IA Scholar Query: Optimization of the multigrid-convergence rate on semi-structured meshes by local Fourier analysis.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 08 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Simulation of Transport in Type-II Superlattice Avalanche Photodiodes
https://scholar.archive.org/work/k2bqhkvr6fb2vdh6xpi7rxcmty
Improvements in semiconductor technology have allowed for growing commercial and institutional use of photodiodes as photodetectors due to their compact size, high reliability, and low cost. Of particular interest are low-excess noise avalanche photodiodes (APDs) which have increased sensitivity as a result of avalanche gain. The desire for APDs for short-wavelength (1-2 micron) and mid-wavelength (2-5 micron) photodetection is driven by applications which include fiber-optic communication, gas detection and monitoring, thermal imaging, and 3D light detection and ranging (LiDAR). We design and assess APDs fabricated from band-engineered InAlAs/InAsSb type-II superlattices which are nanostructures that permit the tuning of APD properties via electronic structure design. Complementary to the experimental growth, characterization, and fabrication of novel semiconductor material systems and devices are theoretical tools to design and simulate electronic devices. The design of type-II superlattices begins with a detailed calculation of the electronic band structure and carrier scattering rates. In this thesis, we consider the next step and report on the development of microscopic transport simulation methods used to devise band-engineering strategies which enable the design of type-II superlattices for use as high-performance APDs. The result is an ensemble Monte Carlo algorithm suitable for the simulation of Boltzmann transport within the full zone electronic structure of type-II superlattices. Within the uniform field model we demonstrate orders-of-magnitude improvement of the hole-to-electron impact ionization ratio---a key metric in the determination of APD performance---for InAlAs/InAsSb superlattices as compared with bulk InAs. We associate superlattice design features like miniband widths and the placement of Brillouin zone boundaries with electron and hole impact ionization coefficients and hence device properties like excess noise and bandwidth. This analysis reveals the InAlAs/InAsSb type-II superlattice as [...]Martin M Winslowwork_k2bqhkvr6fb2vdh6xpi7rxcmtyTue, 08 Nov 2022 00:00:00 GMTSolving linear systems of the form (A + γ UU^T) x = b by preconditioned iterative methods
https://scholar.archive.org/work/aip26lppobfvrkfvlpmsp4r3fu
We consider the iterative solution of large linear systems of equations in which the coefficient matrix is the sum of two terms, a sparse matrix A and a possibly dense, rank deficient matrix of the form γ UU^T, where γ > 0 is a parameter which in some applications may be taken to be 1. The matrix A itself can be singular, but we assume that the symmetric part of A is positive semidefinite and that A+γ UU^T is nonsingular. Linear systems of this form arise frequently in fields like optimization, fluid mechanics, computational statistics, and others. We investigate preconditioning strategies based on an alternating splitting approach combined with the use of the Sherman-Morrison-Woodbury matrix identity. The potential of the proposed approach is demonstrated by means of numerical experiments on linear systems from different application areas.Michele Benzi, Chiara Facciowork_aip26lppobfvrkfvlpmsp4r3fuMon, 07 Nov 2022 00:00:00 GMTFourier Neural Solver for large sparse linear algebraic systems
https://scholar.archive.org/work/7s2eb7j6djfi3psm5onmvr23tm
Large sparse linear algebraic systems can be found in a variety of scientific and engineering fields, and many scientists strive to solve them in an efficient and robust manner. In this paper, we propose an interpretable neural solver, the Fourier Neural Solver (FNS), to address them. FNS is based on deep learning and Fast Fourier transform. Because the error between the iterative solution and the ground truth involves a wide range of frequency modes, FNS combines a stationary iterative method and frequency space correction to eliminate different components of the error. Local Fourier analysis reveals that the FNS can pick up on the error components in frequency space that are challenging to eliminate with stationary methods. Numerical experiments on the anisotropy diffusion equation, convection-diffusion equation, and Helmholtz equation show that FNS is more efficient and more robust than the state-of-the-art neural solver.Chen Cui, Kai Jiang, Yun Liu, Shi Shuwork_7s2eb7j6djfi3psm5onmvr23tmSat, 08 Oct 2022 00:00:00 GMTA high-order fast direct solver for surface PDEs
