6,566 results on '"Numerical Analysis"'
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2. OPTIMAL L² ERROR ANALYSIS OF A LOOSELY COUPLED FINITE ELEMENT SCHEME FOR THIN-STRUCTURE INTERACTIONS.
- Author
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BUYANG LI, WEIWEI SUN, YUPEI XIE, and WENSHAN YU
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FINITE element method , *FLUID-structure interaction , *NUMERICAL analysis , *VELOCITY , *FLUIDS - Abstract
Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluid-structure interactions (FSIs) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions for the FSI problems in the standard L² norm. In this article, we propose a new stable fully discrete kinematically coupled scheme for the incompressible FSI thin-structure model and establish a new approach for the numerical analysis of FSI problems in terms of a newly introduced coupled nonstationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the L² norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field. [ABSTRACT FROM AUTHOR]
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- 2024
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3. KERNEL INTERPOLATION OF HIGH DIMENSIONAL SCATTERED DATA.
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SHAO-BO LIN, XIANGYU CHANG, and XINGPING SUN
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NUMERICAL analysis , *INTERPOLATION , *OPERATOR theory , *DATA distribution , *STOCHASTIC approximation , *KERNEL (Mathematics) , *INTEGRAL operators - Abstract
Data sites selected from modeling high-dimensional problems often appear scattered in nonpaternalistic ways. Except for sporadic-clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These features defy any theoretical treatment that requires local or global quasi-uniformity of distribution of data sites. Incorporating a recently developed application of integral operator theory in machine learning, we propose and study in the current article a new framework to analyze kernel interpolation of high-dimensional data, which features bounding stochastic approximation error by the spectrum of the underlying kernel matrix. Both theoretical analysis and numerical simulations show that spectra of kernel matrices are reliable and stable barometers for gauging the performance of kernel-interpolation methods for high-dimensional data. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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4. INVERSE WAVE-NUMBER-DEPENDENT SOURCE PROBLEMS FOR THE HELMHOLTZ EQUATION.
- Author
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HONGXIA GUO and GUANGHUI HU
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INVERSE problems , *NUMERICAL analysis , *FREQUENCY-domain analysis , *FOURIER transforms , *FACTORIZATION - Abstract
This paper is concerned with the multi-frequency factorization method for imaging the support of a wave-number-dependent source function. It is supposed that the source function is given by the inverse Fourier transform of some time-dependent source with a priori given radiating period. Using the multi-frequency far-field data at a fixed observation direction, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the observation direction. The far-field data from sparse observation directions can be used to recover a Θ-convex polygon of the support. The inversion algorithm is proven valid even with multi-frequency near-field data in three dimensions. The connections to time-dependent inverse source problems are discussed in the near-field case. Numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. This paper provides numerical analysis for a frequency-domain approach to recover the support of an admissible class of time-dependent sources. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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5. GENERALIZED DIMENSION TRUNCATION ERROR ANALYSIS FOR HIGH-DIMENSIONAL NUMERICAL INTEGRATION: LOGNORMAL SETTING AND BEYOND.
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GUTH, PHILIPP A. and KAARNIOJA, VESA
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NUMERICAL analysis , *NUMERICAL integration , *MONTE Carlo method , *RANDOM fields , *RANDOM variables , *TAYLOR'S series - Abstract
Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional integrals of the PDE output over a sequence of stochastic variables. In practical computations, one typically needs to discretize the problem in several ways: approximating an infinite-dimensional input random field with a finite-dimensional random field, spatial discretization of the PDE using, e.g., finite elements, and approximating high-dimensional integrals using cubatures such as quasi--Monte Carlo methods. In this paper, we focus on the error resulting from dimension truncation of an input random field. We show how Taylor series can be used to derive theoretical dimension truncation rates for a wide class of problems and we provide a simple checklist of conditions that a parametric mathematical model needs to satisfy in order for our dimension truncation error bound to hold. Some of the novel features of our approach include that our results are applicable to nonaffine parametric operator equations, dimensionally truncated conforming finite element discretized solutions of parametric PDEs, and even compositions of PDE solutions with smooth nonlinear quantities of interest. As a specific application of our method, we derive an improved dimension truncation error bound for elliptic PDEs with lognormally parameterized diffusion coefficients. Numerical examples support our theoretical findings. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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6. NUMERICAL ANALYSIS FOR CONVERGENCE OF A SAMPLE-WISE BACKPROPAGATION METHOD FOR TRAINING STOCHASTIC NEURAL NETWORKS.
- Author
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ARCHIBALD, RICHARD, FENG BAO, YANZHAO CAO, and HUI SUN
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STOCHASTIC control theory , *NUMERICAL analysis , *CONVOLUTIONAL neural networks , *STOCHASTIC differential equations , *CONDITIONAL expectations - Abstract
The aim of this paper is to carry out convergence analysis and algorithm implementation of a novel sample-wise backpropagation method for training a class of stochastic neural networks (SNNs). The preliminary discussion on such an SNN framework was first introduced in [Archibald et al., Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), pp. 2807--2835]. The structure of the SNN is formulated as a discretization of a stochastic differential equation (SDE). A stochastic optimal control framework is introduced to model the training procedure, and a sample-wise approximation scheme for the adjoint backward SDE is applied to improve the efficiency of the stochastic optimal control solver, which is equivalent to the backpropagation for training the SNN. The convergence analysis is derived by introducing a novel joint conditional expectation for the gradient process. Under the convexity assumption, our result indicates that the number of SNN training steps should be proportional to the square of the number of layers in the convex optimization case. In the implementation of the sample-based SNN algorithm with the benchmark MNIST dataset, we adopt the convolution neural network (CNN) architecture and demonstrate that our sample-based SNN algorithm is more robust than the conventional CNN. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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7. NUMERICAL METHODS AND ANALYSIS OF COMPUTING QUASIPERIODIC SYSTEMS.
