scholarly journals Improving Levenberg-Marquardt Algorithm Inversion Result Using Singular Value Decomposition

2016 ◽  
Vol 5 (2) ◽  
pp. 20
Author(s):  
Widodo Widodo ◽  
Durra Handri Saputera
Keyword(s):  
Singular Value Decomposition ◽  
Singular Value ◽  
Parameter Analysis ◽  
Model Parameters ◽  
Inversion Result ◽  
One Dimensional ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt ◽  
Value Decomposition ◽  
Lm Algorithm

Inversion is a process to determine model parameters from data. In geophysics this process is very important because subsurface image is obtained from this process. There are many inversion algorithms that have been introduced and applied in geophysics problems; one of them is Levenberg-Marquardt (LM) algorithm. In this paper we will present one of LM algorithm application in one-dimensional magnetotelluric (MT) case. The LM algorithm used in this study is improved version of LM algorithm using singular value decomposition (SVD). The result from this algorithm is then compared with the algorithm without SVD in order to understand how much it has been improved. To simplify the comparison, simple synthetic model is used in this study. From this study, the new algorithm can improve the result of the original LM algorithm. In addition, SVD is allowing more parameter analysis to be done in its process. The algorithm created from this study is then used in our modeling program, called MAT1DMT.

Singular value decomposition for cross‐well tomography

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1655-1661 ◽  
Author(s):  
Reinaldo J. Michelena
Keyword(s):  
Singular Value Decomposition ◽  
Null Space ◽  
Vertical Component ◽  
Singular Values ◽  
Singular Value ◽  
Velocity Estimation ◽  
One Dimensional ◽  
Anisotropic Models ◽  
Value Decomposition ◽  
Frequency Variations

I perform singular value decomposition (SVD) on the matrices that result in tomographic velocity estimation from cross‐well traveltimes in isotropic and anisotropic media. The slowness model is parameterized in four ways: One‐dimensional (1-D) isotropic, 1-D anisotropic, two‐dimensional (2-D) isotropic, and 2-D anisotropic. The singular value distribution is different for the different parameterizations. One‐dimensional isotropic models can be resolved well but the resolution of the data is poor. One‐dimensional anisotropic models can also be resolved well except for some variations in the vertical component of the slowness that are not sensitive to the data. In 2-D isotropic models, “pure” lateral variations are not sensitive to the data, and when anisotropy is introduced, the result is that the horizontal and vertical component of the slowness cannot be estimated with the same spatial resolution because the null space is mostly related to horizontal and high frequency variations in the vertical component of the slowness. Since the distribution of singular values varies depending on the parametrization used, the effect of conventional regularization procedures in the final solution may also vary. When the model is isotropic, regularization translates into smoothness, and when the model is anisotropic regularization not only smooths but may also alter the anisotropy in the solution.

Resolution and significant contributions of tidal forcing in flexible harmonic grouping computed using Singular Value Decomposition

2020 ◽  
Author(s):  
Adam Ciesielski ◽  
Thomas Forbriger
Keyword(s):  
Time Series ◽  
Singular Value Decomposition ◽  
Cross Talk ◽  
Singular Value ◽  
Model Parameters ◽  
Tidal Analysis ◽  
Tidal Forcing ◽  
Tidal Harmonics ◽  
Tidal Signal ◽  
Value Decomposition