https://scholar.archive.org/work/u5dbpgjg5zgzzjwufayl4zxf5m
We introduce a fast direct solver for variable-coefficient elliptic partial differential equations on surfaces based on the hierarchical Poincaré-Steklov method. The method takes as input an unstructured, high-order quadrilateral mesh of a surface and discretizes surface differential operators on each element using a high-order spectral collocation scheme. Elemental solution operators and Dirichlet-to-Neumann maps tangent to the surface are precomputed and merged in a pairwise fashion to yield a hierarchy of solution operators that may be applied in 𝒪(N log N) operations for a mesh with N elements. The resulting fast direct solver may be used to accelerate high-order implicit time-stepping schemes, as the precomputed operators can be reused for fast elliptic solves on surfaces. On a standard laptop, precomputation for a 12th-order surface mesh with over 1 million degrees of freedom takes 17 seconds, while subsequent solves take only 0.25 seconds. We apply the method to a range of problems on both smooth surfaces and surfaces with sharp corners and edges, including the static Laplace-Beltrami problem, the Hodge decomposition of a tangential vector field, and some time-dependent nonlinear reaction-diffusion systems.Daniel Fortunatowork_u5dbpgjg5zgzzjwufayl4zxf5mFri, 30 Sep 2022 00:00:00 GMTEfficient multigrid reduction-in-time for method-of-lines discretizations of linear advection
https://scholar.archive.org/work/5bswe6esdbhdhmhaab62ur7cqm
Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes known as characteristic components.We propose an alternative coarse-grid that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order.H. De Sterck and R. D. Falgout and O. A. Krzysik and J. B. Schroderwork_5bswe6esdbhdhmhaab62ur7cqmWed, 14 Sep 2022 00:00:00 GMTRobust field-level inference with dark matter halos
https://scholar.archive.org/work/upuwo62g4bfujmwvgg2wq7mtqy
We train graph neural networks on halo catalogues from Gadget N-body simulations to perform field-level likelihood-free inference of cosmological parameters. The catalogues contain ≲5,000 halos with masses ≳ 10^10 h^-1M_⊙ in a periodic volume of (25 h^-1 Mpc)^3; every halo in the catalogue is characterized by several properties such as position, mass, velocity, concentration, and maximum circular velocity. Our models, built to be permutationally, translationally, and rotationally invariant, do not impose a minimum scale on which to extract information and are able to infer the values of Ω_ m and σ_8 with a mean relative error of ∼6%, when using positions plus velocities and positions plus masses, respectively. More importantly, we find that our models are very robust: they can infer the value of Ω_ m and σ_8 when tested using halo catalogues from thousands of N-body simulations run with five different N-body codes: Abacus, CUBEP^3M, Enzo, PKDGrav3, and Ramses. Surprisingly, the model trained to infer Ω_ m also works when tested on thousands of state-of-the-art CAMELS hydrodynamic simulations run with four different codes and subgrid physics implementations. Using halo properties such as concentration and maximum circular velocity allow our models to extract more information, at the expense of breaking the robustness of the models. This may happen because the different N-body codes are not converged on the relevant scales corresponding to these parameters.Helen Shao, Francisco Villaescusa-Navarro, Pablo Villanueva-Domingo, Romain Teyssier, Lehman H. Garrison, Marco Gatti, Derek Inman, Yueying Ni, Ulrich P. Steinwandel, Mihir Kulkarni, Eli Visbal, Greg L. Bryan, Daniel Angles-Alcazar, Tiago Castro, Elena Hernandez-Martinez, Klaus Dolagwork_upuwo62g4bfujmwvgg2wq7mtqyWed, 14 Sep 2022 00:00:00 GMTComputing f-Divergences and Distances of High-Dimensional Probability Density Functions – Low-Rank Tensor Approximations
https://scholar.archive.org/work/2rzfdf5y65dkdkggaybghrfgpy