- Author
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KAI JIANG, SHIFENG LI, and PINGWEN ZHANG
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COMPUTER systems , *NUMERICAL analysis , *PERIODIC functions , *COMPUTATIONAL complexity , *FAST Fourier transforms , *FOURIER transforms - Abstract
Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is a great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256 (2014), pp. 428--440], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of the PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of the quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that the PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudospectral) method. Then, we analyze the computational complexity of the PM and QSM in calculating quasiperiodic systems. The PM can use a fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of the PM, QSM, and periodic approximation method in solving the linear time-dependent quasiperiodic Schr\"odinger equation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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8. NUMERICAL INTEGRATION OF SCHRÖDINGER MAPS VIA THE HASIMOTO TRANSFORM.
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BANICA, VALERIA, MAIERHOFER, GEORG, and SCHRATZ, KATHARINA
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NUMERICAL integration , *NUMERICAL analysis - Abstract
We introduce a numerical approach to computing the SchrÖdinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schr\"odinger (NLS) equation. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators for the SM equation. Our approach consists of two parts: an integration of the NLS equation followed by the numerical evaluation of the Hasimoto transform. Motivated by the desire to study rough solutions to the SM equation, we also introduce a new symmetric low-regularity integrator for the NLS equation. This is combined with our novel fast low-regularity Hasimoto (FLowRH) transform, based on a tailored analysis of the resonance structures in the Magnus expansion and a fast realization based on block-Toeplitz partitions, to yield an efficient low-regularity integrator for the SM equation. This scheme in particular allows us to obtain approximations to the SM in a more general regime (i.e., under lower-regularity assumptions) than previously proposed methods. The favorable properties of our methods are exhibited both in theoretical convergence analysis and in numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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9. A TANGENTIAL AND PENALTY-FREE FINITE ELEMENT METHOD FOR THE SURFACE STOKES PROBLEM.
- Author
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DEMLOW, ALAN and NEILAN, MICHAEL
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FINITE element method , *STOKES equations , *DEGREES of freedom , *FLUID flow , *NUMERICAL analysis , *VECTOR fields - Abstract
Surface Stokes and Navier--Stokes equations are used to model fluid flow on surfaces. They have attracted significant recent attention in the numerical analysis literature because approximation of their solutions poses significant challenges not encountered in the Euclidean context. One challenge comes from the need to simultaneously enforce tangentiality and H1 conformity (continuity) of discrete vector fields used to approximate solutions in the velocity-pressure formulation. Existing methods in the literature all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. Missing so far is a robust and systematic construction of surface Stokes finite element spaces which employ nodal degrees of freedom, including MINI, Taylor--Hood, Scott--Vogelius, and other composite elements which can lead to divergence-conforming or pressure-robust discretizations. In this paper we construct surface MINI spaces whose velocity fields are tangential. They are not H1-conforming, but do lie in H(div) and do not require penalization to achieve optimal convergence rates. We prove stability and optimal-order energy-norm convergence of the method and demonstrate optimal-order convergence of the velocity field in L2 via numerical experiments. The core advance in the paper is the construction of nodal degrees of freedom for the velocity field. This technique also may be used to construct surface counterparts to many other standard Euclidean Stokes spaces, and we accordingly present numerical experiments indicating optimal-order convergence of nonconforming tangential surface Taylor--Hood P2 P1 elements. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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10. ANALYSIS AND NUMERICAL APPROXIMATION OF STATIONARY SECOND-ORDER MEAN FIELD GAME PARTIAL DIFFERENTIAL INCLUSIONS.
- Author
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OSBORNE, YOHANCE A. P. and SMEARS, IAIN
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NUMERICAL analysis , *DIFFERENTIAL games , *FEEDBACK control systems , *FOKKER-Planck equation , *CONVEX functions , *DIFFERENTIAL inclusions , *FINITE element method - Abstract
The formulation of mean field games (MFG) typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. We develop the analysis and numerical analysis of stationary MFG for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we propose a generalization of the MFG system as a partial differential inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We establish the existence of a weak solution to the MFG PDI system, and we further prove uniqueness under a similar monotonicity condition to the one considered by Lasry and Lions. We then propose a monotone finite element discretization of the problem, and we prove strong H1-norm convergence of the approximations of the value function and strong Lq-norm convergence of the approximations of the density function. We illustrate the performance of the numerical method in numerical experiments featuring nonsmooth solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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11. A PROBABILISTIC SCHEME FOR SEMILINEAR NONLOCAL DIFFUSION EQUATIONS WITH VOLUME CONSTRAINTS.
- Author
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MINGLEI YANG, GUANNAN ZHANG, DEL-CASTILLO-NEGRETE, DIEGO, and YANZHAO CAO
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HEAT equation , *KIRKENDALL effect , *STOCHASTIC differential equations , *INTEGRO-differential equations , *NUMERICAL analysis , *TRANSPORT equation - Abstract
This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integrodifferential operator consists of both local convection-diffusion and nonlocal diffusion operators. Our numerical scheme is based on the direct approximation of the nonlinear Feynman-Kac formula that establishes a link between nonlinear PIDEs and stochastic differential equations. The exploitation of the Feynman-Kac representation avoids solving dense linear systems arising from nonlocal operators. Compared with existing stochastic approaches, our method can achieve first-order convergence after balancing the temporal and spatial discretization errors, which is a significant improvement of existing probabilistic/stochastic methods for nonlocal diffusion problems. Error analysis of our numerical scheme is established. The effectiveness of our approach is shown in two numerical examples. The first example considers a three-dimensional nonlocal diffusion equation to numerically verify the error analysis results. The second example presents a physics problem motivated by the study of heat transport in magnetically confined fusion plasmas. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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12. A NUMERICAL ENERGY REDUCTION APPROACH FOR SEMILINEAR DIFFUSION-REACTION BOUNDARY VALUE PROBLEMS BASED ON STEADY-STATE ITERATIONS.
