Дисертації з теми "Alternating direction methods of multipliers"
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Selvatici, Elena. "Variational formulation for Granular Contact Dynamics simulation via the Alternating Direction Method of Multipliers." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021.
Знайти повний текст джерелаGu, Yan. "STUDIES ON ALTERNATING DIRECTION METHOD OF MULTIPLIERS WITH ADAPTIVE PROXIMAL TERMS FOR CONVEX OPTIMIZATION PROBLEMS." Kyoto University, 2020. http://hdl.handle.net/2433/259758.
Повний текст джерелаFécamp, Vivien. "Recalage/Fusion d'images multimodales à l'aide de graphes d'ordres supérieurs." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLC005/document.
Повний текст джерелаThe main objective of this thesis is the exploration of higher order Markov Random Fields for image registration, specifically to encode the knowledge of global transformations, like rigid transformations, into the graph structure. Our main framework applies to 2D-2D or 3D-3D registration and use a hierarchical grid-based Markov Random Field model where the hidden variables are the displacements vectors of the control points of the grid.We first present the construction of a graph that allows to perform linear registration, which means here that we can perform affine registration, rigid registration, or similarity registration with the same graph while changing only one potential. Our framework is thus modular regarding the sought transformation and the metric used. Inference is performed with Dual Decomposition, which allows to handle the higher order hyperedges and which ensures the global optimum of the function is reached if we have an agreement among the slaves. A similar structure is also used to perform 2D-3D registration.Second, we fuse our former graph with another structure able to perform deformable registration. The resulting graph is more complex and another optimisation algorithm, called Alternating Direction Method of Multipliers is needed to obtain a better solution within reasonable time. It is an improvement of Dual Decomposition which speeds up the convergence. This framework is able to solve simultaneously both linear and deformable registration which allows to remove a potential bias created by the standard approach of consecutive registrations
Tang, Shuhan. "Spectral Analysis Using Multitaper Whittle Methods with a Lasso Penalty." The Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1586863604571678.
Повний текст джерелаAl, Sarray Basad. "Estimation et choix de modèle pour les séries temporelles par optimisation convexe." Besançon, 2016. http://www.theses.fr/2016BESA2084.
Повний текст джерела[…] this study presents some of machine learning and convex methodes for ARMA model selection and estimation based on the conversion between ARMA –AR models and ARMA-State Space Models. Also in this study, for a time series decomposition and time series components analysis some of convex methods are implemented and simulated. The results show the ability of convex methods of analysing and modelling a given series
Ojha, Abhi. "Coupled Natural Gas and Electric Power Systems." Thesis, Virginia Tech, 2017. http://hdl.handle.net/10919/78666.
Повний текст джерелаMaster of Science
Velay, Maxime. "Méthodes d’optimisation distribuée pour l’exploitation sécurisée des réseaux électriques interconnectés." Thesis, Université Grenoble Alpes (ComUE), 2018. http://www.theses.fr/2018GREAT063/document.
