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Christoph Schwab - One of the best experts on this subject based on the ideXlab platform.
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high dimensional adaptive sparse polynomial Interpolation and applications to parametric pdes
Foundations of Computational Mathematics, 2014Co-Authors: Abdellah Chkifa, Albert Cohen, Christoph SchwabAbstract:We consider the problem of Lagrange polynomial Interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an Interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional Interpolation Scheme and sparsification. We derive bounds on the Lebesgue constants for this Interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE's, we have shown in Chkifa et al. (Model. Math. Anal. Numer. 47(1):253---280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our Interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our Interpolation Scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE's.
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High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs
Foundations of Computational Mathematics, 2014Co-Authors: Abdellah Chkifa, Albert Cohen, Christoph SchwabAbstract:We consider the problem of Lagrange polynomial Interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an Interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional Interpolation Scheme and sparsification. We derive bounds on the Lebesgue constants for this Interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE’s, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253–280, 2013 ) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our Interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our Interpolation Scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE’s.
Abdellah Chkifa - One of the best experts on this subject based on the ideXlab platform.
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high dimensional adaptive sparse polynomial Interpolation and applications to parametric pdes
Foundations of Computational Mathematics, 2014Co-Authors: Abdellah Chkifa, Albert Cohen, Christoph SchwabAbstract:We consider the problem of Lagrange polynomial Interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an Interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional Interpolation Scheme and sparsification. We derive bounds on the Lebesgue constants for this Interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE's, we have shown in Chkifa et al. (Model. Math. Anal. Numer. 47(1):253---280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our Interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our Interpolation Scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE's.
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High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs
Foundations of Computational Mathematics, 2014Co-Authors: Abdellah Chkifa, Albert Cohen, Christoph SchwabAbstract:We consider the problem of Lagrange polynomial Interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an Interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional Interpolation Scheme and sparsification. We derive bounds on the Lebesgue constants for this Interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE’s, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253–280, 2013 ) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our Interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our Interpolation Scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE’s.
J Choi - One of the best experts on this subject based on the ideXlab platform.
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minimum temperature mapping over complex terrain by estimating cold air accumulation potential
Agricultural and Forest Meteorology, 2006Co-Authors: Uran Chung, K H Hwang, B S Hwang, J ChoiAbstract:Nocturnal cold air pools, which frequently form in complex terrain under anticyclonic systems, are a hazard toflowering budsof temperate fruit trees, but the spatial resolution to anticipate them exceeds the current weather forecast scale. To supplement the insufficient spatial resolution of official minimum temperature forecasts, a spatial Interpolation Scheme was developed by incorporating local geographic potential for cold air accumulation. Air temperature was measured at 10-min intervals at eight locations within a 2.1 km � 2.1 km hilly orchard area in South Korea from September 2002 to August 2003. Minimum temperature data for suspected radiative cooling nights were collected, and deviations from estimations by a conventional spatial Interpolation were calculated. A digital elevation model with a 10 m cell size was used to calculate the cold air accumulation at eight locations. Zonalaverages ofthecoldair accumulationwere computedforeach locationbyincreasing thecellradius from 1to10.Temperature deviations were regressed to a common logarithm of the smoothed averages of cold air accumulation to derive a linear relationship between the local temperature deviation and the site topography. The highest coefficient of determination was found at a cell radius of 5, which corresponds to an approximately 1 ha boundary surrounding the point of interest. Nocturnal temperature profiles were observed and the inversion cap height was determined using a tethered balloon system. Empirical equations describing the potential effects of cold air accumulation and of theinversion profile onminimum temperaturewere combined witha conventionallapse ratecorrected inverse distance weighting Interpolation Scheme. This new Interpolation Scheme was successfully validated with an independent data set, showing a strong feasibility for development of a site-specific frost warning system for mountainous areas. # 2006 Elsevier B.V. All rights reserved.
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minimum temperature mapping over complex terrain by estimating cold air accumulation potential
Agricultural and Forest Meteorology, 2006Co-Authors: Uran Chung, K H Hwang, B S Hwang, J Choi, H H Seo, J T Lee, Jin I YunAbstract:Abstract Nocturnal cold air pools, which frequently form in complex terrain under anticyclonic systems, are a hazard to flowering buds of temperate fruit trees, but the spatial resolution to anticipate them exceeds the current weather forecast scale. To supplement the insufficient spatial resolution of official minimum temperature forecasts, a spatial Interpolation Scheme was developed by incorporating local geographic potential for cold air accumulation. Air temperature was measured at 10-min intervals at eight locations within a 2.1 km × 2.1 km hilly orchard area in South Korea from September 2002 to August 2003. Minimum temperature data for suspected radiative cooling nights were collected, and deviations from estimations by a conventional spatial Interpolation were calculated. A digital elevation model with a 10 m cell size was used to calculate the cold air accumulation at eight locations. Zonal averages of the cold air accumulation were computed for each location by increasing the cell radius from 1 to 10. Temperature deviations were regressed to a common logarithm of the smoothed averages of cold air accumulation to derive a linear relationship between the local temperature deviation and the site topography. The highest coefficient of determination was found at a cell radius of 5, which corresponds to an approximately 1 ha boundary surrounding the point of interest. Nocturnal temperature profiles were observed and the inversion cap height was determined using a tethered balloon system. Empirical equations describing the potential effects of cold air accumulation and of the inversion profile on minimum temperature were combined with a conventional lapse rate-corrected inverse distance weighting Interpolation Scheme. This new Interpolation Scheme was successfully validated with an independent data set, showing a strong feasibility for development of a site-specific frost warning system for mountainous areas.