- Author
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AMREIN, MARIO, HEID, PASCAL, and WIHLER, THOMAS P.
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BOUNDARY value problems , *MATHEMATICAL sequences , *NUMERICAL analysis , *FINITE element method - Abstract
We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems, where the nonlinear reaction terms need to be neither monotone nor convex. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy reduction approach. More specifically, this procedure aims to generate a sequence of numerical approximations, which results from the iterative solution of related (stabilized) linearized discrete problems, and tends to a critical point of the underlying energy functional in a stable way. Simultaneously, the finite-dimensional approximation spaces are adaptively refined. This is implemented in terms of a new mesh refinement strategy in the context of finite element discretizations, which again relies on the energy structure of the problem under consideration. In particular, in contrast to more traditional approaches, it does not involve any a posteriori error estimators, and is based on local energy reduction indicators instead. In combination, the resulting adaptive algorithm consists of an iterative linearization procedure on a sequence of hierarchically refined discrete spaces, which we prove to converge toward a solution of the continuous problem in an appropriate sense. Numerical experiments demonstrate the robustness and reliability of our approach for a series of examples. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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13. PROJECTION METHODS FOR NEURAL FIELD EQUATIONS.
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AVITABILE, DANIELE
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NONLINEAR evolution equations , *COMPUTATIONAL neuroscience , *NUMERICAL analysis , *INTEGRAL equations , *COLLOCATION methods , *EQUATIONS , *SPECTRAL theory , *MATHEMATICAL bounds - Abstract
Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields and derive a priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neuroscience applications. We find that the convergence rate of a projection scheme for a neural field is determined to a great extent by the convergence rate of the projection operator. This abstract analysis, which unifies the treatment of collocation and Galerkin schemes, is carried out in operator form, without resorting to quadrature rules for the integral term, which are introduced only at a later stage, and whose choice depends on the choice of the projector. Using an elementary time stepper as an example, we demonstrate that the error in a time stepper has two separate contributions: one from the projector and one from the time discretization. We give examples of concrete projection methods: two collocation schemes (piecewise linear and spectral collocation) and two Galerkin schemes (finite element and spectral Galerkin); for each of them we derive error bounds from the general theory, introduce several discrete variants, provide implementation details, and present reproducible convergence tests. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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14. STRATIFIED RADIATIVE TRANSFER IN A FLUID AND NUMERICAL APPLICATIONS TO EARTH SCIENCE.
- Author
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GOLSE, FRANÇOIS and PIRONNEAU, OLIVIER R.
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EARTH sciences , *RADIATIVE transfer , *RADIATIVE transfer equation , *FLUIDS , *NUMERICAL analysis , *ATMOSPHERE - Abstract
New mathematical results are given for the radiative transfer equations alone and coupled with the temperature equation of a fluid: existence, uniqueness, a maximum principle, and a convergent monotone iterative scheme. Numerical tests for Earth's atmosphere and the heating of a pool by the Sun are included. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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15. CONVERGENCE ANALYSIS OF DISCRETE HIGH-INDEX SADDLE DYNAMICS.
- Author
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YUE LUO, XIANGCHENG ZHENG, XIANGLE CHENG, and LEI ZHANG
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SADDLERY , *NUMERICAL analysis - Abstract
Saddle dynamics is a time continuous dynamics to efficiently compute the any-index saddle points and construct the solution landscape. In practice, the saddle dynamics needs to be discretized for numerical computations, while the corresponding numerical analyses are rarely studied in the literature, especially for the high-index cases. In this paper we propose the convergence analysis of discrete high-index saddle dynamics. To be specific, we prove the local linear convergence rates of numerical schemes of high-index saddle dynamics, which indicates that the local curvature in the neighborhood of the saddle point and the accuracy of computing the eigenfunctions are main factors that affect the convergence of discrete saddle dynamics. The proved results serve as compensations for the convergence analysis of high-index saddle dynamics and are substantiated by numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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16. NUMERICAL ANALYSIS OF A DISCONTINUOUS GALERKIN METHOD FOR THE BORRVALL--PETERSSON TOPOLOGY OPTIMIZATION PROBLEM.
- Author
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PAPADOPOULOS, IOANNIS P. A.
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GALERKIN methods , *NUMERICAL analysis , *STOKES flow , *FINITE element method , *TOPOLOGY , *STOKES equations - Abstract
Divergence-free discontinuous Galerkin (DG) finite element methods offer a suitable discretization for the pointwise divergence-free numerical solution of Borrvall and Petersson's model for the topology optimization of fluids in Stokes flow [T. Borrvall and J. Petersson, Internat. J. Numer. Methods Fluids, 41 (2003), pp. 77--107]. The convergence results currently found in the literature only consider H¹-conforming discretizations for the velocity. In this work, we extend the numerical analysis of Papadopoulos and S\"uli to divergence-free DG methods with an interior penalty [I. P. A. Papadopoulos and E. S\"uli, J. Comput. Appl. Math., 412 (2022), 114295]. We show that, given an isolated minimizer of the infinite-dimensional problem, there exists a sequence of DG finite element solutions, satisfying necessary first-order optimality conditions, that strongly converges to the minimizer. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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17. ADAPTIVE MULTILEVEL MONTE CARLO FOR PROBABILITIES.
- Author
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HAJI-ALI, ABDUL-LATEEF, SPENCE, JONATHAN, and TECKENTRUP, ARETHA L.