Повний текст джерелаOur societies are more dependent on electricity than ever, thus any disturbance in the power transmission and delivery has major economic and social impact. The reliability and security of power systems are then crucial to keep, for power system operators, in addition to minimizing the system operating cost. Moreover, transmission systems are interconnected to decrease the cost of operation and improve the system security. One of the main challenges for transmission system operators is therefore to coordinate with interconnected power systems, which raises scalability, interoperability and privacy issues. Hence, this thesis is concerned with how TSOs can operate their networks in a decentralized way but coordinating their operation with other neighboring TSOs to find a cost-effective scheduling that is globally secure.The main focus of this thesis is the security of power systems, this is why the evolution of the main characteristics of the blackouts that are failures in power system security, of the period 2005-2016 is studied. The approach consists in determining what the major characteristics of the incidents of the past 10 years are, to identify what should be taken into account to mitigate the risk of incidents. The evolution have been studied and compared with the characteristics of the blackouts before 2005. The study focuses on the pre-conditions that led to those blackouts and on the cascades, and especially the role of the cascade speed. Some important features are extracted and later integrated in our work.An algorithm that solve the preventive Security Constrained Optimal Power Flow (SCOPF) problem in a fully distributed manner, is thus developed. The preventive SCOPF problem consists in adding constraints that ensure that, after the loss of any major device of the system, the new steady-state reached, as a result of the primary frequency control, does not violate any constraint. The developed algorithm uses a fine-grained decomposition and is implemented under the multi-agent system paradigm based on two categories of agents: devices and buses. The agents are coordinated with the Alternating Direction method of multipliers in conjunction with a consensus problem. This decomposition provides the autonomy and privacy to the different actors of the system and the fine-grained decomposition allows to take the most of the decomposition and provides a good scalability regarding the size of the problem. This algorithm also have the advantage of being robust to any disturbance of the system, including the separation of the system into regions.Then, to account for the uncertainty of production brought by wind farms forecast error, a two-step distributed approach is developed to solve the Chance-Constrained Optimal Power Flow problem, in a fully distributed manner. The wind farms forecast errors are modeled by independent Gaussian distributions and the mismatches with the initials are assumed to be compensated by the primary frequency response of generators. The first step of this algorithm aims at determining the sensitivity factors of the system, needed to formulate the problem. The results of this first step are inputs of the second step that is the CCOPF. An extension of this formulation provides more flexibility to the problem and consists in including the possibility to curtail the wind farms. This algorithm relies on the same fine-grained decomposition where the agents are again coordinated by the ADMM and a consensus problem. In conclusion, this two-step algorithm ensures the privacy and autonomy of the different system actors and it is de facto parallel and adapted to high performance platforms
Guiducci, Martina. "Metodo delle Direzioni Alternate per la ricostruzione di immagini Poissoniane." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/19501/.
Повний текст джерелаRecupero, Giuseppe Antonio. "Un modello variazionale non convesso per il denoising di superfici." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23210/.
Повний текст джерелаScrivanti, Gabriele Luca Giovanni. "Nonsmooth Nonconvex Variational Reconstruction for Electrical Impedance Tomography." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/20700/.
Повний текст джерелаChen, Zhouye. "Reconstruction of enhanced ultrasound images from compressed measurements." Thesis, Toulouse 3, 2016. http://www.theses.fr/2016TOU30222/document.
Повний текст джерелаThe interest of compressive sampling in ultrasound imaging has been recently extensively evaluated by several research teams. Following the different application setups, it has been shown that the RF data may be reconstructed from a small number of measurements and/or using a reduced number of ultrasound pulse emissions. According to the model of compressive sampling, the resolution of reconstructed ultrasound images from compressed measurements mainly depends on three aspects: the acquisition setup, i.e. the incoherence of the sampling matrix, the image regularization, i.e. the sparsity prior, and the optimization technique. We mainly focused on the last two aspects in this thesis. Nevertheless, RF image spatial resolution, contrast and signal to noise ratio are affected by the limited bandwidth of the imaging transducer and the physical phenomenon related to Ultrasound wave propagation. To overcome these limitations, several deconvolution-based image processing techniques have been proposed to enhance the ultrasound images. In this thesis, we first propose a novel framework for Ultrasound imaging, named compressive deconvolution, to combine the compressive sampling and deconvolution. Exploiting an unified formulation of the direct acquisition model, combining random projections and 2D convolution with a spatially invariant point spread function, the benefit of this framework is the joint data volume reduction and image quality improvement. An optimization method based on the Alternating Direction Method of Multipliers is then proposed to invert the linear model, including two regularization terms expressing the sparsity of the RF images in a given basis and the generalized Gaussian statistical assumption on tissue reflectivity functions. It is improved afterwards by the method based on the Simultaneous Direction Method of Multipliers. Both algorithms are evaluated on simulated and in vivo data. With regularization techniques, a novel approach based on Alternating Minimization is finally developed to jointly estimate the tissue reflectivity function and the point spread function. A preliminary investigation is made on simulated data
Zhang, Mo. "Vers une méthode de restauration aveugle d’images hyperspectrales." Thesis, Rennes 1, 2018. http://www.theses.fr/2018REN1S132.
