Pub Date : 2009-05-27DOI: 10.1137/S0040585X97983687
S. Graf, H. Luschgy
For a probability distribution P on ${bf R}^d$ and $nin{bf N}$ consider $e_n = inf pi (P,Q)$, where $pi$ denotes the Prokhorov metric and the infimum is taken over all discrete probabilities Q with $|mbox{supp}(Q) | le n$. We study solutions Q of this minimization problem, stability properties, and consistency of empirical estimators. For some classes of distributions we determine the exact rate of convergence to zero of the nth quantization error $e_n$ as $n rightarrowinfty$.
对于${bf R}^d$和$nin{bf N}$上的概率分布P，考虑$e_n = inf pi (P,Q)$，其中$pi$表示Prokhorov度量，最小值被取为$|mbox{supp}(Q) | le n$的所有离散概率Q。我们研究了这个最小化问题的解Q、稳定性和经验估计量的一致性。对于某些类型的分布，我们确定第n个量化误差$e_n$收敛到零的确切速率为$n rightarrowinfty$。
{"title":"Quantization for Probability Measures in the Prohorov Metric","authors":"S. Graf, H. Luschgy","doi":"10.1137/S0040585X97983687","DOIUrl":"https://doi.org/10.1137/S0040585X97983687","url":null,"abstract":"For a probability distribution P on ${bf R}^d$ and $nin{bf N}$ consider $e_n = inf pi (P,Q)$, where $pi$ denotes the Prokhorov metric and the infimum is taken over all discrete probabilities Q with $|mbox{supp}(Q) | le n$. We study solutions Q of this minimization problem, stability properties, and consistency of empirical estimators. For some classes of distributions we determine the exact rate of convergence to zero of the nth quantization error $e_n$ as $n rightarrowinfty$.","PeriodicalId":142744,"journal":{"name":"Universität Trier, Mathematik/Informatik, Forschungsbericht","volume":"38 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114015426","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 present simultaneous pseudo-timestepping as an efficient method for aerodynamic shape optimization. In this method, instead of solving the necessary optimality conditions by iterative techniques, pseudo-time embedded nonstationary system is integrated in time until a steady state is reached. The main advantages of this method are that it requires no additional globalization techniques and that a preconditioner can be used for convergence acceleration which stems from the reduced SQP method. The important issue of this method is the trade-off between the accuracy of the forward and adjoint solver and its impact on the computational cost to approach an optimum solution is addressed. The method is applied to a test case of drag reduction for an RAE2822 airfoil, keeping it’s thickness constant. The optimum overall cost of computation that is achieved in this method is less than 4 times that of the forward simulation run. Nomenclature (x, y) ∈ R :cartesian coordinates H :total enthalpy (ξ, η) ∈ [0, 1] :generalized coordinates M :Mach number Ω :flow field domain )∞ :values at free stream ∂Ω :flow field boundary γ :ratio of specific heats ~n := ( nx ny ) :unit outward normal Cref :chord length α :angle of attack CD :drag coefficient ρ :density I :cost unction u :x-component of velocity w :vector of state variables v :y-component of velocity q :vector of design variables p :pressure λ :vector of adjoint variables E :total energy J :Jacobian Cp :pressure coefficient B :reduced Hessian
{"title":"An Efficient Method for Aerodynamic Shape Optimization","authors":"S. Hazra","doi":"10.2514/6.2004-4628","DOIUrl":"https://doi.org/10.2514/6.2004-4628","url":null,"abstract":"We present simultaneous pseudo-timestepping as an efficient method for aerodynamic shape optimization. In this method, instead of solving the necessary optimality conditions by iterative techniques, pseudo-time embedded nonstationary system is integrated in time until a steady state is reached. The main advantages of this method are that it requires no additional globalization techniques and that a preconditioner can be used for convergence acceleration which stems from the reduced SQP method. The important issue of this method is the trade-off between the accuracy of the forward and adjoint solver and its impact on the computational cost to approach an optimum solution is addressed. The method is applied to a test case of