Very often, in the course of uncertainty quantification tasks or data analysis, one has to deal with high-dimensional random variables (RVs). A high-dimensional RV can be described by its probability density (pdf) and/or by the corresponding probability characteristic functions (pcf), or by a polynomial chaos (PCE) or similar expansion. Here the interest is mainly to compute characterisations like the entropy, or relations between two distributions, like their Kullback-Leibler divergence. These are all computed from the pdf, which is often not available directly, and it is a computational challenge to even represent it in a numerically feasible fashion in case the dimension d is even moderately large. In this regard, we propose to represent the density by a high order tensor product, and approximate this in a low-rank format. We show how to go from the pcf or functional representation to the pdf. This allows us to reduce the computational complexity and storage cost from an exponential to a linear. The characterisations such as entropy or the f-divergences need the possibility to compute point-wise functions of the pdf. This normally rather trivial task becomes more difficult when the pdf is approximated in a low-rank tensor format, as the point values are not directly accessible any more. The data is considered as an element of a high order tensor space. The considered algorithms are independent of the representation of the data as a tensor. All that we require is that the data can be considered as an element of an associative, commutative algebra with an inner product. Such an algebra is isomorphic to a commutative sub-algebra of the usual matrix algebra, allowing the use of matrix algorithms to accomplish the mentioned tasks.Alexander Litvinenko, Youssef Marzouk, Hermann G. Matthies, Marco Scavino, Alessio Spantiniwork_2rzfdf5y65dkdkggaybghrfgpyWed, 07 Sep 2022 00:00:00 GMTDynamic deformables
https://scholar.archive.org/work/ail7xnlyuzeblargndqwropb3e
Simulating dynamic deformation has been an integral component of Pixar's storytelling since Boo's shirt in Monsters, Inc. (2001). Recently, several key transformations have been applied to Pixar's core simulator Fizt that improve its speed, robustness, and generality. Starting with Coco (2017), improved collision detection and response were incorporated into the cloth solver, then with Cars 3 (2017) 3D solids were introduced, and in Onward (2020) clothing is allowed to interact with a character's body with two-way coupling. The 3D solids are based on a fast, compact, and powerful new formulation that we have published over the last few years at SIGGRAPH. Under this formulation, the construction and eigendecomposition of the force gradient, long considered the most onerous part of the implementation, becomes fast and simple. We provide a detailed, self-contained, and unified treatment here that is not available in the technical papers. We also provide, for the first time, open-source C++ implementations of many of the described algorithms. This new formulation is only a starting point for creating a simulator that is up challenges of a production environment. One challenge is performance: we discuss our current best practices for accelerating system assembly and solver performance. Another challenge that requires considerable attention is robust collision detection and response. Much has been written about collision detection approaches such as proximity-queries, continuous collisions and global intersection analysis. We discuss our strategies for using these techniques, which provides us with valuable information that is needed to handle challenging scenarios.Theodore Kim, David Eberlework_ail7xnlyuzeblargndqwropb3eTue, 02 Aug 2022 00:00:00 GMTMultigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective
https://scholar.archive.org/work/ah6ffh5ocvauxcu6qs2u3evgcy
A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach.H. De Sterck, S. Friedhoff, O. A. Krzysik, Scott P. MacLachlanwork_ah6ffh5ocvauxcu6qs2u3evgcyTue, 02 Aug 2022 00:00:00 GMTLearning Relaxation for Multigrid
https://scholar.archive.org/work/5ca2tqi6mrfynhwfaz3rdum6my
During the last decade, Neural Networks (NNs) have proved to be extremely effective tools in many fields of engineering, including autonomous vehicles, medical diagnosis and search engines, and even in art creation. Indeed, NNs often decisively outperform traditional algorithms. One area that is only recently attracting significant interest is using NNs for designing numerical solvers, particularly for discretized partial differential equations. Several recent papers have considered employing NNs for developing multigrid methods, which are a leading computational tool for solving discretized partial differential equations and other sparse-matrix problems. We extend these new ideas, focusing on so-called relaxation operators (also called smoothers), which are an important component of the multigrid algorithm that has not yet received much attention in this context. We explore an approach for using NNs to learn relaxation parameters for an ensemble of diffusion operators with random coefficients, for Jacobi type smoothers and for 4Color GaussSeidel smoothers. The latter yield exceptionally efficient and easy to parallelize Successive Over Relaxation (SOR) smoothers. Moreover, this work demonstrates that learning relaxation parameters on relatively small grids using a two-grid method and Gelfand's formula as a loss function can be implemented easily. These methods efficiently produce nearly-optimal parameters, thereby significantly improving the convergence rate of multigrid algorithms on large grids.Dmitry Kuznichovwork_5ca2tqi6mrfynhwfaz3rdum6myMon, 25 Jul 2022 00:00:00 GMTEfficient implicit spectral/hp element DG techniques for compressible flows
https://scholar.archive.org/work/66r3qpsjmvg55cjrpqqzxhgone
In the simulation of stiff problems, such as fluid flows at high Reynolds numbers, the efficiency of explicit time integration is significantly limited by the need to use very small time steps. To alleviate this limitation and to accelerate compressible flow simulations based on high-order spectral/$hp$ element methods, an implicit time integration method is developed using singly diagonally implicit Runge-Kutta temporal discretization schemes combined with a Jacobian-free Newton Krylov (JFNK) method. This thesis studies several topics influencing the efficiency, accuracy and robustness of the solver. Firstly, an efficient partially matrix-free block relaxed Jacobi (BRJ) preconditioner is proposed, in which the Jacobian matrix and preconditioning matrices are properly approximated based on studies of their influences on convergence. The preconditioner only forms and stores the diagonal part of the Jacobian matrix while the off-diagonal operators are calculated on the fly. Used together with techniques such as using single precision data, the BRJ can largely reduce the memory consumption when compared with matrix-based ones like incomplete LU factorization preconditioners (ILU). To further accelerate the solver, influences of different parts of the flux Jacobian on the preconditioning effects are studied and terms with minor influences are neglected. This reduces the computational cost of the BRJ preconditioner by about 3 times while maintaining similar preconditioning effects. Secondly, adaptive strategies for a suitable choice of some free parameters are designed to maintain temporal accuracy and relatively high efficiency. The several free parameters in the implicit solver have significant influences on the accuracy, efficiency and stability. Therefore, designing proper strategies in choosing them is essential for developing a robust general purpose solver. Based on the idea of constructing proper relations between the temporal, spatial and iterative errors, adaptive strategies are designed for determining the [...]Zhenguo Yan, Spencer Sherwin, Joaquim Peirowork_66r3qpsjmvg55cjrpqqzxhgoneFri, 15 Jul 2022 00:00:00 GMTWhy diffusion-based preconditioning of Richards equation works: spectral analysis and computational experiments at very large scale
https://scholar.archive.org/work/rarsysr2ijad7f4biptirnuuni
We consider here a cell-centered finite difference approximation of the Richards equation in three dimensions, averaging for interface values the hydraulic conductivity K=K(p), a highly nonlinear function, by arithmetic, upstream, and harmonic means. The nonlinearities in the equation can lead to changes in soil conductivity over several orders of magnitude and discretizations with respect to space variables often produce stiff systems of differential equations. A fully implicit time discretization is provided by backward Euler one-step formula; the resulting nonlinear algebraic system is solved by an inexact Newton Armijo-Goldstein algorithm, requiring the solution of a sequence of linear systems involving Jacobian matrices. We prove some new results concerning the distribution of the Jacobians eigenvalues and the explicit expression of their entries. Moreover, we explore some connections between the saturation of the soil and the ill-conditioning of the Jacobians. The information on eigenvalues justifies the effectiveness of some preconditioner approaches which are widely used in the solution of Richards equation. We also propose a new software framework to experiment with scalable and robust preconditioners suitable for efficient parallel simulations at very large scales. Performance results on a literature test case show that our framework is very promising in the advance towards realistic simulations at extreme scale.Daniele Bertaccini, Pasqua D'Ambra, Fabio Durastante, Salvatore Filipponework_rarsysr2ijad7f4biptirnuuniFri, 15 Jul 2022 00:00:00 GMTPreconditioned iterative methods for optimal control problems with time-dependent PDEs as constraints