Повний текст джерелаWe propose in this thesis manuscript to develop a blind restoration method of single component blurred and noisy images where no prior knowledge is required. This manuscript is composed of three chapters: the first chapter focuses on state-of-art works. The optimization approaches for resolving the restoration problem are discussed first. Then, the main methods of restoration, so-called semi-blind ones because requiring a minimum of a priori knowledge are analysed. Five of these methods are selected for evaluation. The second chapter is devoted to comparing the performance of the methods selected in the previous chapter. The main objective criteria for evaluating the quality of the restored images are presented. Of these criteria, the l1 norm for the estimation error is selected. The comparative study conducted on a database of monochromatic images, artificially degraded by two blurred functions with different support size and three levels of noise, revealed the most two relevant methods. The first one is based on a single-scale alternating approach where both the PSF and the image are estimated alternatively. The second one uses a multi-scale hybrid approach, which consists first of alternatingly estimating the PSF and a latent image, then in a sequential next step, restoring the image. In the comparative study performed, the benefit goes to the latter. The performance of both these methods will be used as references to then compare the newly designed method. The third chapter deals with the developed method. We have sought to make the hybrid approach retained in the previous chapter as blind as possible while improving the quality of estimation of both the PSF and the restored image. The contributions covers a number of points. A first series concerns the redefinition of the scales that of the initialization of the latent image at each scale level, the evolution of the parameters for the selection of the relevant contours supporting the estimation of the PSF and finally the definition of a blind stop criterion. A second series of contributions concentrates on the blind estimation of the two regularization parameters involved in order to avoid having to fix them empirically. Each parameter is associated with a separate cost function either for the PSF estimation or for the estimation of a latent image. In the sequential step that follows, we refine the estimation of the support of the PSF estimated in the previous alternated step, before exploiting it in the process of restoring the image. At this level, the only a priori knowledge necessary is a higher bound of the support of the PSF. The different evaluations performed on monochromatic and hyperspectral images artificially degraded by several motion-type blurs with different support sizes, show a clear improvement in the quality of restoration obtained by the newly designed method in comparison to the best two state-of-the-art methods retained
Boussaid, Haithem. "Efficient inference and learning in graphical models for multi-organ shape segmentation." Thesis, Châtenay-Malabry, Ecole centrale de Paris, 2015. http://www.theses.fr/2015ECAP0002/document.
Повний текст джерелаThis thesis explores the use of discriminatively trained deformable contour models (DCMs) for shape-based segmentation in medical images. We make contributions in two fronts: in the learning problem, where the model is trained from a set of annotated images, and in the inference problem, whose aim is to segment an image given a model. We demonstrate the merit of our techniques in a large X-Ray image segmentation benchmark, where we obtain systematic improvements in accuracy and speedups over the current state-of-the-art. For learning, we formulate training the DCM scoring function as large-margin structured prediction and construct a training objective that aims at giving the highest score to the ground-truth contour configuration. We incorporate a loss function adapted to DCM-based structured prediction. In particular, we consider training with the Mean Contour Distance (MCD) performance measure. Using this loss function during training amounts to scoring each candidate contour according to its Mean Contour Distance to the ground truth configuration. Training DCMs using structured prediction with the standard zero-one loss already outperforms the current state-of-the-art method [Seghers et al. 2007] on the considered medical benchmark [Shiraishi et al. 2000, van Ginneken et al. 2006]. We demonstrate that training with the MCD structured loss further improves over the generic zero-one loss results by a statistically significant amount. For inference, we propose efficient solvers adapted to combinatorial problems with discretized spatial variables. Our contributions are three-fold:first, we consider inference for loopy graphical models, making no assumption about the underlying graph topology. We use an efficient decomposition-coordination algorithm to solve the resulting optimization problem: we decompose the model’s graph into a set of open, chain-structured graphs. We employ the Alternating Direction Method of Multipliers (ADMM) to fix the potential inconsistencies of the individual solutions. Even-though ADMMis an approximate inference scheme, we show empirically that our implementation delivers the exact solution for the considered examples. Second,we accelerate optimization of chain-structured graphical models by using the Hierarchical A∗ search algorithm of [Felzenszwalb & Mcallester 2007] couple dwith the pruning techniques developed in [Kokkinos 2011a]. We achieve a one order of magnitude speedup in average over the state-of-the-art technique based on Dynamic Programming (DP) coupled with Generalized DistanceTransforms (GDTs) [Felzenszwalb & Huttenlocher 2004]. Third, we incorporate the Hierarchical A∗ algorithm in the ADMM scheme to guarantee an efficient optimization of the underlying chain structured subproblems. The resulting algorithm is naturally adapted to solve the loss-augmented inference problem in structured prediction learning, and hence is used during training and inference. In Appendix A, we consider the case of 3D data and we develop an efficientmethod to find the mode of a 3D kernel density distribution. Our algorithm has guaranteed convergence to the global optimum, and scales logarithmically in the volume size by virtue of recursively subdividing the search space. We use this method to rapidly initialize 3D brain tumor segmentation where we demonstrate substantial acceleration with respect to a standard mean-shift implementation. In Appendix B, we describe in more details our extension of the Hierarchical A∗ search algorithm of [Felzenszwalb & Mcallester 2007] to inference on chain-structured graphs
Jeddi, Babak. "A coordinated energy management scheme in a residential neighborhood under given market framework." Thesis, Queensland University of Technology, 2020. https://eprints.qut.edu.au/200710/1/Babak_Jeddi_Thesis.pdf.
