In this paper we will study an elastic dissipation model for a cantilevered beam. This problem for a cantilevered beam has been formulated by D.L. Russell as an open problem in [1, 2]. To determine the relationship between the damping rates and the frequencies we will use a recently developed, adapted form of the method of separation of variables. It will be shown that the dissipation model as proposed by D.L. Russell for the cantilevered beam will not always generate damping. Moreover, it will be shown that some solutions can become unbounded.
{"title":"On an elastic dissipation model for a cantilevered beam","authors":"W. T. Horssen, Zarubinskaya","doi":"10.1090/QAM/1999837","DOIUrl":"https://doi.org/10.1090/QAM/1999837","url":null,"abstract":"In this paper we will study an elastic dissipation model for a cantilevered beam. This problem for a cantilevered beam has been formulated by D.L. Russell as an open problem in [1, 2]. To determine the relationship between the damping rates and the frequencies we will use a recently developed, adapted form of the method of separation of variables. It will be shown that the dissipation model as proposed by D.L. Russell for the cantilevered beam will not always generate damping. Moreover, it will be shown that some solutions can become unbounded.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2003-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133501705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-04-01DOI: 10.1016/S0168-9274(01)00118-0
C. Vuik, A. Segal, L. E. Yaakoubi, E. Dufour
{"title":"A comparison of various deflation vectors applied to elliptic problems with discontinuous coefficients","authors":"C. Vuik, A. Segal, L. E. Yaakoubi, E. Dufour","doi":"10.1016/S0168-9274(01)00118-0","DOIUrl":"https://doi.org/10.1016/S0168-9274(01)00118-0","url":null,"abstract":"","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2002-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127301424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2001-07-01DOI: 10.1175/1520-0493(2001)129<1718:VREKF>2.0.CO;2
A. Heemink, M. Verlaan, A. Segers
A number of algorithms to solve large-scale Kalman filtering problems have been introduced recently. The ensemble Kalman filter represents the probability density of the state estimate by a finite number of randomly generated system states. Another algorithm uses a singular value decomposition to select the leading eigenvectors of the covariance matrix of the state estimate and to approximate the full covariance matrix by a reduced-rank matrix. Both algorithms, however, still require a huge amount of computer resources. In this paper the authors propose to combine the two algorithms and to use a reduced-rank approximation of the covariance matrix as a variance reductor for the ensemble Kalman filter. If the leading eigenvectors explain most of the variance, which is the case for most applications, the computational burden to solve the filtering problem can be reduced significantly (up to an order of magnitude).
{"title":"Variance reduced ensemble Kalman filtering","authors":"A. Heemink, M. Verlaan, A. Segers","doi":"10.1175/1520-0493(2001)129<1718:VREKF>2.0.CO;2","DOIUrl":"https://doi.org/10.1175/1520-0493(2001)129<1718:VREKF>2.0.CO;2","url":null,"abstract":"A number of algorithms to solve large-scale Kalman filtering problems have been introduced recently. The ensemble Kalman filter represents the probability density of the state estimate by a finite number of randomly generated system states. Another algorithm uses a singular value decomposition to select the leading eigenvectors of the covariance matrix of the state estimate and to approximate the full covariance matrix by a reduced-rank matrix. Both algorithms, however, still require a huge amount of computer resources. In this paper the authors propose to combine the two algorithms and to use a reduced-rank approximation of the covariance matrix as a variance reductor for the ensemble Kalman filter. If the leading eigenvectors explain most of the variance, which is the case for most applications, the computational burden to solve the filtering problem can be reduced significantly (up to an order of magnitude).","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133495985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1615/INTJMULTCOMPENG.V6.I1.20
J. M. Tang, C. Vuik
Traditional Krylov iterative solvers, such as the preconditioned conjugate gradient method, can be accelerated by incorporating a second level preconditioner. We use deflation as a second level preconditioner, which is very efficient in many applications. In this paper, we give some theoretical results for the general deflation method applied to singular matrices, which provides us more insights into the properties and the behavior of the method. Moreover, we discuss stability issues of the deflation method and consider some ideas for a more stable method. In the numerical experiments, we apply the deflation method and its stabilized variant to singular linear systems derived from two-phase bubbly flow problems. Due to the appearance of bubbles, those linear systems are ill-conditioned, and therefore, they are usually hard to solve using traditional preconditioned Krylov iterative methods. We show that our deflation methods can be very efficient to solve the linear systems. Finally, we also investigate numerically the stability of these methods by examining the corresponding inner-outer iterations in more detail.