<p>We present the results of our studies of singular value decomposition (SVD) of the forward operator in tidal analysis. Using the resolution matrix and the ratio between singular values, we distinguish significant contributions that compose the tidal signal and we study cross-talk within and between tidal groups. Using all harmonics from the tidal catalogue we investigate the resolution matrix properties with decreasing amplitude of harmonics. We demonstrate the loss of resolution even for harmonics of large amplitude with decreasing time-series length. Our further investigation shows the cross-talk from atmospherically induced gravity variation into a tidal signal (expected and unexpected, e.g. S1, Fi1, Sig1). We investigate the ability to determine the ratio of gravimetric factors of degree 2 and degree 3 tides from the specific tidal gravity signal recordings.</p><p><span>The main interest of tidal analysis is the accurate and precise determination of tidal parameters, which are amplitude (gravimetric) factor and phase lag, the quantities describing the Earth response to the tidal forcing. Tidal catalogues </span><span>define the tide generating potential in terms </span><span>of harmonics. Widely used software, like ETERNA or Baytap-G, uses a-priori grouping of harmonics which is based on reasonable considerations like the Rayleigh criterion of spectral resolution. Wave grouping is a model parameteri</span><span>s</span><span>ation used to make the analysis problem overdetermined by using assumptions regarding the model parameters (e.g. credo of smoothness, known free-core resonance parameters, known ratio between response to degree 2 and degree 3 forcing). </span><span>If</span><span> those assumptions are incorrect, this can lead to artefacts which might go unnoticed. This presents a limitation for example in the search for causes of temporal variation of tidal parameters, as reported recently. SVD of the unparameterised problem allows us to investigate these limitations.</span></p><p><span>In our analysis, SVD is a factorisation of a linear regression matrix. The regression matrix consists of tidal harmonics in-phase and quadrature signal for rigid Earth tide (tidal forcing to Earth surface). We compute time series for each harmonic present in Tamura tidal catalogue </span><span>by </span><span>using a modified version of "Predict" (ETERNA package). Resulting values can be, but do not need </span><span>to</span><span> be, grouped prior to SVD analysis. Other than with conventional programs, wave groups can not only be defined along the frequency axis. They can as well be used to separate harmonics of degree 2 and degree 3. SVD allows us to study the significance of tidal harmonics, cross-talk between harmonics or groups and matrix null space. Thus, we can discriminate the parameters with small singular value, which do not significantly contribute to the predicted tidal data or are noise-sensitive.</span></p>

DESIGN OF 2-D DIGITAL FILTERS BASED ON THE OPTIMAL DECOMPOSITION OF MAGNITUDE SPECIFICATIONS

1993 ◽  
Vol 03 (03) ◽  
pp. 733-756 ◽  
Author(s):  
T.-B. DENG ◽  
M. KAWAMATA ◽  
T. HIGUCHI
Keyword(s):  
Singular Value Decomposition ◽  
Digital Filters ◽  
Matrix Decomposition ◽  
Decomposition Methods ◽  
Singular Value ◽  
The Other ◽  
Design Technique ◽  
One Dimensional ◽  
Rms Error ◽  
Value Decomposition

The optimal decomposition (OD) technique for decomposing 2-D magnitude specifications into 1-D ones is proposed. Differing from the conventional matrix decomposition methods such as the singular value decomposition (SVD), the OD of a 2-D magnitude specification matrix results in 1-D magnitude specifications which are always non-negative while the root mean-squared (rms) error in decomposition is minimum. Therefore, the OD is more suitable for designing 2-D digital filters (2DDFs) through designing one-dimensional digital filters (1DDFs) than the conventional matrix decomposition methods. Based on the OD, the problem of designing a recursive 2DDF can be reduced to one of designing a pair of 1DDFs, one 1-input/multi-output, and the other multi-input/ 1-output. Consequently, 2DDFs can easily be obtained by designing 1DDFs using the well-established 1-D design techniques. Three design examples are presented to illustrate that this OD-based 2DDF design technique is extremely efficient.

Structural Physical Parameter Identification of Nonlinear Parametric System in Time Domain

2011 ◽  
Vol 94-96 ◽  
pp. 1040-1043
Author(s):  
Xiang Jian Wang ◽  
Jie Cui
Keyword(s):  
Singular Value Decomposition ◽  
Parameter Identification ◽  
Physical Parameter ◽  
High Precision ◽  
Time Domain ◽  
Singular Value ◽  
Parametric System ◽  
Levenberg Marquardt ◽  
Shear Type ◽  
Value Decomposition

The modified Levenberg-Marquardt (mLM) method is introduced for nonlinear parametric system, such as stiffness proportional damping and Rayleigh proportional damping. Since the mLM method is sensitive to the initial values of parameter, a SVD-mLM method is proposed with combination of singular value decomposition (SVD). Parameter identification of five-storey shear-type is simulated with incomplete output. The results show that the identified parameters have high precision, and the proposed method is effective and robust on noise.

