اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn generic frequency decomposition. Part 1: vectorial decomposition
177
0
0.0
(
0
)
تأليف
Sossio Vergara
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The famous Fourier theorem states that, under some restrictions, any periodic function (or real world signal) can be obtained as a sum of sinusoids, and hence, a technique exists for decomposing a signal into its sinusoidal components. From this theory an entire branch of research has flourished: from the Short-Time or Windowed Fourier Transform to the Wavelets, the Frames, and lately the Generic Frequency Analysis. The aim of this paper is to take the Frequency Analysis a step further. It will be shown that keeping the same reconstruction algorithm as the Fourier Theorem but changing to a new computing method for the analysis phase allows the generalization of the Fourier Theorem to a large class of nonorthogonal bases. New methods and algorithms can be employed in function decomposition on such generic bases. It will be shown that these algorithms are a generalization of the Fourier analysis, i.e. they are reduced to the familiar Fourier tools when using orthogonal bases. The differences between this tool and the wavelets and frames theories will be discussed. Examples of analysis and reconstruction of functions using the given algorithms and nonorthogonal bases will be given. In this first part the focus will be on vectorial decomposition, while the second part will be on phased decomposition. The phased decomposition thanks to a single function basis has many interesting consequences and applications.
قيم البحث
اقرأ أيضاً
The CANDECOMP/PARAFAC (CP) decomposition is a leading method for the analysis of multiway data. The standard alternating least squares algorithm for the CP decomposition (CP-ALS) involves a series of highly overdetermined linear least squares problem
s. We extend randomized least squares methods to tensors and show the workload of CP-ALS can be drastically reduced without a sacrifice in quality. We introduce techniques for efficiently preprocessing, sampling, and computing randomized least squares on a dense tensor of arbitrary order, as well as an efficient sampling-based technique for checking the stopping condition. We also show more generally that the Khatri-Rao product (used within the CP-ALS iteration) produces conditions favorable for direct sampling. In numerical results, we see improvements in speed, reductions in memory requirements, and robustness with respect to initialization.
In this paper, we propose a computationally efficient iterative algorithm for proper orthogonal decomposition (POD) using random sampling based techniques. In this algorithm, additional rows and columns are sampled and a merging technique is used to
update the dominant POD modes in each iteration. We derive bounds for the spectral norm of the error introduced by a series of merging operations. We use an existing theorem to get an approximate measure of the quality of subspaces obtained on convergence of the iteration. Results on various datasets indicate that the POD modes and/or the subspaces are approximated with excellent accuracy with a significant runtime improvement over computing the truncated SVD. We also propose a method to compute the POD modes of large matrices that do not fit in the RAM using this iterative sampling and merging algorithms.
This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. In this work, we consider the trapezoidal family of schemes for integrating the or
dinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Method 1 (d-continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (Modified d-continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 (v-continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the ``energy method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs).
In a previous paper [1] it was discussed the viability of functional analysis using as a basis a couple of generic functions, and hence vectorial decomposition. Here we complete the paradigm exploiting one of the analysis methodologies developed ther
e, but applied to phase coordinates, so needing only one function as a basis. It will be shown that, thanks to the novel iterative analysis, any function satisfying a rather loose requisite is ontologically a basis. This in turn generalizes the polar version of the Fourier theorem to an ample class of nonorthogonal bases. The main advantage of this generalization is that it inherits some of the properties of the original Fourier theorem. As a result the new transform has a wide range of applications and some remarkable consequences. The new tool will be compared with wavelets and frames. Examples of analysis and reconstruction of functions using the developed algorithms and generic bases will be given. Some of the properties, and applications that can promptly benefit from the theory, will be discussed. The implementation of a matched filter for noise suppression will be used as an example of the potential of the theory.
The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree $n$ are classified
. By duality, corresponding results are obtained for valuations on finite-valued convex functions.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