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RANDOM variables , *PROBABILITY theory , *NUMERICAL analysis , *COMPUTATIONAL complexity - Abstract
We consider the numerical approximation of P[G], where the d-dimensional random variable G cannot be sampled directly, but there is a hierarchy of increasingly accurate approximations N which can be sampled. The cost of standard Monte Carlo estimation scales poorly with accuracy in this setup since it compounds the approximation and sampling cost. A direct application of multilevel Monte Carlo (MLMC) improves this cost scaling slightly, but returns suboptimal computational complexities since estimation of the probability involves a discontinuous functional of Gl. We propose a general adaptive framework which is able to return the MLMC complexities seen for smooth or Lipschitz functionals of Gl. Our assumptions and numerical analysis are kept general allowing the methods to be used for a wide class of problems. We present numerical experiments on nested simulation for risk estimation, where G = E[X|Y] is approximated by an inner Monte Carlo estimate. Further experiments are given for digital option pricing, involving an approximation of a d-dimensional SDE. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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18. CONVERGENCE IN TOTAL VARIATION OF THE EULER--MARUYAMA SCHEME APPLIED TO DIFFUSION PROCESSES WITH MEASURABLE DRIFT COEFFICIENT AND ADDITIVE NOISE.
- Author
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BENCHEIKH, OUMAIMA and JOURDAIN, BENJAMIN
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STOCHASTIC differential equations , *LEBESGUE measure , *NUMERICAL analysis , *DIFFUSION coefficients , *LINEAR orderings , *POLYNOMIAL chaos - Abstract
We are interested in the Euler--Maruyama discretization of a stochastic differential equation in dimension d with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order 1/2 in total variation distance. When the drift has a spatial divergence in the sense of distributions with pth power integrable with respect to the Lebesgue measure in space uniformly in time for some p≥ d, the order of convergence at the terminal time improves to 1 up to some logarithmic factor. In dimension d = 1, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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19. NUMERICAL ANALYSIS FOR MAXWELL OBSTACLE PROBLEMS IN ELECTRIC SHIELDING.
- Author
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HENSEL, MAURICE and YOUSEPT, IRWIN
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NUMERICAL analysis , *FARADAY effect , *FINITE element method , *EVOLUTION equations , *MAGNETIC fields - Abstract
This paper proposes and examines a finite element method (FEM) for a Maxwell obstacle problem in electric shielding. The model is given by a coupled system comprising the Faraday equation and an evolutionary variational inequality (VI) of Ampère-Maxwell-type. Based on the leapfrog (Yee) time-stepping and the Nédélec edge elements, we set up a fully discrete FEM where the obstacle is discretized in such a way that no additional nonlinear solver is required for the computation of the discrete VI. While the L²-stability is achieved for the discrete solutions and the associated difference quotients, the scheme only guarantees the L¹-stability for the discrete magnetic curl field in the obstacle region. The lack of the global L²-stability for the magnetic curl field is justified by the low regularity issue in Maxwell obstacle problems and turns to be the main challenge in the convergence analysis. Our convergence proof consists of two main stages. First, exploiting the L¹-stability in the obstacle region, we derive a convergence result towards a weaker system involving smooth feasible test functions. In the second step, we recover the original system by enlarging the feasible test function set through a specific constraint preserving mollification process in the spirit of Ern and Guermond [Comput. Methods Appl. Math., 16 (2016), pp. 51-75]. This paper is closed by three-dimensional numerical results of the proposed FEM confirming the theoretical convergence result and, in particular, the Faraday shielding effect. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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20. A CONVERGENT INTERACTING PARTICLE METHOD AND COMPUTATION OF KPP FRONT SPEEDS IN CHAOTIC FLOWS.
- Author
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JUNLONG LYU, ZHONGJIAN WANG, XIN, JACK, and ZHIWEN ZHANG
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ADVECTION-diffusion equations , *THREE-dimensional flow , *LINEAR operators , *VARIATIONAL principles , *NUMERICAL analysis , *SPEED - Abstract
In this paper, we study the propagation speeds of reaction-diffusion-advection fronts in time-periodic cellular and chaotic flows with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We first apply the variational principle to reduce the computation of KPP front speeds to a principal eigenvalue problem of a linear advection-diffusion operator with space-time periodic coefficient on a periodic domain. To this end, we develop efficient Lagrangian particle methods to compute the principal eigenvalue through the Feynman-Kac formula. By estimating the convergence rate of Feynman-Kac semigroups and the operator splitting method for approximating the linear advection-diffusion solution operators, we obtain convergence analysis for the proposed numerical method. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing KPP front speeds in time-periodic cellular and chaotic flows, especially the time-dependent Arnold-Beltrami-Childress flow and time-dependent Kolmogorov flow in three-dimensional space. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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21. NUMERICAL ANALYSIS OF RESONANCES BY A SLAB OF SUBWAVELENGTH SLITS BY FOURIER-MATCHING METHOD.
- Author
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JIAXIN ZHOU and WANGTAO LU
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NUMERICAL analysis , *ASYMPTOTIC expansions , *RESONANCE , *SCHAUDER bases , *LINEAR systems , *FOURIER series , *SEPARATION of variables - Abstract
This paper proposes a simple and rigorous Fourier-matching method to study transverse-magnetic-polarized electro-magnetic resonances by a perfectly conducting slab with a finite number of subwavelength slits of width h\ll 1. Since variable separation is applicable in the region outside the slits, by Fourier transforming its governing equation, a magnetic field can be represented in terms of its derivative on the aperture. Next, inside each slit where variable separation is still available, the field can be represented as a Fourier series in terms of a countable set of basis functions with unknown Fourier coefficients. Finally, by matching the two subdomain representations on the aperture, we establish a linear system of an infinite number of equations governing the countable Fourier coefficients; the unknowns are further rescaled to be in the standard l² space. By the asymptotic expansion of each entry of the coefficient matrix, we rigorously show that its certain principal submatrix is invertible so that the infinite-dimensional linear system can be reduced to a finite-dimensional linear system. Resonance frequencies are exactly those frequencies making the linear system rank-deficient. This in turn leads to an asymptotic formula of accuracy O (h³ log h) for computing the resonance frequencies. We emphasize that the new formula is more accurate than all existing results and is the first formula for slits of number more than two to the best of our knowledge. Numerical experiments are carried out finally to validate the proposed formula and demonstrate its accuracy. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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22. A MESHFREE METHOD FOR A PDE-CONSTRAINED OPTIMIZATION PROBLEM.
