Dalia Fishelov - Academia.edu (original) (raw)
Papers by Dalia Fishelov
Lecture notes in computational science and engineering, Nov 29, 2022
Journal of Computational and Applied Mathematics
Neural Computation
Solving the eigenvalue problem for differential operators is a common problem in many scientific ... more Solving the eigenvalue problem for differential operators is a common problem in many scientific fields. Classical numerical methods rely on intricate domain discretization and yield nonanalytic or nonsmooth approximations. We introduce a novel neural network–based solver for the eigenvalue problem of differential self-adjoint operators, where the eigenpairs are learned in an unsupervised end-to-end fashion. We propose several training procedures for solving increasingly challenging tasks toward the general eigenvalue problem. The proposed solver is capable of finding the M smallest eigenpairs for a general differential operator. We demonstrate the method on the Laplacian operator, which is of particular interest in image processing, computer vision, and shape analysis among many other applications. In addition, we solve the Legendre differential equation. Our proposed method simultaneously solves several eigenpairs and can be easily used on free-form domains. We exemplify it on L-s...
Journal of Scientific Computing, 2021
In our book "Navier-Stokes Equations in Planar Domains", Imperial College Press, 2013, we have su... more In our book "Navier-Stokes Equations in Planar Domains", Imperial College Press, 2013, we have suggested a fourth-order compact scheme for the Navier-Stokes equations in streamfunction formulation ∂ t (∆ψ) + (∇ ⊥ ψ) • ∇(∆ψ) = ν∆ 2 ψ. Here we present a new approach for the analysis of a high-order compact scheme for the Navier-Stokes equations in cases where the convective term vanishes, or in cases where the viscous term dominates the convective term. In these cases the Navier-Stokes equations is replaced by the time-dependent Stokes equation ∂ t (∆ψ) = ν∆ 2 ψ. The same type of fourth-order compact schemes that were proposed for the Navier-Stokes equations, may be adopted to the time-dependent Stokes problem. For these methods the truncation error is only of first-order at near-boundary points, but is of fourth order at interior points. We prove that the rate of convergence is actually four, thus the error tends to zero as O(h 4), where h is the size of the mesh.
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications, 2017
We present a high-order finite difference scheme for Navier-Stokes equations in irregular domains... more We present a high-order finite difference scheme for Navier-Stokes equations in irregular domains. The scheme is an extension of a fourth-order scheme for Navier-Stokes equations in streamfunction formulation on a rectangular domain (Ben-Artzi et al., 2010). The discretization offered here contains two types of interior points. The first is regular interior points, where all eight neighboring points of a grid point are inside the domain and not too close to the boundary. The second is interior points where at least one of the closest eight neighbors is outside the computational domain or too close to the boundary. In the second case we design discrete operators which approximate spatial derivatives of the streamfunction on irregular meshes, using discretizations of pure derivatives in the x, y and along the diagonals of the element.
Navier-Stokes Equations in Planar Domains, 2013
SIAM Journal on Scientific and Statistical Computing, 1990
Vortex methods for slightly viscous three-dimensional flow are presented. Vortex methods have bee... more Vortex methods for slightly viscous three-dimensional flow are presented. Vortex methods have been used extensively for two-dimensional problems, though their most efficient extension to threedimensional problems is still under investigation. A method that evaluates the vorticity by exactly differentiating an approximate velocity field is applied. Numerical results are presented for a flow past a semi-infinite plate, and they demonstrate three-dimensional features of the flow and transition to turbulence.
Lecture Notes in Computational Science and Engineering, 2020
Here we present a new approach for the analysis of high-order compact schemes for the clamped pla... more Here we present a new approach for the analysis of high-order compact schemes for the clamped plate problem. A similar model is the Navier-Stokes equation in streamfunction formulation. In our book "Navier-Stokes Equations in Planar Domains", Imperial College Press, 2013, we have suggested fourth-order compact schemes for the Navier-Stokes equations. The same type of schemes may be applied to the clamped plate problem. For these methods the truncation error is only of firstorder at near-boundary points, but is of fourth order at interior points. It is proven that the rate of convergence is actually four, thus the error tends to zero as O(h 4).
In Ben-Artzi et al. (SIAM J Numer Anal 47:3087–3108 (2009), [1]) a Cartesian embedded finite diff... more In Ben-Artzi et al. (SIAM J Numer Anal 47:3087–3108 (2009), [1]) a Cartesian embedded finite difference scheme for biharmonic problems has been introduced. The design of the scheme relies on a 19-dimensional polynomial space. In this paper, we show how to simplify the implementation by introducing a directional decomposition of this space. The boundary is handled via a level-set approach. Numerical results for non convex domains demonstrate the fourth order accuracy of the scheme.
ESAIM: Mathematical Modelling and Numerical Analysis, 2001
We present a new methodology for the numerical resolution of the hydrodynamics of incompressible ... more We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data for some test cases to which we apply the computational scheme.
