract. Based on ideas due to Scovel-Weinstein, I present a general framework for constructing fluid moment closures of the Vlasov-Poisson system that exactly preserve that system's Hamiltonian structure. Notably, the technique applies in any space dimension and produces closures involving arbitrarily-large finite collections of moments. After selecting a desired collection of moments, the Poisson bracket for the closure is uniquely determined. Therefore data-driven fluid closures can be constructed by adjusting the closure Hamiltonian for compatibility with kinetic simulations. The talk is based on a longer discussion in: arXiv:2308.02733.
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Abstract. Dynamical low-rank approximation (DLRA) is an emerging tool for reducing computational costs and provides memory savings when solving high-dimensional problems. In this work, we propose and analyze a semi-implicit dynamical low-rank discontinuous Galerkin (DLR-DG) method for the space homogeneous kinetic equation with a relaxation operator, modeling the emission and absorption of particles by a background medium. Both DLRA and the DG scheme can be formulated as Galerkin equations. To ensure their consistency, a weighted DLRA is introduced so that the resulting DLR-DG solution is a solution to the fully discrete DG scheme in a subspace of the classical DG solution space. Similar to the classical DG method, we show that the proposed DLR-DG method is well-posed. We also identify conditions such that the DLR-DG solution converges to the equilibrium. Numerical results are presented to demonstrate the theoretical findings. This is a joint work with Eirik Endeve, Cory Hauck, and Stefan Schnake.
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Abstract. Inverse source scattering problems are essential in various fields, including antenna synthesis, medical imaging, and earthquake monitoring. In many applications, it is necessary to consider uncertainties in the model, and such problems are known as stochastic inverse problems. Traditional methods require a large number of realizations and information on medium coefficients to achieve accurate reconstruction for inverse random source problems.
To address this issue, we propose a data-assisted approach that uses boundary measurement data to reconstruct the statistical properties of the random source with fewer realizations. We compare the performance of different data-driven algorithms under this framework to enhance the initial approximation obtained from integral equations. Our numerical experiments demonstrate that the data-assisted approach achieves better reconstruction with only 1/10 of the realizations required by traditional methods.
Among the various Image-to-Image translation algorithms that we tested, the pix2pix method outperforms others in reconstructing well-separated inclusions with accurate positions. Our proposed approach results in stable reconstruction with respect to the observation data noise.
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Abstract. During this talk I will introduce and compare several finite element numerical schemes to approximate a chemo-attraction model with consumption effects, which is a nonlinear parabolic system for two variables; the cell density and the concentration of the chemical signal that the cell feel attracted to. I will detail the main properties of each scheme, such as conservation of cells, energy-stability and approximated positivity. Moreover, I will present numerical results to illustrate the efficiency of each of the schemes and to compare them with others classical schemes. This contribution is based on a joint work with Francisco Guillén-Gonzaléz (Universidad de Sevilla,
Spain).
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Abstract. Wave turbulence describes the dynamics of both classical and non-classical nonlinear waves out of thermal equilibrium. In this talk, I will present some of our recent results on wave turbulence theory. In the first part of the talk, I will discuss our rigorous derivation of wave turbulence equations. The second part of the talk is devoted to the analysis of wave turbulence equations as well as some numerical illustrations. The last part concerns some physical applications of wave turbulence theory. The talk is based on my joint work with Gigliola Staffilani (MIT), Avy Soffer (Rutgers), Yves Pomeau (ENS Paris), and Steven Walton (Los Alamos).
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Abstract. Quantum entanglement is the physical phenomenon, the medium, and, most importantly, the resources that enable quantum technologies. In this talk, we discuss recent results in estimating the degree of entanglement over major models of generic states as measured by different entanglement entropies, where the case of von Neumann entropy will be studied in some detail. Main ingredients leading to our results are random matrix theory, orthogonal polynomials, and the observed anomaly cancellation phenomena.
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Abstract. Semi-Lagrangian (SL) schemes are known as a major numerical tool for solving transport equations with many advantages and have been widely deployed in the fields of computational fluid dynamics, plasma physics modeling, numerical weather prediction, among others. In this work, we develop a novel machine learning-assisted approach to accelerate the conventional SL finite volume (FV) schemes. The proposed scheme avoids the expensive tracking of upstream cells but attempts to learn the SL discretization from the data by incorporating specific inductive biases in the neural network, significantly simplifying the algorithm implementation, and leading to improved efficiency. In addition, the method delivers sharp shock transitions and a level of accuracy that would typically require a much finer grid with traditional transport solvers. Furthermore, we present a multi-fidelity version of the method which is designed for scenarios where there is an abundance of low-fidelity data and a limited amount of high-fidelity data. Numerical tests demonstrate the effectiveness and efficiency of the proposed method.