Barrage Text Classification with Improved Active Learning and CNN

2019 ◽  
Vol 23 (6) ◽  
pp. 980-989
Author(s):  
Ningjia Qiu ◽  
◽  
Lin Cong ◽  
Sicheng Zhou ◽  
Peng Wang
Keyword(s):  
Active Learning ◽  
Singular Value Decomposition ◽  
Dimensionality Reduction ◽  
Text Classification ◽  
Singular Value ◽  
Convergence Speed ◽  
Decomposition Algorithm ◽  
Model Parameters ◽  
Value Decomposition ◽  
Improved Algorithm

Traditional convolutional neural networks (CNNs) use a pooling layer to reduce the dimensionality of texts, but lose semantic information. To solve this problem, this paper proposes a convolutional neural network model based on singular value decomposition algorithm (SVD-CNN). First, an improved density-based center point clustering active learning sampling algorithm (DBC-AL) is used to obtain a high-quality training set at a low labelling cost. Second, the method uses the singular value decomposition algorithm for feature extraction and dimensionality reduction instead of a pooling layer, fuses the dimensionality reduction matrix, and completes the barrage text classification task. Finally, the partial sampling gradient descent algorithm (PSGD) is applied to optimize the model parameters, which accelerates the convergence speed of the model while ensuring stability of the model training. To verify the effectiveness of the improved algorithm, several barrage datasets were used to compare the proposed model and common text classification models. The experimental results show that the improved algorithm preserves the semantic features of the text more successfully, ensures the stability of the training process, and improves the convergence speed of the model. Further, the model’s classification performance on different barrage texts is superior to traditional algorithms.

GENERALIZED SINGULAR VALUE DECOMPOSITION AND ITS APPLICATIONS IN MODEL ANALYSIS

2006 ◽  
Vol 09 (02) ◽  
pp. 171-184
Author(s):  
EUGENE V. DULOV ◽  
HUMBERTO SARRIA ZAPATA ◽  
NATALIA A. ANDRIANOVA
Keyword(s):  
Singular Value Decomposition ◽  
Formal Model ◽  
Model Analysis ◽  
Singular Value ◽  
Problem Solver ◽  
Model Parameters ◽  
Stochastic Dynamic ◽  
Generalized Singular Value Decomposition ◽  
Value Decomposition ◽  
Stochastic Dynamic Systems

For a variety of processes we can observe and register their characteristics, making up a sequence of measurement vectors or matrices (rectangular in general). Our goal is to extract some model dependent information using the available information. Such approaches are typical in technology (for a neat chemistry example, see [7,9]) and model analysis like parameter identification of linear stochastic dynamic systems. Since a stochastic nature of financial and economic data is evident, we can extend this data analysis technique to a number of new applications. If we are successful, some kind of adaptive filter can be further constructed (similar to the classic Kalman's one, for example). Inspired with formal model parameters, we can apply this filter to process financial data like stock information to predict and verify how close is a mathematical model to a real-time data. Namely, when provided with a set measurements represented by matrices Ai ∈ Mm,n (ℝ), we have to estimate a problem dependent characteristic matrices [Formula: see text] with P,Q being orthonormal matrices, Bi ∈ Mr (ℝ), r ≤ min {m,n}. Formulated as above, the problem is usually called a generalized singular value decomposition (GSVD) problem and could be solved numerically [1, 2]. These matrices provide some basic information applicable for higher level automated problem solver or human interpretation.

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