- Author
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HOFF, DANIEL and WENDLAND, HOLGER
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NUMERICAL analysis , *RADIAL basis functions , *COLLOCATION methods , *MESHFREE methods - Abstract
We describe a new approximation method for solving a PDE-constrained optimization problem numerically. Our method is based on the adjoint formulation of the optimization problem, leading to a system of weakly coupled, elliptic PDEs. These equations are then solved using kernel-based collocation. We derive an error analysis and give numerical examples. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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23. Dispersion Analysis of CIP-FEM for the Helmholtz Equation
- Author
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Yu Zhou and Haijun Wu
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
24. Geometric Two-Scale Integrators for Highly Oscillatory System: Uniform Accuracy and Near Conservations
- Author
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Wang, Bin and Zhao, Xiaofei
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Numerical Analysis (math.NA) ,65L05, 65L20, 65L70, 65M15 - Abstract
In this paper, we consider a class of highly oscillatory Hamiltonian systems which involve a scaling parameter $\varepsilon\in(0,1]$. The problem arises from many physical models in some limit parameter regime or from some time-compressed perturbation problems. The solution of the model exhibits rapid temporal oscillations with $\mathcal{O}(1)$-amplitude and $\mathcal{O}(1/\varepsilon)$-frequency, which makes classical numerical methods inefficient. We apply the two-scale formulation approach to the problem and propose two new time-symmetric numerical integrators. The methods are proved to have the uniform second order accuracy for all $\varepsilon$ at finite times and some near-conservation laws in long times. Numerical experiments on a H\'{e}non-Heiles model, a nonlinear Schr\"{o}dinger equation and a charged-particle system illustrate the performance of the proposed methods over the existing ones., Comment: 21 pages
- Published
- 2023
25. Optimal \(\boldsymbol{{L^2}}\) Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn–Hilliard–Navier–Stokes System
- Author
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Wentao Cai, Weiwei Sun, Jilu Wang, and Zongze Yang
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
26. A Spectral Decomposition Method to Approximate Dirichlet-to-Neumann Maps in Complicated Waveguides
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Ruming Zhang
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
27. Optimal Rate of Convergence for Approximations of SPDEs with Nonregular Drift
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Oleg Butkovsky, Konstantinos Dareiotis, and Máté Gerencsér
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Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
28. Robust Superlinear Krylov Convergence for Complex Noncoercive Compact-Equivalent Operator Preconditioners
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Owe Axelsson, János Karátson, and Frédéric Magoulès
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Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
29. A Discontinuous Petrov–Galerkin Method for Reissner–Mindlin Plates
- Author
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Thomas Führer, Norbert Heuer, and Antti H. Niemi
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
30. Value-Gradient Based Formulation of Optimal Control Problem and Machine Learning Algorithm
- Author
-
Alain Bensoussan, Jiayue Han, Sheung Chi Phillip Yam, and Xiang Zhou
- Subjects
Numerical Analysis ,Computational Mathematics ,Optimization and Control (math.OC) ,Applied Mathematics ,FOS: Mathematics ,Mathematics - Optimization and Control - Abstract
Optimal control problem is typically solved by first finding the value function through Hamilton-Jacobi equation (HJE) and then taking the minimizer of the Hamiltonian to obtain the control. In this work, instead of focusing on the value function, we propose a new formulation for the gradient of the value function (value-gradient) as a decoupled system of partial differential equations in the context of continuous-time deterministic discounted optimal control problem. We develop an efficient iterative scheme for this system of equations in parallel by utilizing the properties that they share the same characteristic curves as the HJE for the value function. For the theoretical part, we prove that this iterative scheme converges linearly in $L_\alpha^2$ sense for some suitable exponent $\alpha$ in a weight function. For the numerical method, we combine characteristic line method with machine learning techniques. Specifically, we generate multiple characteristic curves at each policy iteration from an ensemble of initial states, and compute both the value function and its gradient simultaneously on each curve as the labelled data. Then supervised machine learning is applied to minimize the weighted squared loss for both the value function and its gradients. Experimental results demonstrate that this new method not only significantly increases the accuracy but also improves the efficiency and robustness of the numerical estimates, particularly with less amount of characteristics data or fewer training steps.
- Published
- 2023
31. Optimal Analysis of Non-Uniform Galerkin-Mixed Finite Element Approximations to the Ginzburg–Landau Equations in Superconductivity
- Author
-
Huadong Gao and Weiwei Sun
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
32. Optimization and Convergence of Numerical Attractors for Discrete-Time Quasi-Linear Lattice System
- Author
-
Yangrong Li, Shuang Yang, Tomás Caraballo, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, and Universidad de Sevilla. FQM314: Análisis Estocástico de Sistemas Diferenciales
- Subjects
semi-continuity of attractors ,Numerical Analysis ,Computational Mathematics ,discrete-time equation ,Applied Mathematics ,numerical attractor ,p-Laplace lattice ,finite-dimensional 16 approximation - Abstract
Existence and connection of numerical attractors for discrete-time p -Laplace lattice systems via the implicit Euler scheme are proved. The numerical attractors are shown to have an optimized bound, which leads to the continuous convergence of the numerical attractors when the graph of the nonlinearity closes to the vertical axis or when the external force vanishes. A new type of Taylor expansion without Fréchet derivatives is established and applied to show the discretization error of order two, which is crucial to prove that the numerical attractors converge upper semicontinuously to the global attractor of the original continuous-time system as the step size of the time goes to zero. It is also proved that the truncated numerical attractors for finitely dimensional systems converge upper semicontinuously to the numerical attractor and the lower semicontinuity holds in special cases.