Повний текст джерелаWang, Fan. "Alternating direction methods for image recovery." HKBU Institutional Repository, 2012. https://repository.hkbu.edu.hk/etd_ra/1406.
Повний текст джерелаAl-Wali, Azzam Ahmad. "Explicit alternating direction methods for problems in fluid dynamics." Thesis, Loughborough University, 1994. https://dspace.lboro.ac.uk/2134/6840.
Повний текст джерелаLoomis, Christopher F. "Alternating Direction Implicit Method with Adaptive Grids for Modeling Chemotaxis in Dictyostelium discoideum." BYU ScholarsArchive, 2015. https://scholarsarchive.byu.edu/etd/5737.
Повний текст джерелаAmmanouil, Rita. "Contributions au démélange non-supervisé et non-linéaire de données hyperspectrales." Thesis, Université Côte d'Azur (ComUE), 2016. http://www.theses.fr/2016AZUR4079/document.
Повний текст джерелаSpectral unmixing has been an active field of research since the earliest days of hyperspectralremote sensing. It is concerned with the case where various materials are found inthe spatial extent of a pixel, resulting in a spectrum that is a mixture of the signatures ofthose materials. Unmixing then reduces to estimating the pure spectral signatures and theircorresponding proportions in every pixel. In the hyperspectral unmixing jargon, the puresignatures are known as the endmembers and their proportions as the abundances. Thisthesis focuses on spectral unmixing of remotely sensed hyperspectral data. In particular,it is aimed at improving the accuracy of the extraction of compositional information fromhyperspectral data. This is done through the development of new unmixing techniques intwo main contexts, namely in the unsupervised and nonlinear case. In particular, we proposea new technique for blind unmixing, we incorporate spatial information in (linear and nonlinear)unmixing, and we finally propose a new nonlinear mixing model. More precisely, first,an unsupervised unmixing approach based on collaborative sparse regularization is proposedwhere the library of endmembers candidates is built from the observations themselves. Thisapproach is then extended in order to take into account the presence of noise among theendmembers candidates. Second, within the unsupervised unmixing framework, two graphbasedregularizations are used in order to incorporate prior local and nonlocal contextualinformation. Next, within a supervised nonlinear unmixing framework, a new nonlinearmixing model based on vector-valued functions in reproducing kernel Hilbert space (RKHS)is proposed. The aforementioned model allows to consider different nonlinear functions atdifferent bands, regularize the discrepancies between these functions, and account for neighboringnonlinear contributions. Finally, the vector-valued kernel framework is used in orderto promote spatial smoothness of the nonlinear part in a kernel-based nonlinear mixingmodel. Simulations on synthetic and real data show the effectiveness of all the proposedtechniques
Shen, Sumin. "Contributions to Structured Variable Selection Towards Enhancing Model Interpretation and Computation Efficiency." Diss., Virginia Tech, 2020. http://hdl.handle.net/10919/96767.