drag reduction for an RAE2822 airfoil, keeping it’s thickness constant. The optimum overall cost of computation that is achieved in this method is less than 4 times that of the forward simulation run. Nomenclature (x, y) ∈ R :cartesian coordinates H :total enthalpy (ξ, η) ∈ [0, 1] :generalized coordinates M :Mach number Ω :flow field domain )∞ :values at free stream ∂Ω :flow field boundary γ :ratio of specific heats ~n := ( nx ny ) :unit outward normal Cref :chord length α :angle of attack CD :drag coefficient ρ :density I :cost unction u :x-component of velocity w :vector of state variables v :y-component of velocity q :vector of design variables p :pressure λ :vector of adjoint variables E :total energy J :Jacobian Cp :pressure coefficient B :reduced Hessian","PeriodicalId":142744,"journal":{"name":"Universität Trier, Mathematik/Informatik, Forschungsbericht","volume":"12 8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126064855","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}
Symbolic Model checking is a widely used technique in sequential verification. As the size of the OBDDs and also the computation time depends on the order of the input variables, the verification may only succeed if a well suited variable order is chosen. Since the characteristics of the represented functions are changing, the variable order has to be adapted dynamically. Unfortunately, dynamic reordering strategies are often very time consuming and sometimes do not provide any improvement of the OBDD representation. This paper presents adaptions of reordering techniques originally intended for combinatorial verification to the specific requirements of symbolic model checking. The techniques are orthogonal in the way that they use either structural information about the OBDDs or semantical information about the represented functions. The application of these techniques substantially accelerates the reordering process and makes it possible to finish computations, that are too time consuming, otherwise.
{"title":"Speeding up symbolic model checking by accelerating dynamic variable reordering","authors":"C. Meinel, Christian Stangier","doi":"10.1145/330855.330954","DOIUrl":"https://doi.org/10.1145/330855.330954","url":null,"abstract":"Symbolic Model checking is a widely used technique in sequential verification. As the size of the OBDDs and also the computation time depends on the order of the input variables, the verification may only succeed if a well suited variable order is chosen. Since the characteristics of the represented functions are changing, the variable order has to be adapted dynamically. Unfortunately, dynamic reordering strategies are often very time consuming and sometimes do not provide any improvement of the OBDD representation. This paper presents adaptions of reordering techniques originally intended for combinatorial verification to the specific requirements of symbolic model checking. The techniques are orthogonal in the way that they use either structural information about the OBDDs or semantical information about the represented functions. The application of these techniques substantially accelerates the reordering process and makes it possible to finish computations, that are too time consuming, otherwise.","PeriodicalId":142744,"journal":{"name":"Universität Trier, Mathematik/Informatik, Forschungsbericht","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125708031","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 : 1998-11-01DOI: 10.1142/S0218202598000561
B. King
In this paper, we discuss the use of nonuniform grids in reduced basis design for developing low order nonlinear feedback controllers for hybrid distributed parameter systems. The reduced basis approach was presented in an earlier paper by Burns and King; therein, all approximations were based upon uniform grids. In this paper, we explore the effect on control design of using nonuniform grids in the fundamental step of approximating the functional controller gains. We illustrate the process using a weakly nonlinear distributed parameter system.