- Published
- 2023
33. Efficient Stability-Preserving Numerical Methods for Nonlinear Coercive Problems in Vector Space
- Author
-
Wansheng Wang and Shoufu Li
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
34. Analysis of a Filtered Time-Stepping Finite Element Method for Natural Convection Problems
- Author
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Jilian Wu, Ning Li, and Xinlong Feng
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
35. An Arbitrary Order and Pointwise Divergence-Free Finite Element Scheme for the Incompressible 3D Navier–Stokes Equations
- Author
-
Marien-Lorenzo Hanot, Institut Montpelliérain Alexander Grothendieck (IMAG), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Numerical Analysis ,Applied Mathematics ,Numerical Analysis (math.NA) ,Finite Element ,exterior calculus ,incompressible Navier-Stokes ,Computational Mathematics ,Mathematics - Analysis of PDEs ,de Rham complex ,35Q30 (Primary) 65N30, 76D07, 76M10 (Secondary) ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Mathematics - Numerical Analysis ,Hodge decomposition ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Analysis of PDEs (math.AP) - Abstract
In this paper we discretize the incompressible Navier-Stokes equations in the framework of finite element exterior calculus. We make use of the Lamb identity to rewrite the equations into a vorticity-velocity-pressure form which fits into the de Rham complex of minimal regularity. We propose a discretization on a large class of finite elements, including arbitrary order polynomial spaces readily available in many libraries. The main advantage of this discretization is that the divergence of the fluid velocity is pointwise zero at the discrete level. This exactness ensures pressure robustness. We focus the analysis on a class of linearized equations for which we prove well-posedness and provide a priori error estimates. The results are validated with numerical simulations., 26 pages, 6 figures, added detailed proofs in appendices
- Published
- 2023
36. Summation-by-Parts Operators for General Function Spaces
- Author
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Jan Glaubitz, Jan Nordström, and Philipp Öffner
- Subjects
mimetic discretization ,Matematik ,Numerical Analysis ,Applied Mathematics ,radial basis functions ,Numerical Analysis (math.NA) ,trigonometric functions ,65M12, 65M60, 65M70, 65D25, 65T40, 65D12 ,Computational Mathematics ,FOS: Mathematics ,Mathematics - Numerical Analysis ,general function spaces ,exponential functions ,summation-by-parts operators ,Mathematics - Abstract
Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions., 22 pages, 6 figures
- Published
- 2023
37. Fluid-Fluid Interaction Problems at High Reynolds Numbers: Reducing the Modeling Error with LES-C
- Author
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Mustafa Aggul, Alexander E. Labovsky, Eda Onal, and Kyle J. Schwiebert
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
38. PROJECTION METHODS FOR NEURAL FIELD EQUATIONS
- Author
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Daniele Avitabile, Department of Mathematics [Amsterdam], Vrije Universiteit Amsterdam [Amsterdam] (VU), Mathématiques pour les Neurosciences (MATHNEURO), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Mathematics, Amsterdam Neuroscience - Cellular & Molecular Mechanisms, and Amsterdam Neuroscience - Systems & Network Neuroscience
- Subjects
Numerical Analysis ,Quantitative Biology::Neurons and Cognition ,mathematical neuroscience ,[SDV.NEU.NB]Life Sciences [q-bio]/Neurons and Cognition [q-bio.NC]/Neurobiology ,Applied Mathematics ,Numerical Analysis (math.NA) ,Dynamical Systems (math.DS) ,integro-differential equations ,Computational Mathematics ,FOS: Mathematics ,projection methods ,Mathematics - Numerical Analysis ,Mathematics - Dynamical Systems ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
International audience; Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields, and derive a-priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neuroscience applications. We find that the convergence rate of a projection scheme for a neural field is determined to a great extent by the convergence rate of the projection operator. This abstract analysis, which unifies the treatment of collocation and Galerkin schemes, is carried out in operator form, without resorting to quadrature rules for the integral term, which are introduced only at a later stage, and whose choice is enslaved by the choice of the projector. Using an elementary timestepper as an example, we demonstrate that the error in a time stepper has two separate contributions: one from the projector, and one from the time discretisation. We give examples of concrete projection methods: two collocation schemes (piecewise-linear and spectral collocation) and two Galerkin schemes (finite elements and spectral Galerkin); for each of them we derive error bounds from the general theory, introduce several discrete variants, provide implementation details, and present reproducible convergence tests.
- Published
- 2023
39. UNSTABILIZED HYBRID HIGH-ORDER METHOD FOR A CLASS OF DEGENERATE CONVEX MINIMIZATION PROBLEMS.
- Author
-
CARSTENSEN, CARSTEN and TRAN, TIEN
- Subjects
- *
CALCULUS of variations , *NUMERICAL analysis , *ENERGY density , *BINDING energy , *TOPOLOGY - Abstract
The relaxation in the calculus of variation motivates the numerical analysis of a class of degenerate convex minimization problems with nonstrictly convex energy densities with some convexity control and two-sided p-growth. The minimizers may be nonunique in the primal variable but lead to a unique stress σ ∈ H (div, Ω; 필). Examples include the p-Laplacian, an optimal design problem in topology optimization, and the convexified double-well problem. The approximation by hybrid high-order methods (HHO) utilizes a reconstruction of the gradients with piecewise Raviart-Thomas or Brezzi-Douglas-Marini finite elements without stabilization on a regular triangulation into simplices. The application of this HHO method to the class of degenerate convex minimization problems allows for a unique H(div) conforming stress approximation σh. The main results are a priori and a posteriori error estimates for the stress error σ-σh in Lebesgue norms and a computable lower energy bound. Numerical benchmarks display higher convergence rates for higher polynomial degrees and include adaptive mesh-refining with the first superlinear convergence rates of guaranteed lower energy bounds. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
