381 research outputs found

    Data-assisted modeling of complex chemical and biological systems

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    Complex systems are abundant in chemistry and biology; they can be multiscale, possibly high-dimensional or stochastic, with nonlinear dynamics and interacting components. It is often nontrivial (and sometimes impossible), to determine and study the macroscopic quantities of interest and the equations they obey. One can only (judiciously or randomly) probe the system, gather observations and study trends. In this thesis, Machine Learning is used as a complement to traditional modeling and numerical methods to enable data-assisted (or data-driven) dynamical systems. As case studies, three complex systems are sourced from diverse fields: The first one is a high-dimensional computational neuroscience model of the Suprachiasmatic Nucleus of the human brain, where bifurcation analysis is performed by simply probing the system. Then, manifold learning is employed to discover a latent space of neuronal heterogeneity. Second, Machine Learning surrogate models are used to optimize dynamically operated catalytic reactors. An algorithmic pipeline is presented through which it is possible to program catalysts with active learning. Third, Machine Learning is employed to extract laws of Partial Differential Equations describing bacterial Chemotaxis. It is demonstrated how Machine Learning manages to capture the rules of bacterial motility in the macroscopic level, starting from diverse data sources (including real-world experimental data). More importantly, a framework is constructed though which already existing, partial knowledge of the system can be exploited. These applications showcase how Machine Learning can be used synergistically with traditional simulations in different scenarios: (i) Equations are available but the overall system is so high-dimensional that efficiency and explainability suffer, (ii) Equations are available but lead to highly nonlinear black-box responses, (iii) Only data are available (of varying source and quality) and equations need to be discovered. For such data-assisted dynamical systems, we can perform fundamental tasks, such as integration, steady-state location, continuation and optimization. This work aims to unify traditional scientific computing and Machine Learning, in an efficient, data-economical, generalizable way, where both the physical system and the algorithm matter

    On uncertainty quantification of eigenvalues and eigenspaces with higher multiplicity

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    We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations with random input data. The considered eigenpairs can be of higher but finite multiplicity. We investigate stochastic quantities of interest of the eigenpairs and discuss why, for multiplicity greater than 1, only the stochastic properties of the eigenspaces are meaningful, but not the ones of individual eigenpairs. To that end, we characterize the Fr\'echet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result, we prove local analyticity of the eigenspaces. Based on the Fr\'echet derivatives of the eigenpairs we discuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. Numerical examples are presented to illustrate the theoretical results

    Microstructure modeling and crystal plasticity parameter identification for predicting the cyclic mechanical behavior of polycrystalline metals

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    Computational homogenization permits to capture the influence of the microstructure on the cyclic mechanical behavior of polycrystalline metals. In this work we investigate methods to compute Laguerre tessellations as computational cells of polycrystalline microstructures, propose a new method to assign crystallographic orientations to the Laguerre cells and use Bayesian optimization to find suitable parameters for the underlying micromechanical model from macroscopic experiments

    NLTGCR: A class of Nonlinear Acceleration Procedures based on Conjugate Residuals

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    This paper develops a new class of nonlinear acceleration algorithms based on extending conjugate residual-type procedures from linear to nonlinear equations. The main algorithm has strong similarities with Anderson acceleration as well as with inexact Newton methods - depending on which variant is implemented. We prove theoretically and verify experimentally, on a variety of problems from simulation experiments to deep learning applications, that our method is a powerful accelerated iterative algorithm.Comment: Under Revie

    A Scalable Two-Level Domain Decomposition Eigensolver for Periodic Schr\"odinger Eigenstates in Anisotropically Expanding Domains

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    Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schr\"odinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in non-uniform, expanding domains. Since the quasi-optimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting two-level additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain's anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency.Comment: 30 pages, 7 figures, 2 table

    CDOpt: A Python Package for a Class of Riemannian Optimization

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    Optimization over the embedded submanifold defined by constraints c(x)=0c(x) = 0 has attracted much interest over the past few decades due to its wide applications in various areas. Plenty of related optimization packages have been developed based on Riemannian optimization approaches, which rely on some basic geometrical materials of Riemannian manifolds, including retractions, vector transports, etc. These geometrical materials can be challenging to determine in general. Existing packages only accommodate a few well-known manifolds whose geometrical materials are easily accessible. For other manifolds which are not contained in these packages, the users have to develop the geometric materials by themselves. In addition, it is not always tractable to adopt advanced features from various state-of-the-art unconstrained optimization solvers to Riemannian optimization approaches. We introduce CDOpt (available at https://cdopt.github.io/), a user-friendly Python package for a class Riemannian optimization. Based on constraint dissolving approaches, Riemannian optimization problems are transformed into their equivalent unconstrained counterparts in CDOpt. Therefore, solving Riemannian optimization problems through CDOpt directly benefits from various existing solvers and the rich expertise gained over decades for unconstrained optimization. Moreover, all the computations in CDOpt related to any manifold in question are conducted on its constraints expression, hence users can easily define new manifolds in CDOpt without any background on differential geometry. Furthermore, CDOpt extends the neural layers from PyTorch and Flax, thus allows users to train manifold constrained neural networks directly by the solvers for unconstrained optimization. Extensive numerical experiments demonstrate that CDOpt is highly efficient and robust in solving various classes of Riemannian optimization problems.Comment: 31 page

    Calculations of Excited Electronic States by Converging on Saddle Points Using Generalized Mode Following