https://scholar.archive.org/work/cvyajxxvcfckffdt4jeujmttuu
In this work, we study fast and robust solvers for optimal control problems with Partial Differential Equations (PDEs) as constraints. Speci cally, we devise preconditioned iterative methods for time-dependent PDE-constrained optimization problems, usually when a higher-order discretization method in time is employed as opposed to most previous solvers. We also consider the control of stationary problems arising in uid dynamics, as well as that of unsteady Fractional Differential Equations (FDEs). The preconditioners we derive are employed within an appropriate Krylov subspace method. The fi rst key contribution of this thesis involves the study of fast and robust preconditioned iterative solution strategies for the all-at-once solution of optimal control problems with time-dependent PDEs as constraints, when a higher-order discretization method in time is employed. In fact, as opposed to most work in preconditioning this class of problems, where a ( first-order accurate) backward Euler method is used for the discretization of the time derivative, we employ a (second-order accurate) Crank-Nicolson method in time. By applying a carefully tailored invertible transformation, we symmetrize the system obtained, and then derive a preconditioner for the resulting matrix. We prove optimality of the preconditioner through bounds on the eigenvalues, and test our solver against a widely-used preconditioner for the linear system arising from a backward Euler discretization. These theoretical and numerical results demonstrate the effectiveness and robustness of our solver with respect to mesh-sizes and regularization parameter. Then, the optimal preconditioner so derived is generalized from the heat control problem to time-dependent convection{diffusion control with Crank- Nicolson discretization in time. Again, we prove optimality of the approximations of the main blocks of the preconditioner through bounds on the eigenvalues, and, through a range of numerical experiments, show the effectiveness and robustness of our approac [...]Santolo Leveque, University Of Edinburgh, John Pearson, Jacek Gondziowork_cvyajxxvcfckffdt4jeujmttuuMon, 27 Jun 2022 00:00:00 GMTAdvanced multi-level and multi-index Monte Carlo methods in uncertainty quantification
https://scholar.archive.org/work/lt5nlpz4fnayfogjjsgcqw6h5y
This thesis examines forward uncertainty propagation with the focus on Monte Carlo sampling methods. The goal of the dissertation is to design and apply various efficient and practical multi-level and multi-index Monte Carlo methods in the context of 2D elliptic PDEs with random input.STANISLAV POLISHCHUKwork_lt5nlpz4fnayfogjjsgcqw6h5yTue, 21 Jun 2022 00:00:00 GMTA Platform for High-Bandwidth, Low-Noise Electrical Nanopore Sensing with Thermal Control
https://scholar.archive.org/work/fy37zthldzb2dfrd6pen7gmf3y
Solid-state nanopores are an emerging class of single-molecule detectors that provide information about molecular identity via the analysis of transient fluctuations in the ionic current flowing across a nanoscale pore in a thin membrane. The transport of biomolecules across a pore is a key step in nanopore-based sensing of DNA, RNA and proteins. The dynamics of biomolecular transport are complex and depend on the strength of many interactions, which can be tuned with temperature. However, temperature is rarely controlled during solid-state nanopore experiments because of the added electrical noise from the temperature control and measurement systems, greatly reducing the signal-to-noise ratio when detecting individual molecules. So far, the use of electric-based heating and cooling strategies has limited the recording bandwidth to the kHz range, restricting the studies to long polymers translocating via the pore relatively slowly. Yet, many molecules translocate through the pore orders of magnitude faster. This research presents the development and testing of an instrument to allow low-noise electrical recording of nanopore signals at MHz