40. AMPLITUDE-BASED GENERALIZED PLANE WAVES: NEW QUASI-TREFFTZ FUNCTIONS FOR SCALAR EQUATIONS IN TWO DIMENSIONS.
- Author
-
IMBERT-GERARD, LISE-MARIE
- Subjects
- *
PLANE wavefronts , *HELMHOLTZ equation , *WAVENUMBER , *NUMERICAL analysis , *EQUATIONS - Abstract
Generalized plane waves (GPWs) were introduced to take advantage of Trefftz methods for problems modeled by variable coefficient equations. Despite the fact that GPWs do not satisfy the Trefftz property, i.e., they are not exact solutions to the governing equation, they instead satisfy a quasi-Trefftz property: They are only approximate solutions. They lead to high-order numerical methods, and this quasi-Trefftz property is critical for their numerical analysis. The present work introduces a new family of GPWs: amplitude-based. The motivation lies in the poor behavior of the phase-based GPW approximations in the preasymptotic regime, which will be tamed by avoiding high-degree polynomials within an exponential. The new ansatz introduces higher-order terms in the amplitude rather than in the phase of a plane wave as was initially proposed. The new functions' construction and the study of their approximation properties are guided by the road map proposed in [L.-M. Imbert-Gérard and G. Sylvand, Numer. Math., to appear]. For the sake of clarity, the first focus is on the two-dimensional Helmholtz equation with spatially varying wave number. The extension to a range of operators allowing for anisotropy in the first- and second-order terms follows. Numerical simulations illustrate the theoretical study of the new quasi-Trefftz functions. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
41. STABILITY-ENHANCED AP IMEX1-LDG METHOD: ENERGY-BASED STABILITY AND RIGOROUS AP PROPERTY.
- Author
-
ZHICHAO PENG, YINGDA CHENG, JING-MEI QIU, and FENGYAN LI
- Subjects
- *
NUMERICAL analysis , *TRANSPORT equation , *KNUDSEN flow , *MECHANICAL properties of condensed matter , *GEOMETRY - Abstract
In our recent work [Z. Peng et al., J. Comput. Phys., 415 (2020), 109485], a family of high-order asymptotic preserving (AP) methods, termed IMEX-LDG methods, are designed to solve some linear kinetic transport equations, including the one-group transport equation in slab geometry and the telegraph equation, in a diffusive scaling. As the Knudsen number ε goes to zero, the limiting schemes are implicit discretizations to the limiting diffusive equation. Both Fourier analysis and numerical experiments imply the methods are unconditionally stable in the diffusive regime when ε « 1. In this paper, we develop an energy approach to establish the numerical stability of the IMEX1-LDG method, the subfamily of the methods that is first-order accurate in time and arbitrary order in space, for the model with general material properties. Our analysis is the first to simultaneously confirm unconditional stability when ε « I and the uniform stability property with respect to e. To capture the unconditional stability, we propose a novel discrete energy and explore various stabilization mechanisms of the method and their relative contributions in different regimes. A general form of the weight function, introduced to obtain the unconditional stability for ε « 1, is also for the first time considered in such stability analysis. Based on uniform stability, a rigorous asymptotic analysis is then carried out to show the AP property. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
42. A PRIORI ERROR ANALYSIS OF A NUMERICAL STOCHASTIC HOMOGENIZATION METHOD.
- Author
-
FISCHER, JULIAN, GALLISTL, DIETMAR, and PETERSEIM, DANIEL
- Subjects
- *
STOCHASTIC analysis , *NUMERICAL analysis , *ORTHOGONAL decompositions , *RANDOM fields , *DECOMPOSITION method , *ERROR analysis in mathematics , *SPECTRAL element method - Abstract
This paper provides an a priori error analysis of a localized orthogonal decomposition method for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in the form of the spectral gap inequality, then the expected L² error of the method can be estimated, up to logarithmic factors, by H + (ε/H)d/2, ε being the small correlation length of the random coefficient and H the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
43. A Two-Level Preconditioned Helmholtz Subspace Iterative Method for Maxwell Eigenvalue Problems
- Author
-
Qigang Liang and Xuejun Xu
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
44. Nonlinear Eigenvalue Methods for Linear Pointwise Stability of Nonlinear Waves
- Author
-
Arnd Scheel
- Subjects
Numerical Analysis ,Computational Mathematics ,Mathematics - Analysis of PDEs ,Applied Mathematics ,FOS: Mathematics ,FOS: Physical sciences ,Mathematics - Numerical Analysis ,Numerical Analysis (math.NA) ,Pattern Formation and Solitons (nlin.PS) ,Nonlinear Sciences - Pattern Formation and Solitons ,Analysis of PDEs (math.AP) - Abstract
We propose an iterative method to find pointwise growth exponential growth rates in linear problems posed on essentially one-dimensional domains. Such pointwise growth rates capture pointwise stability and instability in extended systems and arise as spectral values of a family of matrices that depends analytically on a spectral parameter, obtained via a scattering-type problem. Different from methods in the literature that rely on computing determinants of this nonlinear matrix pencil, we propose and analyze an inverse power method that allows one to locate robustly the closest spectral value to a given reference point in the complex plane. The method finds branch points, eigenvalues, and resonance poles without a priori knowledge., Comment: 22 pages, 3 figures
- Published
- 2023
45. Convergence of Regularized Particle Filters for Stochastic Reaction Networks
- Author
-
Zhou Fang, Ankit Gupta, and Mustafa Khammash
- Subjects
Numerical Analysis ,Computational Mathematics ,FOS: Biological sciences ,Applied Mathematics ,regularized particle filters ,stochastic reaction networks ,multiscale systems ,filtering theory ,Quantitative Biology - Quantitative Methods ,Quantitative Methods (q-bio.QM) ,60J22, 62M20, 65C05, 92-08, 93E11 - Abstract