{"title":"Fast deflation methods with applications to two-phase flows","authors":"J. M. Tang, C. Vuik","doi":"10.1615/INTJMULTCOMPENG.V6.I1.20","DOIUrl":"https://doi.org/10.1615/INTJMULTCOMPENG.V6.I1.20","url":null,"abstract":"Traditional Krylov iterative solvers, such as the preconditioned conjugate gradient method, can be accelerated by incorporating a second level preconditioner. We use deflation as a second level preconditioner, which is very efficient in many applications. In this paper, we give some theoretical results for the general deflation method applied to singular matrices, which provides us more insights into the properties and the behavior of the method. Moreover, we discuss stability issues of the deflation method and consider some ideas for a more stable method. In the numerical experiments, we apply the deflation method and its stabilized variant to singular linear systems derived from two-phase bubbly flow problems. Due to the appearance of bubbles, those linear systems are ill-conditioned, and therefore, they are usually hard to solve using traditional preconditioned Krylov iterative methods. We show that our deflation methods can be very efficient to solve the linear systems. Finally, we also investigate numerically the stability of these methods by examining the corresponding inner-outer iterations in more detail.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123728310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1007/978-3-642-11795-4_27
T. Collignon, M. Gijzen
{"title":"Solving large sparse linear systems efficiently on grid computers using an asynchronous iterative method as a preconditioner","authors":"T. Collignon, M. Gijzen","doi":"10.1007/978-3-642-11795-4_27","DOIUrl":"https://doi.org/10.1007/978-3-642-11795-4_27","url":null,"abstract":"","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128951596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1007/978-3-642-03344-5_4
T. Collignon, M. Gijzen
{"title":"Parallel Scientific Computing on Loosely Coupled Networks of Computers","authors":"T. Collignon, M. Gijzen","doi":"10.1007/978-3-642-03344-5_4","DOIUrl":"https://doi.org/10.1007/978-3-642-03344-5_4","url":null,"abstract":"","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128560117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1007/978-3-642-18560-1_7
F. Vermolen, K. Vuik, G. Segal
{"title":"Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems","authors":"F. Vermolen, K. Vuik, G. Segal","doi":"10.1007/978-3-642-18560-1_7","DOIUrl":"https://doi.org/10.1007/978-3-642-18560-1_7","url":null,"abstract":"","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134090479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper utilizes the saddlepoint approximation as an efficient tool to estimate the portfolio credit loss distribution in the Vasicek model. Value at Risk (VaR), the risk measure chosen in the Basel II Accord for the evaluation of capital requirement, can then be found by inverting the loss distribution. VaR Contribution (VaRC), Expected Shortfall (ES) and ES Contribution (ESC) can all be calculated accurately. Saddlepoint approximation is well known to provide good approximations to very small tail probabilities, which makes it a very suitable technique in the context of portfolio credit loss. The portfolio credit model we employ is the Vasicek one factor model, which has an analytical solution if the portfolio is well diversified. The Vasicek asymptotic formula however fails when the portfolio is dominated by a few loans. We show that saddlepoint approximation is able to handle such exposure concentration. We also point out that the saddlepoint approximation technique can be readily applied to more general Bernoulli mixture models (possibly multi-factor). It can further handle portfolios with random LGD.
{"title":"Higher order saddlepoint approximations in the Vasicek portfolio credit loss model","authors":"X. Huang, C. Oosterlee, J. Weide","doi":"10.21314/JCF.2007.165","DOIUrl":"https://doi.org/10.21314/JCF.2007.165","url":null,"abstract":"This paper utilizes the saddlepoint approximation as an efficient tool to estimate the portfolio credit loss distribution in the Vasicek model. Value at Risk (VaR), the risk measure chosen in the Basel II Accord for the evaluation of capital requirement, can then be found by inverting the loss distribution. VaR Contribution (VaRC), Expected Shortfall (ES) and ES Contribution (ESC) can all be calculated accurately. Saddlepoint approximation is well known to provide good approximations to very small tail probabilities, which makes it a very suitable technique in the context of portfolio credit loss. The portfolio credit model we employ is the Vasicek one factor model, which has an analytical solution if the portfolio is well diversified. The Vasicek asymptotic formula however fails when the portfolio is dominated by a few loans. We show that saddlepoint approximation is able to handle such exposure concentration. We also point out that the saddlepoint approximation technique can be readily applied to more general Bernoulli mixture models (possibly multi-factor). It can further handle portfolios with random LGD.","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128418614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a new framework for modeling systematic risk in LossGiven-Default (LGD) in the context of credit portfolio losses. The class of models is very flexible and accommodates well skewness and heteroscedastic errors. The quantities in the models have simple economic interpretation. Inference of models in this framework can be unified. Moreover, it allows efficient numerical procedures, such as the normal approximation and the saddlepoint approximation, to calculate the portfolio loss distribution, Value at Risk (VaR) and Expected Shortfall (ES).
{"title":"Generalized beta regression models for random loss given default","authors":"Xinzheng Huang, C. Oosterlee","doi":"10.21314/jcr.2011.150","DOIUrl":"https://doi.org/10.21314/jcr.2011.150","url":null,"abstract":"We propose a new framework for modeling systematic risk in LossGiven-Default (LGD) in the context of credit portfolio losses. The class of models is very flexible and accommodates well skewness and heteroscedastic errors. The quantities in the models have simple economic interpretation. Inference of models in this framework can be unified. Moreover, it allows efficient numerical procedures, such as the normal approximation and the saddlepoint approximation, to calculate the portfolio loss distribution, Value at Risk (VaR) and Expected Shortfall (ES).","PeriodicalId":266346,"journal":{"name":"Reports of the Department of Applied Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114813748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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