Повний текст джерелаDoctor of Philosophy
The advances in data-collecting technologies provides great opportunities to access large sample-size data sets with high dimensionality. Variable selection is an important procedure to extract useful knowledge from such complex data. While in many real-data applications, appropriate selection of variables should facilitate the model interpretation and computation efficiency. It is thus important to incorporate domain knowledge of underlying data generation mechanism to select key variables for improving the model performance. However, general variable selection techniques often do not take the underlying data generation mechanism into considerations. This thesis proposal aims to develop statistical modeling methodologies with a focus on the structured variable selection towards better model interpretation and computation efficiency. The proposed approaches have been applied to real-world problems to demonstrate their model performance.
Rouf, Hasan. "Unconditionally stable finite difference time domain methods for frequency dependent media." Thesis, University of Manchester, 2010. https://www.research.manchester.ac.uk/portal/en/theses/unconditionally-stable-finite-difference-time-domain-methods-for-frequency-dependent-media(50e4adf1-d1e4-4ad2-ab2d-70188fb8b7b6).html.
Повний текст джерелаBagli, Maria Chiara. "ll metodo ADMM per la regolarizzazione con Variazione Totale Generalizzata." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/20966/.
Повний текст джерелаKraakmo, Kristina. "Numerical Simulations for the Flow of Rocket Exhaust Through a Granular Medium." Master's thesis, University of Central Florida, 2013. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/5968.
Повний текст джерелаM.S.
Masters
Mathematics
Sciences
Mathematical Science
Hossain, Mohammad Sahadet. "Numerical Methods for Model Reduction of Time-Varying Descriptor Systems." Doctoral thesis, Universitätsbibliothek Chemnitz, 2011. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-74776.
Повний текст джерелаNuswantara, Wolfgang Xaverius Dorojatun Jalma, and Jalma. "Dictionary Learning by Alternating Direction Method of Multipliers." Thesis, 2015. http://ndltd.ncl.edu.tw/handle/45259232160673563018.
Повний текст джерела國立臺灣科技大學
電子工程系
103
This thesis investigates the application of the alternating direction method of multipliers (ADMM) to dictionary learning for noise removal problem. Dictionary learning (DL) is the process of acquiring dictionary that can yield sparse representation of desired signal by learning from training sig-nal, instead of using prespeci ed transformation basis. The prior arts of the dictionary learning algorithms include the method of optimal direction (MOD) and K-SVD. These methods have main drawback in computational complexity.By contrast, the ADMM, which we use to transform the complex problem, into simple update steps, yields lower computational complexity. We further proposed on inexact ADMM that can reduce the computational time, for scenarios with large dictionary size. Simulation results show that the proposed methods can successfully train the dictionary and yields promising performance for the image noise removal problem. In particular, the pro-posed ADMM method can be around 30 times faster than K-SVD, while inexact ADMM method can further reduce the computational time around 40 %.
Wu, Chia-Wei, and 吳家維. "Decentralized Frequency-Based Load Control by Alternating Direction Method of Multipliers." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/08787826934963527482.
Повний текст джерела國立臺灣科技大學
電子工程系
104
In this paper, we consider a distributed load control problem for achieving supply-demand balance and frequency regulation in a power system. While most of the distributed load control schemes require the loads to exchange in- formation through two-way communications, recent results have shown that it is possible to achieve fully distributed control by frequency-based imbalance estimation. However, the existing methods need the load disutility functions or the sum of them to be strongly convex. For a wider range of application scenarios, we propose a new load control algorithm based on the proximal Ja- cobi alternating direction method of multipliers (PJ-ADMM). The proposed algorithm works for arbitrary convex disutility functions. Moreover, we ex- tend the PJ-ADMM based load control algorithm to an asynchronous setup. Asynchronous updates can spare the loads from strict synchronization and are particularly useful when the number of loads is large. Simulation results are presented to show that the proposed algorithms regulate the frequency to the nominal value well, in both synchronous and asynchronous scenarios and with and without imbalance estimation errors.
Singh, Aditya Vikram. "Theoretical and Algorithmic Aspects of Rigid Registration." Thesis, 2019. https://etd.iisc.ac.in/handle/2005/4727.
Повний текст джерелаAhmed, Miraj S. K. "Multiview Registration Using Rank-Constrained Semide nite Programming." Thesis, 2018. https://etd.iisc.ac.in/handle/2005/4761.