{"title":"Nonuniform Grids for Reduced Basis Design of Low Order Feedback Controllers for Nonlinear Continuous Systems","authors":"B. King","doi":"10.1142/S0218202598000561","DOIUrl":"https://doi.org/10.1142/S0218202598000561","url":null,"abstract":"In this paper, we discuss the use of nonuniform grids in reduced basis design for developing low order nonlinear feedback controllers for hybrid distributed parameter systems. The reduced basis approach was presented in an earlier paper by Burns and King; therein, all approximations were based upon uniform grids. In this paper, we explore the effect on control design of using nonuniform grids in the fundamental step of approximating the functional controller gains. We illustrate the process using a weakly nonlinear distributed parameter system.","PeriodicalId":142744,"journal":{"name":"Universität Trier, Mathematik/Informatik, Forschungsbericht","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126915271","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}
{"title":"The Electronic Colloquium on Computational Complexity (ECCC): A Digital Library in Use","authors":"J. Bern, C. Damm, C. Meinel","doi":"10.1007/BFb0026741","DOIUrl":"https://doi.org/10.1007/BFb0026741","url":null,"abstract":"","PeriodicalId":142744,"journal":{"name":"Universität Trier, Mathematik/Informatik, Forschungsbericht","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133121751","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 : 1997-03-01DOI: 10.1137/S1052623495290441
T. Rautert, E. Sachs
We consider the problem of designing feedback control laws when a complete set of state variables is not available. For linear autonomous systems with quadratic performance criterion, the design problem consists of choosing an appropriate matrix of feedback gains according to a certain objective function. In the literature, the performance of quasi-Newton methods has been reported to be substandard. We try to explain some of these observations and to propose structured quasi-Newton updates. These methods, which take into account the special structure of the problem, show considerable improvement in the convergence. Using test examples from optimal output feedback design, we also can verify these results numerically.
{"title":"Computational Design of Optimal Output Feedback Controllers","authors":"T. Rautert, E. Sachs","doi":"10.1137/S1052623495290441","DOIUrl":"https://doi.org/10.1137/S1052623495290441","url":null,"abstract":"We consider the problem of designing feedback control laws when a complete set of state variables is not available. For linear autonomous systems with quadratic performance criterion, the design problem consists of choosing an appropriate matrix of feedback gains according to a certain objective function. In the literature, the performance of quasi-Newton methods has been reported to be substandard. We try to explain some of these observations and to propose structured quasi-Newton updates. These methods, which take into account the special structure of the problem, show considerable improvement in the convergence. Using test examples from optimal output feedback design, we also can verify these results numerically.","PeriodicalId":142744,"journal":{"name":"Universität Trier, Mathematik/Informatik, Forschungsbericht","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1997-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122272181","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 : 1996-11-01DOI: 10.1080/00029890.1996.12004814
T. Burger, P. Gritzmann, V. Klee
Imagine yourself as the commander of a space ship. Liftoff was a piece of cake, and since then you have been gliding merrily along. But then comes the bad news: A Klingon ship is approaching, and you must prepare for the attack. More bad news: Your batteries are running low! The good news is that your solar cells are working and you are close to a bright star. Thus you can recharge your batteries, but you must certainly do that as quickly as possible. You analyze the situation. Since the solar cells are distributed evenly over the surface of the ship, you decide that you should rotate the ship so that its "face area" is maximized with respect to the light source (assuming that you are still so far from the star that the incoming rays are practically parallel). A similar but opposite problem arises when you approach a star that emits harmful radiation. You then want to minimize the exposure to the radiation and therefore to minimize the face area in the direction of the star. In these problems, you are in control of a body in 3, and you want to turn the body so as to maximize or minimize its "shadow area"with respect to a particular direction of projection (the direction of the incoming rays). In a mathematically equivalent formulation, you may regard the body as being fixed and then look for a direction that maximizes or minimizes the area of the body's projection on a plane orthogonal to the direction. Projections belong to the basic tools in many areas of mathematics. While the projection on a given subspace can be expressed as a simple matrux operation applied to the original body, it is not so clear how to find projections that are "optimal"with respect to an application