2018 IEEE International Conference on the Science of Electrical Engineering in Israel (ICSEE), 2018
A common pre-possessing task in machine learning is to complete missing data entries in order to ... more A common pre-possessing task in machine learning is to complete missing data entries in order to form a full dataset. In case the dimension of the input data is high, it is often the case that the rows and columns are correlated. In this work, we construct a multi-scale model that is based on the the dual row-column geometry of the dataset and apply it to imputation. The imputation is carried out within the model construction. Experimental results demonstrate the efficiency of our approach on a publicly available dataset.
An important challenge in Data Mining and Machine Learning is the proper analysis of a given data... more An important challenge in Data Mining and Machine Learning is the proper analysis of a given dataset, especially for understanding and working with functions defined over it. In this paper we propose Auto-adaptive Laplacian Pyramids (ALP) for target function smoothing when the target function can be define on a high-dimensional dataset. The proposed algorithm automatically selects the optimal function resolution (stopping time) adapted to the data defined and its noise. We illustrate its application on a radiation forecasting example.
Applied Mathematics and Computation, 2017
Nonlinear dimensionality reduction methods often include the construction of kernels for embeddin... more Nonlinear dimensionality reduction methods often include the construction of kernels for embedding the high-dimensional data points. Standard methods for extending the embedding coordinates (such as the Nyström method) also rely on spectral decomposition of kernels. It is desirable that these kernels capture most of the data sets' information using only a few leading modes of the spectrum. In this work we propose multi-scale kernels, which are constructed as combinations of Gaussian kernels, to be used for kernel-based extension schemes. We review the kernels' spectral properties and show that their first few modes capture more information compared to the standard Gaussian kernel. Their application is demonstrated on a synthetic data-set and also applied to a real-life example that models daily electricity profiles and predicts the average day-ahead behavior.
Computational Science and Its Applications – ICCSA 2017, 2017
A challenging problem in machine learning is handling missing data, also known as imputation. Sim... more A challenging problem in machine learning is handling missing data, also known as imputation. Simple imputation techniques complete the missing data by the mean or the median values. A more sophisticated approach is to use regression to predict the missing data from the complete input columns. In case the dimension of the input data is high, dimensionality reduction methods may be applied to compactly describe the complete input. Then, a regression from the low-dimensional space to the incomplete data column can be constructed from imputation. In this work, we propose a two-step algorithm for data completion. The first step utilizes a non-linear manifold learning technique, named diffusion maps, for reducing the dimension of the data. This method faithfully embeds complex data while preserving its geometric structure. The second step is the Laplacian pyramids multi-scale method, which is applied for regression. Laplacian pyramids construct kernels of decreasing scales to capture finer modes of the data. Experimental results demonstrate the efficiency of our approach on a publicly available dataset.
Lecture notes in computational science and engineering, Nov 29, 2022
Journal of Computational and Applied Mathematics
Neural Computation
Solving the eigenvalue problem for differential operators is a common problem in many scientific ... more Solving the eigenvalue problem for differential operators is a common problem in many scientific fields. Classical numerical methods rely on intricate domain discretization and yield nonanalytic or nonsmooth approximations. We introduce a novel neural network–based solver for the eigenvalue problem of differential self-adjoint operators, where the eigenpairs are learned in an unsupervised end-to-end fashion. We propose several training procedures for solving increasingly challenging tasks toward the general eigenvalue problem. The proposed solver is capable of finding the M smallest eigenpairs for a general differential operator. We demonstrate the method on the Laplacian operator, which is of particular interest in image processing, computer vision, and shape analysis among many other applications. In addition, we solve the Legendre differential equation. Our proposed method simultaneously solves several eigenpairs and can be easily used on free-form domains. We exemplify it on L-s...
Journal of Scientific Computing, 2021
In our book "Navier-Stokes Equations in Planar Domains", Imperial College Press, 2013, we have su... more In our book "Navier-Stokes Equations in Planar Domains", Imperial College Press, 2013, we have suggested a fourth-order compact scheme for the Navier-Stokes equations in streamfunction formulation ∂ t (∆ψ) + (∇ ⊥ ψ) • ∇(∆ψ) = ν∆ 2 ψ. Here we present a new approach for the analysis of a high-order compact scheme for the Navier-Stokes equations in cases where the convective term vanishes, or in cases where the viscous term dominates the convective term. In these cases the Navier-Stokes equations is replaced by the time-dependent Stokes equation ∂ t (∆ψ) = ν∆ 2 ψ. The same type of fourth-order compact schemes that were proposed for the Navier-Stokes equations, may be adopted to the time-dependent Stokes problem. For these methods the truncation error is only of first-order at near-boundary points, but is of fourth order at interior points. We prove that the rate of convergence is actually four, thus the error tends to zero as O(h 4), where h is the size of the mesh.
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications, 2017
We present a high-order finite difference scheme for Navier-Stokes equations in irregular domains... more We present a high-order finite difference scheme for Navier-Stokes equations in irregular domains. The scheme is an extension of a fourth-order scheme for Navier-Stokes equations in streamfunction formulation on a rectangular domain (Ben-Artzi et al., 2010). The discretization offered here contains two types of interior points. The first is regular interior points, where all eight neighboring points of a grid point are inside the domain and not too close to the boundary. The second is interior points where at least one of the closest eight neighbors is outside the computational domain or too close to the boundary. In the second case we design discrete operators which approximate spatial derivatives of the streamfunction on irregular meshes, using discretizations of pure derivatives in the x, y and along the diagonals of the element.
