We propose in this paper a data driven state estimation scheme for generating nonlinear reduced models for parametric families of PDEs, directly providingdata-to-state maps, represented in terms of Deep Neural Network...We propose in this paper a data driven state estimation scheme for generating nonlinear reduced models for parametric families of PDEs, directly providingdata-to-state maps, represented in terms of Deep Neural Networks. A major constituentis a sensor-induced decomposition of a model-compliant Hilbert space warranting approximation in problem relevant metrics. It plays a similar role as in a ParametricBackground Data Weak framework for state estimators based on Reduced Basis concepts. Extensive numerical tests shed light on several optimization strategies that areto improve robustness and performance of such estimators.展开更多
This paper is concerned with preconditioners for interior penalty discontinuous Galerkin discretizations of second-order elliptic boundary value problems.We extend earlier related results in[7]in the following sense.S...This paper is concerned with preconditioners for interior penalty discontinuous Galerkin discretizations of second-order elliptic boundary value problems.We extend earlier related results in[7]in the following sense.Several concrete realizations of splitting the nonconforming trial spaces into a conforming and(remaining)nonconforming part are identified and shown to give rise to uniformly bounded condition numbers.These asymptotic results are complemented by numerical tests that shed some light on their respective quantitative behavior.展开更多
This paper is concerned with Bernstein polynomials on k-simploids by which we mean a crossproduct of k lower dimensional simplices. Specifically, we show that if the Bernstein polynomials ofa given function f on a k-s...This paper is concerned with Bernstein polynomials on k-simploids by which we mean a crossproduct of k lower dimensional simplices. Specifically, we show that if the Bernstein polynomials ofa given function f on a k-simploid form a decreasing sequence then f+l, where l is any correspondingtensor product of affine functions. achieves its maximum on the boundary of the k-simploid. Thisextends recent results in [1] for bivariate Bernstein polynomials on triangles. Unlike the approachused in [1] our approach is based on semigroup techniques and the maximum principle for secondorder elliptic operators. Furthermore, we derive analogous results for cube spline surfaces.展开更多
基金This work was supported by National Science Foundation under grant DMS-2012469.
文摘We propose in this paper a data driven state estimation scheme for generating nonlinear reduced models for parametric families of PDEs, directly providingdata-to-state maps, represented in terms of Deep Neural Networks. A major constituentis a sensor-induced decomposition of a model-compliant Hilbert space warranting approximation in problem relevant metrics. It plays a similar role as in a ParametricBackground Data Weak framework for state estimators based on Reduced Basis concepts. Extensive numerical tests shed light on several optimization strategies that areto improve robustness and performance of such estimators.
基金This work has been supported in part by the French-German PROCOPE contract 11418YBby the European Commission Human Potential Programme under contract HPRN-CT-2002-00286“Breaking Complexity”,by the SFB 401 and the Leibniz Pro-gramme funded by DFG.
文摘This paper is concerned with preconditioners for interior penalty discontinuous Galerkin discretizations of second-order elliptic boundary value problems.We extend earlier related results in[7]in the following sense.Several concrete realizations of splitting the nonconforming trial spaces into a conforming and(remaining)nonconforming part are identified and shown to give rise to uniformly bounded condition numbers.These asymptotic results are complemented by numerical tests that shed some light on their respective quantitative behavior.
基金This work was partially supported by NATO Grant No.DJ RG 639/84
文摘This paper is concerned with Bernstein polynomials on k-simploids by which we mean a crossproduct of k lower dimensional simplices. Specifically, we show that if the Bernstein polynomials ofa given function f on a k-simploid form a decreasing sequence then f+l, where l is any correspondingtensor product of affine functions. achieves its maximum on the boundary of the k-simploid. Thisextends recent results in [1] for bivariate Bernstein polynomials on triangles. Unlike the approachused in [1] our approach is based on semigroup techniques and the maximum principle for secondorder elliptic operators. Furthermore, we derive analogous results for cube spline surfaces.