bandwidth as a function of temperature. Initial experiments using this custom-built instrument for the study of linear DNA polymers confirm previously observed translocation behaviours, while providing a higher temporal resolution. Overall results show that high-speed nanopore experiments are possible while controlling the temperature up to 70 °C, opening up exciting opportunities to study the unfolding of proteins toward single-molecule protein sequencing and the passage of DNA nanostructures for different bioassays. Future work will focus on realizing microfluidic flow cells and nanopore performance at higher temperature for longer recording times.Dmytro Lomovtsev, University, Mywork_fy37zthldzb2dfrd6pen7gmf3yMon, 20 Jun 2022 00:00:00 GMTVascular fluid-structure interaction: unified continuum formulation, image-based mesh generation pipeline, and scalable fully implicit solver technology
https://scholar.archive.org/work/h7gmatnwubbexexd2qiwdjnsbq
We propose a computational framework for vascular fluid-structure interaction (FSI), focusing on biomechanical modeling, geometric modeling, and solver technology. The biomechanical model is constructed based on the unified continuum formulation. We highlight that the chosen time integration scheme differs from existing implicit FSI integration methods in that it is indeed second-order accurate, does not suffer from the overshoot phenomenon, and optimally dissipates high-frequency modes in both subproblems. We propose a pipeline for generating subject-specific meshes for FSI analysis for anatomically realistic geometric modeling. Unlike most existing methodologies that operate directly on the wall surface mesh, our pipeline starts from the image segmentation stage. With high-quality surface meshes obtained, the volumetric meshes are then generated, guaranteeing a boundary-layered mesh in the fluid subdomain and a matching mesh across the fluid-solid interface. In the last, we propose a combined suite of nonlinear and linear solver technologies. Invoking a segregated algorithm within the Newton-Raphson iteration, the problem reduces to solving two linear systems in the multi-corrector stage. The first linear system can be addressed by the algebraic multigrid (AMG) method. The matrix related to the balance equations presents a two-by-two block structure in both subproblems. Using the Schur complement reduction (SCR) technique reduces the problem to solving matrices of smaller sizes of the elliptic type, and the AMG method again becomes a natural candidate. The benefit of the unified formulation is demonstrated in parallelizing the solution algorithms as the number of unknowns matches in both subdomains. We use the Greenshields-Weller benchmark as well as a patient-specific vascular model to demonstrate the robustness, efficiency, and scalability of the overall FSI solver technology.Ju Liu and Jiayi Huang and Qingshuang Lu and Yujie Sunwork_h7gmatnwubbexexd2qiwdjnsbqSun, 12 Jun 2022 00:00:00 GMTSpectral analysis and fast methods for large matrices arising from PDE approximation
https://scholar.archive.org/work/2hkuqayz5ndabgtwn2mjftjwpi
The main goal of this thesis is to show the crucial role that plays the symbol in analysing the spectrum the sequence of matrices resulting from PDE approximation and in designing a fast method to solve the associated linear problem. In the first part, we study the spectral properties of the matrices arising from ℙ_k Lagrangian Finite Elements approximation of second order elliptic differential problem with Dirichlet boundary conditions and where the operator is div(-a(𝐱) ∇·), with a continuous and positive over Ω, Ω being an open and bounded subset of ℝ^d, d≥ 1. We investigate the spectral distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on Ω=(0,1)^2 and we give a brief account in the case of variable coefficients and more general domains. While in the second part, we design a fast method of multigrid type for the resolution of linear systems arising from the ℚ_k Finite Elements approximation of the same considered problem in one and higher dimensional. The analysis is performed in one dimension, while the numerics are carried out also in higher dimension d≥ 2, demonstrating an optimal behavior in terms of the dependency on the matrix size and a robustness with respect to the dimensionality d and to the polynomial degree k.Ryma Imene Rahlawork_2hkuqayz5ndabgtwn2mjftjwpiFri, 10 Jun 2022 00:00:00 GMTLeveraging topology, geometry, and symmetries for efficient Machine Learning