Navier-Stokes Equations in Planar Domains, 2013
SIAM Journal on Scientific and Statistical Computing, 1990
Vortex methods for slightly viscous three-dimensional flow are presented. Vortex methods have bee... more Vortex methods for slightly viscous three-dimensional flow are presented. Vortex methods have been used extensively for two-dimensional problems, though their most efficient extension to threedimensional problems is still under investigation. A method that evaluates the vorticity by exactly differentiating an approximate velocity field is applied. Numerical results are presented for a flow past a semi-infinite plate, and they demonstrate three-dimensional features of the flow and transition to turbulence.
Lecture Notes in Computational Science and Engineering, 2020
Here we present a new approach for the analysis of high-order compact schemes for the clamped pla... more Here we present a new approach for the analysis of high-order compact schemes for the clamped plate problem. A similar model is the Navier-Stokes equation in streamfunction formulation. In our book "Navier-Stokes Equations in Planar Domains", Imperial College Press, 2013, we have suggested fourth-order compact schemes for the Navier-Stokes equations. The same type of schemes may be applied to the clamped plate problem. For these methods the truncation error is only of firstorder at near-boundary points, but is of fourth order at interior points. It is proven that the rate of convergence is actually four, thus the error tends to zero as O(h 4).
In Ben-Artzi et al. (SIAM J Numer Anal 47:3087–3108 (2009), [1]) a Cartesian embedded finite diff... more In Ben-Artzi et al. (SIAM J Numer Anal 47:3087–3108 (2009), [1]) a Cartesian embedded finite difference scheme for biharmonic problems has been introduced. The design of the scheme relies on a 19-dimensional polynomial space. In this paper, we show how to simplify the implementation by introducing a directional decomposition of this space. The boundary is handled via a level-set approach. Numerical results for non convex domains demonstrate the fourth order accuracy of the scheme.
ESAIM: Mathematical Modelling and Numerical Analysis, 2001
We present a new methodology for the numerical resolution of the hydrodynamics of incompressible ... more We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data for some test cases to which we apply the computational scheme.
2018 IEEE International Conference on the Science of Electrical Engineering in Israel (ICSEE), 2018
A common pre-possessing task in machine learning is to complete missing data entries in order to ... more A common pre-possessing task in machine learning is to complete missing data entries in order to form a full dataset. In case the dimension of the input data is high, it is often the case that the rows and columns are correlated. In this work, we construct a multi-scale model that is based on the the dual row-column geometry of the dataset and apply it to imputation. The imputation is carried out within the model construction. Experimental results demonstrate the efficiency of our approach on a publicly available dataset.
An important challenge in Data Mining and Machine Learning is the proper analysis of a given data... more An important challenge in Data Mining and Machine Learning is the proper analysis of a given dataset, especially for understanding and working with functions defined over it. In this paper we propose Auto-adaptive Laplacian Pyramids (ALP) for target function smoothing when the target function can be define on a high-dimensional dataset. The proposed algorithm automatically selects the optimal function resolution (stopping time) adapted to the data defined and its noise. We illustrate its application on a radiation forecasting example.
Applied Mathematics and Computation, 2017
Nonlinear dimensionality reduction methods often include the construction of kernels for embeddin... more Nonlinear dimensionality reduction methods often include the construction of kernels for embedding the high-dimensional data points. Standard methods for extending the embedding coordinates (such as the Nyström method) also rely on spectral decomposition of kernels. It is desirable that these kernels capture most of the data sets' information using only a few leading modes of the spectrum. In this work we propose multi-scale kernels, which are constructed as combinations of Gaussian kernels, to be used for kernel-based extension schemes. We review the kernels' spectral properties and show that their first few modes capture more information compared to the standard Gaussian kernel. Their application is demonstrated on a synthetic data-set and also applied to a real-life example that models daily electricity profiles and predicts the average day-ahead behavior.
Computational Science and Its Applications – ICCSA 2017, 2017
A challenging problem in machine learning is handling missing data, also known as imputation. Sim... more A challenging problem in machine learning is handling missing data, also known as imputation. Simple imputation techniques complete the missing data by the mean or the median values. A more sophisticated approach is to use regression to predict the missing data from the complete input columns. In case the dimension of the input data is high, dimensionality reduction methods may be applied to compactly describe the complete input. Then, a regression from the low-dimensional space to the incomplete data column can be constructed from imputation. In this work, we propose a two-step algorithm for data completion. The first step utilizes a non-linear manifold learning technique, named diffusion maps, for reducing the dimension of the data. This method faithfully embeds complex data while preserving its geometric structure. The second step is the Laplacian pyramids multi-scale method, which is applied for regression. Laplacian pyramids construct kernels of decreasing scales to capture finer modes of the data. Experimental results demonstrate the efficiency of our approach on a publicly available dataset.