Albert Cohen - One of the best experts on this subject based on the ideXlab platform.
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high dimensional adaptive sparse polynomial Interpolation and applications to parametric pdes
Foundations of Computational Mathematics, 2014Co-Authors: Abdellah Chkifa, Albert Cohen, Christoph SchwabAbstract:We consider the problem of Lagrange polynomial Interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an Interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional Interpolation Scheme and sparsification. We derive bounds on the Lebesgue constants for this Interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE's, we have shown in Chkifa et al. (Model. Math. Anal. Numer. 47(1):253---280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our Interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our Interpolation Scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE's.
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High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs
Foundations of Computational Mathematics, 2014Co-Authors: Abdellah Chkifa, Albert Cohen, Christoph SchwabAbstract:We consider the problem of Lagrange polynomial Interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an Interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional Interpolation Scheme and sparsification. We derive bounds on the Lebesgue constants for this Interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE’s, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253–280, 2013 ) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our Interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our Interpolation Scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE’s.
Uran Chung - One of the best experts on this subject based on the ideXlab platform.
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minimum temperature mapping over complex terrain by estimating cold air accumulation potential
Agricultural and Forest Meteorology, 2006Co-Authors: Uran Chung, K H Hwang, B S Hwang, J ChoiAbstract:Nocturnal cold air pools, which frequently form in complex terrain under anticyclonic systems, are a hazard toflowering budsof temperate fruit trees, but the spatial resolution to anticipate them exceeds the current weather forecast scale. To supplement the insufficient spatial resolution of official minimum temperature forecasts, a spatial Interpolation Scheme was developed by incorporating local geographic potential for cold air accumulation. Air temperature was measured at 10-min intervals at eight locations within a 2.1 km � 2.1 km hilly orchard area in South Korea from September 2002 to August 2003. Minimum temperature data for suspected radiative cooling nights were collected, and deviations from estimations by a conventional spatial Interpolation were calculated. A digital elevation model with a 10 m cell size was used to calculate the cold air accumulation at eight locations. Zonalaverages ofthecoldair accumulationwere computedforeach locationbyincreasing thecellradius from 1to10.Temperature deviations were regressed to a common logarithm of the smoothed averages of cold air accumulation to derive a linear relationship between the local temperature deviation and the site topography. The highest coefficient of determination was found at a cell radius of 5, which corresponds to an approximately 1 ha boundary surrounding the point of interest. Nocturnal temperature profiles were observed and the inversion cap height was determined using a tethered balloon system. Empirical equations describing the potential effects of cold air accumulation and of theinversion profile onminimum temperaturewere combined witha conventionallapse ratecorrected inverse distance weighting Interpolation Scheme. This new Interpolation Scheme was successfully validated with an independent data set, showing a strong feasibility for development of a site-specific frost warning system for mountainous areas. # 2006 Elsevier B.V. All rights reserved.
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minimum temperature mapping over complex terrain by estimating cold air accumulation potential
Agricultural and Forest Meteorology, 2006Co-Authors: Uran Chung, K H Hwang, B S Hwang, J Choi, H H Seo, J T Lee, Jin I YunAbstract:Abstract Nocturnal cold air pools, which frequently form in complex terrain under anticyclonic systems, are a hazard to flowering buds of temperate fruit trees, but the spatial resolution to anticipate them exceeds the current weather forecast scale. To supplement the insufficient spatial resolution of official minimum temperature forecasts, a spatial Interpolation Scheme was developed by incorporating local geographic potential for cold air accumulation. Air temperature was measured at 10-min intervals at eight locations within a 2.1 km × 2.1 km hilly orchard area in South Korea from September 2002 to August 2003. Minimum temperature data for suspected radiative cooling nights were collected, and deviations from estimations by a conventional spatial Interpolation were calculated. A digital elevation model with a 10 m cell size was used to calculate the cold air accumulation at eight locations. Zonal averages of the cold air accumulation were computed for each location by increasing the cell radius from 1 to 10. Temperature deviations were regressed to a common logarithm of the smoothed averages of cold air accumulation to derive a linear relationship between the local temperature deviation and the site topography. The highest coefficient of determination was found at a cell radius of 5, which corresponds to an approximately 1 ha boundary surrounding the point of interest. Nocturnal temperature profiles were observed and the inversion cap height was determined using a tethered balloon system. Empirical equations describing the potential effects of cold air accumulation and of the inversion profile on minimum temperature were combined with a conventional lapse rate-corrected inverse distance weighting Interpolation Scheme. This new Interpolation Scheme was successfully validated with an independent data set, showing a strong feasibility for development of a site-specific frost warning system for mountainous areas.
