Papers by kishore kumar Naraparaju
arXiv (Cornell University), Jul 14, 2017
In this article we consider the iterative schemes to compute the canonical (CP) approximation of ... more In this article we consider the iterative schemes to compute the canonical (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research on the QTT method [16] developed for the tensor train (TT) approximation of the quantized images of function related data. In the QTT approach the target vector of length 2 L is reshaped to a L th order tensor with two entries in each mode (Quantized representation) and then approximated by the QTT tenor including 2r 2 L parameters, where r is the maximal TT rank. In what follows, we consider the Alternating Least-Squares (ALS) iterative scheme to compute the rank-r CP approximation of the quantized vectors, which requires only 2rL ≪ 2 L parameters for storage. In the earlier papers [17] such a representation was called Q Can format, while in this paper we abbreviate it as the QCP representation. We test the ALS algorithm to calculate the QCP approximation on various functions, and in all cases we observed the exponential error decay in the QCP rank. The main idea for recovering a discretized function in the rank-r QCP format using the reduced number the functional samples, calculated only at O(2rL) grid points, is presented. The special version of ALS scheme for solving the arising minimization problem is described. This approach can be viewed as the sparse QCP-interpolation method that allows to recover all 2rL representation parameters of the rank-r QCP tensor. Numerical examples show the efficiency of the QCP-ALS type iteration and indicate the exponential convergence rate in r.
Nonconforming spectral element method: a friendly introduction in one dimension and a short review in higher dimensions
Computational & Applied Mathematics, Apr 1, 2023
Nonconforming spectral/Hp method for elliptic interface problem
by Naraparaju Kishore Kuma
Journal of Scientific Computing, 2012
In this paper a numerical method based on least-squares approximation is proposed for elliptic in... more In this paper a numerical method based on least-squares approximation is proposed for elliptic interface problems in two dimensions, where the interface is smooth. The underlying method is spectral element method. In the least-squares formulation a functional is minimized as defined in (4.1). The jump in the solution and its normal derivative across the interface are enforced (in an appropriate Sobolev norm) in the functional. The solution is obtained by solving the normal equations using preconditioned conjugate gradient method. Essentially the method is nonconforming, so a block diagonal matrix is constructed as a preconditioner based on the stability estimate where each diagonal block is decoupled. A conforming solution is obtained by making a set of corrections to the nonconforming solution as in [24] and an error estimate in H 1-norm is given which shows the exponential convergence of the proposed method.
This paper deals with the approximation of d-dimensional tensors, as discrete representations of ... more This paper deals with the approximation of d-dimensional tensors, as discrete representations of arbitrary functions f (x 1 ,. .. , x d) on [0, 1] d , in the so-called Tensor Chain format. The main goal of this paper is to show that the construction of a Tensor Chain approximation is possible using Skeleton/Cross Approximation type methods. The complete algorithm is described, computational issues are discussed in detail and the complexity of the algorithm is shown to be linear in d. Some numerical examples are given to validate the theoretical results.
In this article we consider the iterative schemes to compute the canonical (CP) approximation of ... more In this article we consider the iterative schemes to compute the canonical (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research on the QTT method [16] developed for the tensor train (TT) approximation of the quantized images of function related data. In the QTT approach the target vector of length $2^{L}$ is reshaped to a $L^{th}$ order tensor with two entries in each mode (Quantized representation) and then approximated by the QTT tenor including $2r^2 L$ parameters, where $r$ is the maximal TT rank. In what follows, we consider the Alternating Least-Squares (ALS) iterative scheme to compute the rank-$r$ CP approximation of the quantized vectors, which requires only $2 r L\ll 2^L$ parameters for storage. In the earlier papers [17] such a representation was called Q$_{Can}$ format, while in this paper we abbreviate it as the QCP representation. We test the ALS algorithm to ...
fMathematik in den Naturwissenschaften Leipzig
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This paper deals with the approximation of d-dimensional tensors, as discrete representations of ... more This paper deals with the approximation of d-dimensional tensors, as discrete representations of arbitrary functions f (x 1 ,. .. , x d) on [0, 1] d , in the so-called Tensor Chain format. The main goal of this paper is to show that the construction of a Tensor Chain approximation is possible using Skeleton/Cross Approximation type methods. The complete algorithm is described, computational issues are discussed in detail and the complexity of the algorithm is shown to be linear in d. Some numerical examples are given to validate the theoretical results.
The solution of elliptic boundary value problems often leads to singularities due to nonsmoothnes... more The solution of elliptic boundary value problems often leads to singularities due to nonsmoothness of the domains on which the problem is posed. This paper studies the performance of the nonconforming hp/spectral element method for elliptic problems on non smooth domains. This paper deals with monotone singularities of type r α and r α log δ r as well as the oscillating singularities of type r α sin(ε log r).
Computing and Visualization in Science, 2011
In this article we present a generalized version of the Cross Approximation for 3d-tensors. The g... more In this article we present a generalized version of the Cross Approximation for 3d-tensors. The given tensor a ∈ R n×n×n is represented as a matrix of vectors and 2d adaptive Cross Approximation is applied in a nested way to get the tensor decomposition. The main focus lies on theoretical issues of the construction such as the desired interpolation property or the explicit formulas for the vectors in the decomposition. The computational complexity of the proposed algorithm is shown to be linear in n.
A note on tensor chain approximation
Computing and Visualization in Science, 2012
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Papers by kishore kumar Naraparaju