Filtering for stochastic reaction networks (SRNs) is an important problem in systems/synthetic biology aiming to estimate the state of unobserved chemical species. A good solution to it can provide scientists valuable information about the hidden dynamic state and enable optimal feedback control. Usually, the model parameters need to be inferred simultaneously with state variables, and a conventional particle filter can fail to solve this problem accurately due to sample degeneracy. In this case, the regularized particle filter (RPF) is preferred to the conventional ones, as the RPF can mitigate sample degeneracy by perturbing particles with artificial noise. However, the artificial noise introduces an additional bias to the estimate, and, thus, it is questionable whether the RPF can provide reliable results for SRNs. In this paper, we aim to identify conditions under which the RPF converges to the exact filter in the filtering problem determined by a bimolecular network. First, we establish computationally efficient RPFs for SRNs on different scales using different dynamical models, including the continuous-time Markov process, tau-leaping model, and piecewise deterministic process. Then, by parameter sensitivity analyses, we show that the established RPFs converge to the exact filters if all reactions leading to an increase of the molecular population have linearly growing propensities and some other mild conditions are satisfied simultaneously. This ensures the performance of the RPF for a large class of SRNs, and several numerical examples are presented to illustrate our results., Comment: 28 pages, 6 figures
- Published
- 2023
46. Preintegration via Active Subspace
- Author
-
Sifan Liu and Art B. Owen
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
47. Agglomeration-Based Geometric Multigrid Schemes for the Virtual Element Method
- Author
-
Paola F. Antonietti, Stefano Berrone, Martina Busetto, and Marco Verani
- Subjects
Numerical Analysis ,Computational Mathematics ,agglomeration ,virtual element method ,Applied Mathematics ,FOS: Mathematics ,geometric multigrid algorithms, agglomeration, virtual element method, elliptic problems, polygonal meshes ,Mathematics - Numerical Analysis ,Numerical Analysis (math.NA) ,polygonal meshes ,elliptic problems ,geometric multigrid algorithms - Abstract
In this paper we analyse the convergence properties of two-level, W-cycle and V-cycle agglomeration-based geometric multigrid schemes for the numerical solution of the linear system of equations stemming from the lowest order $C^0$-conforming Virtual Element discretization of two-dimensional second-order elliptic partial differential equations. The sequence of agglomerated tessellations are nested, but the corresponding multilevel virtual discrete spaces are generally non-nested thus resulting into non-nested multigrid algorithms. We prove the uniform convergence of the two-level method with respect to the mesh size and the uniform convergence of the W-cycle and the V-cycle multigrid algorithms with respect to the mesh size and the number of levels. Numerical experiments confirm the theoretical findings.
- Published
- 2023
48. Rank-Adaptive Time Integration of Tree Tensor Networks
- Author
-
Ceruti, Gianluca, Lubich, Christian, and Sulz, Dominik
- Subjects
tensor differential equation ,Numerical Analysis ,Computational Mathematics ,tree tensor network ,rank adaptivity ,Applied Mathematics ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,MathematicsofComputing_NUMERICALANALYSIS ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Numerical Analysis (math.NA) ,dynamical low-rank approximation - Abstract
A rank-adaptive integrator for the approximate solution of high-order tensor differential equations by tree tensor networks is proposed and analyzed. In a recursion from the leaves to the root, the integrator updates bases and then evolves connection tensors by a Galerkin method in the augmented subspace spanned by the new and old bases. This is followed by rank truncation within a specified error tolerance. The memory requirements are linear in the order of the tensor and linear in the maximal mode dimension. The integrator is robust to small singular values of matricizations of the connection tensors. Up to the rank truncation error, which is controlled by the given error tolerance, the integrator preserves norm and energy for Schro"\dinger equations, and it dissipates the energy in gradient systems. Numerical experiments with a basic quantum spin system illustrate the behavior of the proposed algorithm.
- Published
- 2023
49. Convergence Analysis of Newton–Schur Method for Symmetric Elliptic Eigenvalue Problem
- Author
-
Nian Shao and Wenbin Chen
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics - Published
- 2023
50. Analysis of Preintegration Followed by Quasi–Monte Carlo Integration for Distribution Functions and Densities
- Author
-
Gilbert, Alexander D., Kuo, Frances Y., and Sloan, Ian H.
- Subjects
Numerical Analysis ,Computational Mathematics ,Applied Mathematics ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Numerical Analysis (math.NA) - Abstract
In this paper, we analyse a method for approximating the distribution function and density of a random variable that depends in a non-trivial way on a possibly high number of independent random variables, each with support on the whole real line. Starting with the integral formulations of the distribution and density, the method involves smoothing the original integrand by preintegration with respect to one suitably chosen variable, and then applying a suitable quasi-Monte Carlo (QMC) method to compute the integral of the resulting smoother function. Interpolation is then used to reconstruct the distribution or density on an interval. The preintegration technique is a special case of conditional sampling, a method that has previously been applied to a wide range of problems in statistics and computational finance. In particular, the pointwise approximation studied in this work is a specific case of the conditional density estimator previously considered in L'Ecuyer et al., arXiv:1906.04607. Our theory provides a rigorous regularity analysis of the preintegrated function, which is then used to show that the errors of the pointwise and interpolated estimators can both achieve nearly first-order convergence. Numerical results support the theory.
- Published
- 2023
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