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    Variational calculations of excited electronic states are carried out by finding saddle points on the surface that describes how the energy of the system varies as a function of the electronic degrees of freedom. This approach has several advantages over commonly used methods especially in the context of density functional calculations, as collapse to the ground state is avoided and yet, the orbitals are variationally optimized for the excited state. This optimization makes it possible to describe excitations with large charge transfer where calculations based on ground state orbitals are problematic, as in linear response time-dependent density functional theory. A generalized mode following method is presented where an nthn^{\text{th}}-order saddle point is found by inverting the components of the gradient in the direction of the eigenvectors of the nn lowest eigenvalues of the electronic Hessian matrix. This approach has the distinct advantage of following a chosen excited state through atomic configurations where the symmetry of the single determinant wave function is broken, as demonstrated in calculations of potential energy curves for nuclear motion in the ethylene and dihydrogen molecules. The method is implemented using a generalized Davidson algorithm and an exponential transformation for updating the orbitals within a generalized gradient approximation of the energy functional. Convergence is found to be more robust than for a direct optimization approach previously shown to outperform standard self-consistent field approaches, as illustrated here for charge transfer excitations in nitrobenzene and N-phenylpyrrole, involving calculations of 4th4^{\text{th}}- and 6th6^{\text{th}}-order saddle points, respectively. Finally, calculations of a diplatinum and silver complex are presented, illustrating the applicability of the method to excited state energy curves of large molecules.Comment: 57 pages, 12 figures, submitted to the Journal of Chemical Theory and Computatio

    Non-linear S2 Acceleration for Multidimensional Problems with Unstructured Meshes

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    The SN transport equation is popularly used to describe the distribution of neutrons in many applications including nuclear reactors. The topic of this research is a non-linear acceleration method for accelerating convergence of the scalar flux when the SN equation is solved iteratively. The SN angular flux iterate is used to compute average direction cosines in each octant. These direction cosines define a vector in each octant that may not have a unit length. Nonetheless, these eight average directions are used to form an S2-like equation that serves as the low-order equation in a nonlinear acceleration scheme. The acronym NL-S2 will be used to denote this non-linear S2-like equation. This method is investigated for use accelerating k-eigenvalue calculations and in this case, a k-eigenvalue can be converged on the low order system. NL-S2 is simple to discretize consistently with the SN equation and when this is done the scalar flux solution for the NL-S2 equation is the same as that for the SN equation. A primary motivation for this investigation of NL-S2 acceleration is that an SN style sweeper might be effective for inverting the NL-S2 “streaming plus collision” operator. However, the NL-S2 system, while looking similar to an S2 equation, has some significant differences. For any mesh other than one consisting entirely of rectangles or rectangular cuboids, the NL-S2 system will have many cyclic dependencies coupling cells. The NL-S2 method has been investigated in a number of other works, however all previous investigations focused either on one-dimensional problems or two-dimensional problems using a structured mesh. In this work, several methods for using an SN style sweeper were investigated for the NL-S2 system. It is found that modifications can be made to the NL-S2 linear system that drastically reduce the amount of off-diagonal matrix coefficients. The modified NL-S2 system is equivalent to the original at convergence of the scalar flux solution. An SN style sweeper is shown to be effective for this modified NL-S2 streaming plus collision operator. Acceleration of k-eigenvalue calculations is investigated for the well known two-dimensional C5G7 benchmark as well as a C5G7 like three-dimensional problem. A pincell problem containing a large void in the center is also investigated and NL-S2 acceleration is found to not be significantly impacted by the void. Our results indicate that NL-S2 acceleration is an effective alternative to traditional diffusion-based methods

    Iterative Methods for Neutron and Thermal Radiation Transport Problems

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    We develop, analyze, and test iterative methods for three kinds of multigroup transport problems: (1) k-eigenvalue neutronics, (2) thermal radiation transport, and (3) problems with “upscattering,” in which particles can gain energy from collisions. For k-eigenvalue problems, many widely used methods to accelerate power iteration use “low-order” equations that contain nonlinear functionals of the transport solution. The nonlinear functionals require that the transport discretization produce strictly positive solutions, and the low-order problems are often more difficult to solve than simple diffusion problems. Similar iterative methods have been proposed that avoid nonlinearities and employ simple diffusion operators in their low-order problems. However, due partly to theoretical concerns, such methods have been largely overlooked by the reactor analysis community. To address theoretical questions, we present analyses showing that a power-like iteration process applied to the linear low-order problem (which looks like a k-eigenvalue problem with a fixed source) provides rapid acceleration and produces the correct transport eigenvalue and eigenvector. We also provide numerical results that support the existing body of evidence that these methods give rapid iterative convergence, similar to methods that use nonlinear functionals. Thermal-radiation problems solve for radiation intensity and material temperature using coupled equations that are nonlinear in temperature. Some of the most powerful iterative methods in use today solve the coupled equations using a low-order equation in place of the transport equation, where the low-order equation contains nonlinear functionals of the transport solution. The nonlinear functionals need to be updated only a few times before the system converges. We develop, analyze, and test a new method that works in the same way but employs a simple diffusion low-order operator without nonlinear functionals. Our analysis and results show rapid iterative convergence, comparable to methods that use nonlinear functionals in more complicated low-order equations. For problems with upscattering, we have investigated the importance of linearly anisotropic scattering for problems dominated by scattering in Graphite. Our results show that the linearly anisotropic scattering encountered in problems of practical interest does not degrade the effec-tiveness of the iterative acceleration method. Additionally, we have tested a method devised by Hanuš and Ragusa using the semi-consistent Continuous/Discontinuous Finite Element Method (CDFEM) diffusion discretization we have devised, in place of the Modified Interior Penalty (MIP) discretization they employed. Our results with CDFEM show an increased number of transport iterations compared to MIP when there are cells with high-aspect ratio, but a reduction in overall runtime due to reduced degrees of freedom of the CDFEM operator compared to the MIP operator
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