that one may have in mind. Problems of this kind occur in a great variety of situations with a similarly great variety of (more or less explicit) criteria for what is a good projection. Examples include the analysis of statistical, astronomical or linguistic data, and also the design and analysis of algorithms for manifold applications. We do not want to elaborate on these applications here; the goal of this paper really is to present some of the (as we hope the reader will agree) beautiful mathematics underlying the special projection problems of maximizing or minimizing the "shadow area" and their higherdimensional analogues involving orthogonal projections of a body in 114Z1 onto an (n1)-dimensional subspace. We assume that the body in question is an ndimensional convex polytope. When n = 3, this seems to be a reasonable assumption in the case of the space ship (see Figure 1). It is not hard to see that when n = 2 (so that we are projecting a convex polygon P onto various lines), the maximum projection-length is equal to P's diameter and the minimum projection-length is equal to P's width (the minimum distance between two parallel supporting lines of P) (see Figure 2). Thus the n-dimensional task considered here is one of several ways of extending to 114Z1 the classical Eu
{"title":"Polytope Projection and Projection Polytopes","authors":"T. Burger, P. Gritzmann, V. Klee","doi":"10.1080/00029890.1996.12004814","DOIUrl":"https://doi.org/10.1080/00029890.1996.12004814","url":null,"abstract":"Imagine yourself as the commander of a space ship. Liftoff was a piece of cake, and since then you have been gliding merrily along. But then comes the bad news: A Klingon ship is approaching, and you must prepare for the attack. More bad news: Your batteries are running low! The good news is that your solar cells are working and you are close to a bright star. Thus you can recharge your batteries, but you must certainly do that as quickly as possible. You analyze the situation. Since the solar cells are distributed evenly over the surface of the ship, you decide that you should rotate the ship so that its \"face area\" is maximized with respect to the light source (assuming that you are still so far from the star that the incoming rays are practically parallel). A similar but opposite problem arises when you approach a star that emits harmful radiation. You then want to minimize the exposure to the radiation and therefore to minimize the face area in the direction of the star. In these problems, you are in control of a body in 3, and you want to turn the body so as to maximize or minimize its \"shadow area\"with respect to a particular direction of projection (the direction of the incoming rays). In a mathematically equivalent formulation, you may regard the body as being fixed and then look for a direction that maximizes or minimizes the area of the body's projection on a plane orthogonal to the direction. Projections belong to the basic tools in many areas of mathematics. While the projection on a given subspace can be expressed as a simple matrux operation applied to the original body, it is not so clear how to find projections that are \"optimal\"with respect to an application that one may have in mind. Problems of this kind occur in a great variety of situations with a similarly great variety of (more or less explicit) criteria for what is a good projection. Examples include the analysis of statistical, astronomical or linguistic data, and also the design and analysis of algorithms for manifold applications. We do not want to elaborate on these applications here; the goal of this paper really is to present some of the (as we hope the reader will agree) beautiful mathematics underlying the special projection problems of maximizing or minimizing the \"shadow area\" and their higherdimensional analogues involving orthogonal projections of a body in 114Z1 onto an (n1)-dimensional subspace. We assume that the body in question is an ndimensional convex polytope. When n = 3, this seems to be a reasonable assumption in the case of the space ship (see Figure 1). It is not hard to see that when n = 2 (so that we are projecting a convex polygon P onto various lines), the maximum projection-length is equal to P's diameter and the minimum projection-length is equal to P's width (the minimum distance between two parallel supporting lines of P) (see Figure 2). Thus the n-dimensional task considered here is one of several ways of extending to 114Z1 the classical Eu","PeriodicalId":142744,"journal":{"name":"Universität Trier, Mathematik/Informatik, Forschungsbericht","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1996-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130215018","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 : 1996-09-24DOI: 10.1007/3-540-61739-6_42
Christian Fecht, H. Seidl
{"title":"An Even Faster Solver for General Systems of Equations","authors":"Christian Fecht, H. Seidl","doi":"10.1007/3-540-61739-6_42","DOIUrl":"https://doi.org/10.1007/3-540-61739-6_42","url":null,"abstract":"","PeriodicalId":142744,"journal":{"name":"Universität Trier, Mathematik/Informatik, Forschungsbericht","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1996-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125725254","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 : 1995-05-01DOI: 10.1112/PLMS/S3-70.3.691
