Model Discovery and Verification With Symbolic, Hybrid Symbolic-Numeric and Parallel Computation

使用符号、混合符号数值和并行计算进行模型发现和验证

• 批准号: 0830347
• 负责人: Erich Kaltofen
• 金额: $ 30万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0830347&HistoricalAwards=false
• 关键词: Model; Discovery; Verification; Symbolic; Hybrid

ABSTRACT:A mathematical model for a natural (e.g., the heart beat of ahuman) or man-made process (e.g., the radio wave of a wirelesssignal) is a mathematical expression or an algorithm that onevaluation of system parameters (such as a point in time) yieldsa model value (e.g., the amplitude of a wave). Models arecreated by understanding the process, which suggests the formof the expression, and by observing and measuring an actualprocess. From those data points the best model is fitted bya computation. Erich Kaltofen studies both how to fit to datacertain models, such as fractions of sparse polynomials, andthen how to certify that the computation has produced the bestpossible model. The algorithms for creation of best fits andsubsequent certification of optimality can be compute-intensiveand require multi-processor computing environments.Erich Kaltofen and his students and collaborators will designalgorithms for symbolic models such as sparse multivariaterational functions and formulas with very large and evenparametric exponents. Our algorithms can work with both exactand approximate data, the latter by hybrid symbolic/numerictechniques. Computation with floating point scalars requiresa new kind of probabilistic analysis when randomization isapplied, and we will make use of recent results on estimatingthe spectra and condition numbers of random matrices. Oneapplication of such randomization is the efficient solutionof highly under- and overdetermined dense linear systems.A new alternative to error analysis is the exact validation viasymbolic computation of the global optimality of our approximatesolutions. Semidefinite programming and Newton refinementare used to compute a numerical sum-of-squares representation,which is converted to an exact rational identity for a nearbyrational lower bound. Since the exact certificates leave nodoubt, the numeric heuristics need not be fully analyzed.We will search for rationalizations that can validate verylarge sums-of-squares and hence apply to large inputs. We willdevelop parallel and distribute computing tools for the arisingsymbolic and hybrid symbolic-numeric computation tasks.

摘要：自然的数学模型（例如，Ahuman的心跳）或人造过程（例如，无线信号的无线电波）是数学表达式或算法，即系统参数的一个值（例如，时间）产生模型值（例如，波的幅度）。 通过了解该过程的模型，这表明表达式的形式，并观察和测量实际程序。 从这些数据点来看，最佳模型拟合了ByA计算。 Erich Kaltofen研究了如何适合数据模型，例如稀疏多项式的分数，然后如何证明该计算产生了最佳的模型。 创建最佳拟合和最佳认证的算法可以进行计算，需要多处理计算机。ErichKaltofen和他的学生和合作者将为符号模型（例如稀疏的多发性功能和非常大的ex​​terants）设计符号模型。 我们的算法可以与精确的数据一起使用，后者通过混合符号/numerictechniques来使用。 随机分配随机化时，使用浮点标量计算需要新型的概率分析，我们将利用最新结果来估计随机矩阵的光谱和条件数量。 这种随机化的启用是高度不足和过度确定的致密线性系统的有效解决方案。误差分析的新替代方案是我们近似值的全局最优性的确切验证viasymbolic计算。 半决赛编程和Newton Realementare用于计算数值平衡表示表示，该表示将转换为附近下层界限的精确合理身份。 由于确切的证书离开了Nodoubt，因此不需要对数字启发式方法进行全面分析。我们将搜索可以验证非常长的平方和规则的合理化，因此适用于大型输入。 我们将平行开发并分发用于Arisingsymbolic和混合符号数字计算任务的计算工具。