Повний текст джерела(10716096), Shreyansh Rakeshkuma Shethia. "Fast Tracking ADMM for Distributed Optimization and Convergence under Time-Varying Networks." Thesis, 2021.
Знайти повний текст джерелаRichter, Robin. "Cartoon-Residual Image Decompositions with Application in Fingerprint Recognition." Doctoral thesis, 2019. http://hdl.handle.net/21.11130/00-1735-0000-0005-12CB-2.
Повний текст джерелаSu, Yu-Shun, and 蘇郁舜. "Application of Alternating Direction of Multiplier Method on dimensional reduction." Thesis, 2018. http://ndltd.ncl.edu.tw/handle/5df46s.
Повний текст джерела國立中興大學
應用數學系所
106
Dimensionality reduction helps us not only reduce the use of capacity of computer disk, but also speed up the machine learning algorithms. Most importantly, dimensionality reduction helps us visualize the data so that we can find the structure in low rank structure and the underlying information lurking behind it. Among all the data visualization techniques, Principle Component Analysis(PCA) is the most popular method in uncovering the linear low rank structure. Maximum Variance Unfolding(MVU) and Locally Linear Embedding(LLE) are the classical methods for dimensionality reduction of nonlinear structures. In this paper, we use the concept of neighbor selection of MVU and LLE in the Alternating Direction method of Multipliers(ADMM) and accomplish dimensionality reduction by maximizing Frobenius norm of the data. The output of LLE can be used as an initial value of ADMM iteration to improve the result of distortion and lost distant by LLE to get a better result. In addition, we make a series of experiments to demonstrate their performance, including swissroll and nine-dimensional models, as well as exploring how to choose the better parameters beta of ADMM algorithm.
Deng, Wei. "Recovering Data with Group Sparsity by Alternating Direction Methods." Thesis, 2012. http://hdl.handle.net/1911/64676.
Повний текст джерелаSchomburg, Helen. "New Algorithms for Local and Global Fiber Tractography in Diffusion-Weighted Magnetic Resonance Imaging." Doctoral thesis, 2017. http://hdl.handle.net/11858/00-1735-0000-0023-3F8B-F.
Повний текст джерелаSathinarain, Melisha. "Numerical investigation of the parabolic mixed-derivative diffusion equation via alternating direction implicit methods." Thesis, 2013. http://hdl.handle.net/10539/13016.
Повний текст джерелаIn this dissertation, we investigate the parabolic mixed derivative diffusion equation modeling the viscous and viscoelastic effects in a non-Newtonian viscoelastic fluid. The model is analytically considered using Fourier and Laplace transformations. The main focus of the dissertation, however, is the implementation of the Peaceman-Rachford Alternating Direction Implicit method. The one-dimensional parabolic mixed derivative diffusion equation is extended to a two-dimensional analog. In order to do this, the two-dimensional analog is solved using a Crank-Nicholson method and implemented according to the Peaceman- Rachford ADI method. The behaviour of the solution of the viscoelastic fluid model is analysed by investigating the effects of inertia and diffusion as well as the viscous behaviour, subject to the viscosity and viscoelasticity parameters. The two-dimensional parabolic diffusion equation is then implemented with a high-order method to unveil more accurate solutions. An error analysis is executed to show the accuracy differences between the numerical solutions of the general ADI and high-order compact methods. Each of the methods implemented in this dissertation are investigated via the von-Neumann stability analysis to prove stability under certain conditions.
Yang, Song-ming, and 楊松銘. "Improved Accuracy for Alternating Direction Methods for Parabolic Equations Based on Mixed Finite Element Procedures." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/75141617022757458217.
Повний текст джерела國立中山大學
應用數學系研究所
91
Classical alternating direction (AD) methods for parabolic equations, based on some standard implicit time stepping procedure such as Crank-Nicolson, can have errors associated with the AD perturbations that are much larger than the errors associated with the underlying time stepping procedure . We plan to show that minor modifications in the AD procedures can virtually eliminate the perturbation errors at an minor additional computational cost. A mixed finite element method is applied in the spactial variables. Similar to the finite difference and finite element methods in spactial variables, we plan to have the same accuracy in time. A convergence analysis can also be shown .