https://scholar.archive.org/work/ei2jbnij4zedhiyq536b7idpda
First and foremost, I want to thank Pierre Vandergheynst for giving me the opportunity to pursue a PhD in his laboratory and for providing me with an environment where my creativity could flourish. Pierre gave me resources, freedom, and trust without asking for anything in return. Next, I want to thank Xavier Bresson for guiding me through both technical and non-technical matters. Xavier was an adviser and mentor until he left for his own lab. I would not be who I am without him. I am grateful to my thesis committee-Martin Jaggi, Pascal Frossard, Max Welling, Yan LeCun-for their interest and enthusiasm: it was an honor to discuss my research with you. Over the years, I had the pleasure to interact with and learn from many people. I thank Nathanaël Perraudin for his patience teaching me: I had so many questions! A unique collaborator and friend. Along with Vassilis Kalofolias and Johan Paratte, I thank this trio for bringing me into this journey at the LTS2 laboratory and getting me started. To them and to all the marvelous companions at the LTS2 and beyond, I thank you for the time we shared turning these years into a memorable experience. It was wonderful to grow together. Special thanks go to my office-mates, Nauman Shahid, Rodrigo Pena, Volodymir Miz, Konstantinos Pitas, and Youngjoo Seo: they made our office a lovely place and the table football a source of much after-lunch excitement. I thank my collaborators, includingMichaël Defferrardwork_ei2jbnij4zedhiyq536b7idpdaTue, 31 May 2022 00:00:00 GMTLocal Compatibility Boundary Conditions for High-Order Accurate Finite-Difference Approximations of PDEs
https://scholar.archive.org/work/y2ribn6ru5asljbauo4cdemb6e
We describe a new approach to derive numerical approximations of boundary conditions for high-order accurate finite-difference approximations. The approach, called the Local Compatibility Boundary Condition (LCBC) method, uses boundary conditions and compatibility boundary conditions derived from the governing equations, as well as interior and boundary grid values, to construct a local polynomial, whose degree matches the order of accuracy of the interior scheme, centered at each boundary point. The local polynomial is then used to derive a discrete formula for each ghost point in terms of the data. This approach leads to centered approximations that are generally more accurate and stable than one-sided approximations. Moreover, the stencil approximations are local since they do not couple to neighboring ghost-point values which can occur with traditional compatibility conditions. The local polynomial is derived using continuous operators and derivatives which enables the automatic construction of stencil approximations at different orders of accuracy. The LCBC method is developed here for problems governed by second-order partial differential equations, and it is verified for a wide range of sample problems, both time-dependent and time-independent, in two space dimensions and for schemes up to sixth-order accuracy.Nour G. Al Hassanieh, Jeffrey W. Banks, William D. Henshaw, Donald W. Schwendemanwork_y2ribn6ru5asljbauo4cdemb6eThu, 26 May 2022 00:00:00 GMTPredicting Physics in Mesh-reduced Space with Temporal Attention
https://scholar.archive.org/work/vebpddse6nc7be5nwab2kzbday
Graph-based next-step prediction models have recently been very successful in modeling complex high-dimensional physical systems on irregular meshes. However, due to their short temporal attention span, these models suffer from error accumulation and drift. In this paper, we propose a new method that captures long-term dependencies through a transformer-style temporal attention model. We introduce an encoder-decoder structure to summarize features and create a compact mesh representation of the system state, to allow the temporal model to operate on a low-dimensional mesh representations in a memory efficient manner. Our method outperforms a competitive GNN baseline on several complex fluid dynamics prediction tasks, from sonic shocks to vascular flow. We demonstrate stable rollouts without the need for training noise and show perfectly phase-stable predictions even for very long sequences. More broadly, we believe our approach paves the way to bringing the benefits of attention-based sequence models to solving high-dimensional complex physics tasks.Xu Han and Han Gao and Tobias Pfaff and Jian-Xun Wang and Li-Ping Liuwork_vebpddse6nc7be5nwab2kzbdayThu, 26 May 2022 00:00:00 GMT