P. Gritzmann, V. Klee, J. Westwater
As the terms are used here, a body in R is a compact convex set with non-empty interior, and a polytope is a body that has only finitely many extreme points. The class of all bodies whose interior includes the origin 0 is denoted by %%. A set X is symmetric if X = -X. The ray-oracle of a body C e "#({ is the function 0c which, accepting as input an arbitrary ray R issuing from 0, produces the point at which R intersects the boundary of C. This paper is concerned with a few central aspects of the following general question: given certain information about C, what additional information can be obtained by questioning the ray-oracle, and how efficiently can it be obtained? It is assumed that infinite-precision real arithmetic and the usual vector operations in U are available at no cost, so the efficiency of an algorithm is measured solely in terms of its number of calls to the ray-oracle. The paper discusses two main problems, the first of which—the containment problem—arose from a question in abstract numerical analysis. Here the goal is to construct a polytope P (not necessarily in any sense a small one) that contains C, where this requires precise specification of the vertices of P. There are some sharp positive results for the case in which d = 2 and C is known not to be too asymmetric, but the main result on the containment problem is negative. It asserts that when d 2 3 and the body is known only to be rotund and symmetric, there is no algorithm for the containment problem. This is the case even when there is available a certain master oracle whose questionanswering power far exceeds that of the ray-oracle. However, it turns out that even when there is no additional information about C, the following relaxation of the containment problem admits an algorithmic solution based solely on the ray-oracle: construct a polytope containing C or conclude that the centred condition number of C exceeds a prescribed bound. In the other main problem—the reconstruction problem— it is known only that C is itself a polytope and the problem is to construct C with the aid of a finite number of calls to the ray-oracle. That is accomplished with a number of calls that depends on the number of faces (and hence on the 'combinatorial complexity') of C.
{"title":"Polytope Containment and Determination by Linear Probes","authors":"P. Gritzmann, V. Klee, J. Westwater","doi":"10.1112/PLMS/S3-70.3.691","DOIUrl":"https://doi.org/10.1112/PLMS/S3-70.3.691","url":null,"abstract":"As the terms are used here, a body in R is a compact convex set with non-empty interior, and a polytope is a body that has only finitely many extreme points. The class of all bodies whose interior includes the origin 0 is denoted by %%. A set X is symmetric if X = -X. The ray-oracle of a body C e \"#({ is the function 0c which, accepting as input an arbitrary ray R issuing from 0, produces the point at which R intersects the boundary of C. This paper is concerned with a few central aspects of the following general question: given certain information about C, what additional information can be obtained by questioning the ray-oracle, and how efficiently can it be obtained? It is assumed that infinite-precision real arithmetic and the usual vector operations in U are available at no cost, so the efficiency of an algorithm is measured solely in terms of its number of calls to the ray-oracle. The paper discusses two main problems, the first of which—the containment problem—arose from a question in abstract numerical analysis. Here the goal is to construct a polytope P (not necessarily in any sense a small one) that contains C, where this requires precise specification of the vertices of P. There are some sharp positive results for the case in which d = 2 and C is known not to be too asymmetric, but the main result on the containment problem is negative. It asserts that when d 2 3 and the body is known only to be rotund and symmetric, there is no algorithm for the containment problem. This is the case even when there is available a certain master oracle whose questionanswering power far exceeds that of the ray-oracle. However, it turns out that even when there is no additional information about C, the following relaxation of the containment problem admits an algorithmic solution based solely on the ray-oracle: construct a polytope containing C or conclude that the centred condition number of C exceeds a prescribed bound. In the other main problem—the reconstruction problem— it is known only that C is itself a polytope and the problem is to construct C with the aid of a finite number of calls to the ray-oracle. That is accomplished with a number of calls that depends on the number of faces (and hence on the 'combinatorial complexity') of C.","PeriodicalId":142744,"journal":{"name":"Universität Trier, Mathematik/Informatik, Forschungsbericht","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1995-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131444059","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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