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Abstract. The discretization of the Euler equations of gas dynamics ("compressible hydrodynamics") in a moving material frame is at the heart of many multi-physics simulation codes. The Arbitrary Lagrangian-Eulerian (ALE) framework is frequently applied in these settings in the form of a Lagrange phase, where the hydrodynamics equations are solved on a moving mesh, followed by a three-part "advection phase" involving mesh optimization, field remap and multi-material zone treatment.
This talk presents a general Lagrangian framework [1] for discretization of compressible shock hydrodynamics using high-order finite elements. The use of high-order polynomial spaces to define both the mapping and the reference basis functions in the Lagrange phase leads to improved robustness and symmetry preservation properties, better representation of the mesh curvature that naturally develops with the material motion and significant reduction in mesh imprinting. We will discuss the application of the curvilinear technology to the “advection phase” of ALE, including a DG-advection approach for conservative and monotonic high-order finite element interpolation (remap), as well as to coupled physics, such as electromagnetic diffusion. We will also review progress in robust and efficient algorithms for high-order mesh optimization, matrix-free preconditioning, high-order time integration and matrix-free monotonicity, which are critical components for the successful use of high-order methods in the compressible ALE settings.
In addition to their mathematical benefits, high-order finite element discretizations are a natural fit for modern HPC hardware, because their order can be used to tune the performance, by increasing the FLOPs/bytes ratio, or to adjust the algorithm for different hardware. In this direction, we will present some of our work on scalable high-order finite element software that combines the modular finite element library MFEM [2], the high-order shock hydrodynamics code BLAST [3] and its miniapp Laghos [4], where we will demonstrate the benefits of our approach with respect to strong scaling and GPU acceleration. Finally, we will give a brief update on related efforts in the co-design Center for Efficient Exascale Discretizations (CEED) in the Exascale Computing Project (ECP) of the DOE [5].
[1] "High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics", V. Dobrev and Tz. Kolev and R. Rieben, SIAM Journal on Scientific Computing, (34) 2012, pp.B606-B641.
[2] MFEM: Modular finite element library, http://mfem.org.
[3] BLAST: High-order shock hydrodynamics, http://llnl.gov/casc/blast.
[4] Laghos: Lagrangian high-order solver, https://github.com/CEED/Laghos.
[5] Center for Efficient Exascale Discretizations, http://ceed.exascaleproject.org.
When: 4:00 pm (Lubbock's local time is GMT -5)
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Abstract. Reductions in computational expense are often achieved through low-fidelity surrogates using data from fine-scale dynamical models. This reliance on summarized information presents challenges for predictive simulation, however, since summarization generally fails to commute with fundamental model properties such as conservation or involution which guarantee stability of the fine-scale system. This talk explores consequences of this idea and presents some recent methods for ensuring that structural properties such as conservation laws and dissipation inequalities are respected by data-driven surrogate models independent of their predictive accuracy.
When: 4:00 pm (Lubbock's local time is CST)
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Abstract. The time-independent Boltzmann neutron transport equation is an integro-differential equation for the neutron angular flux describing the location in space, energy, and direction of neutrons in a physical system of interest such as a nuclear reactor. Discretization of the neutron transport equation is driven by the physics of neutron interactions in matter, and in most cases of interest to nuclear engineers, generates large linear systems of equations. Recently, the tensor train (TT) technique has been shown to provide a useful low-rank approximation framework for high dimensional problems. Here we present a new approach to solve the neutron transport equation in Cartesian geometry using the TT representation of
the governing equations. The TT representation is explicitly constructed from the discretization of the governing equations using diamond differencing in space, multigroup-in-energy, and discrete ordinate collocation in angle. The proposed methodology is verified/validated by solving the canonical example of the k-effective eigenvalue neutron transport problem, which describes the criticality of a nuclear system such as a reactor. We compare and contrast accuracy and time complexity of the proposed TT approach to results obtained from PARTISN, the Los Alamos National Laboratory neutral particle transport code.
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