擬插值在函式逼近理論及其套用中起著重要的作用。擬插值的一個最大優點在於它不需要求解任何線性方程組就能夠直接給出逼近函式。
基本介紹
- 中文名：擬插值的若干理論及其套用
- 類別：論文
- 摘要：擬插值理論及其套用
- 關鍵字：擬插值、Strang-Fix條件
中文摘要,外文摘要,
中文摘要
而且，一些擬插值甚至還具有保形性(例如MQ擬插值、B-樣條擬插值等)。另外，和插值相比，擬插值還具有計算穩定、計算量小等其它優點。特別地，當採樣信息含有噪聲時，擬插值還能夠過濾噪聲。因而，擬插值無論在理論上還是在實際套用中都受到了廣泛的研究。
但是，大多數對擬插值的研究往往針對採樣信息是離散函式值(或離散函式值的有限個線性組合)的情形。由於在實際套用中，採樣信息通常更多的是線性泛函信息(某種微分方程右端項的離散值...>> 詳細
但是，大多數對擬插值的研究往往針對採樣信息是離散函式值(或離散函式值的有限個線性組合)的情形。由於在實際套用中，採樣信息通常更多的是線性泛函信息(某種微分方程右端項的離散值...>> 詳細
擬插值在函式逼近理論及其套用中起著重要的作用。擬插值的一個最大優點在於它不需要求解任何線性方程組就能夠直接給出逼近函式。而且，一些擬插值甚至還具有保形性(例如MQ擬插值、B-樣條擬插值等)。另外，和插值相比，擬插值還具有計算穩定、計算量小等其它優點。特別地，當採樣信息含有噪聲時，擬插值還能夠過濾噪聲。因而，擬插值無論在理論上還是在實際套用中都受到了廣泛的研究。
但是，大多數對擬插值的研究往往針對採樣信息是離散函式值(或離散函式值的有限個線性組合)的情形。由於在實際套用中，採樣信息通常更多的是線性泛函信息(某種微分方程右端項的離散值)而不是離散函式值(例如方程數值解、遙感、地震數據等)。因此，為了使得擬插值能夠套用到更多領域，討論針對線性泛函信息的擬插值將更有意義。
另一方面，當用擬插值求解微分方程數值解時，往往需要用擬插值逼近高階導數。這將降低數值解的逼近階。因此，為了得到具有高階逼近階的數值解，就需要構造這樣的一個擬插僮格式：它可以由方程右端項的離散值及邊界條件(初始條件或者其它的條件)直接給出方程的數值解，而不需要逼近高階導數。
基於以上兩點，本論文重點研究針對線性泛函信息的擬插值的構造及其套用。
我們構造一個針對線性泛函信息的擬插值格式並給出它的誤差估計。根據誤差估計，可以找到一個擬插值運算元，使得它能夠提供與微分方程右端項的光滑階一致的最佳逼近階。
作為套用，我們用它分別去求解微分方程數值和構造動力系統中的Lyapunov函式。
理論和數值結果都表明這個格式能夠克服無格線配置法的缺陷：求解大型線性方程組、計算不穩定等。
在一些場合，採樣信息往往具有周期性，例如信號處理、醫學圖像處理等。這時，顯然用周期函式來擬合這種信息將更加合理。
三角B-樣條擬插值可以擬合周期採樣信息。但是，由於三角B-樣條擬插值以三角B-樣條基為核函式，它的光滑階低。因此，如果用三角B-樣條擬插值來逼近高階導數，就需要用高階的三角B-樣條基為核函式。這即意味著需要計算高階的廣義差商，從而導致計算不穩定。
為了克服三角B-樣條擬插值的缺陷，本論文最後構造了一個無窮次光滑的周期擬插值格式．這個格式不僅可以擬合周期採樣信息，而且還可以逼近高階導數。另外，由於這個格式的核函式的構造只需要二階廣義差商，它避免了高階廣義差商的不穩定性。
作為套用，我們用它分別逼近一個函式自身、一階導數、二階導數，以及求解與時間有關的偏微分方程。數值結果表明我們構造的擬插值格式不僅能夠很好地逼近函式，而且對導數也有較好的逼近性質。
關鍵字：擬插值；Strang-Fix條件；徑向基函式；無格線配置法；MQ三角樣條擬插值；微分方程數值解；動力系統；Lyapunov函式。
但是，大多數對擬插值的研究往往針對採樣信息是離散函式值(或離散函式值的有限個線性組合)的情形。由於在實際套用中，採樣信息通常更多的是線性泛函信息(某種微分方程右端項的離散值)而不是離散函式值(例如方程數值解、遙感、地震數據等)。因此，為了使得擬插值能夠套用到更多領域，討論針對線性泛函信息的擬插值將更有意義。
另一方面，當用擬插值求解微分方程數值解時，往往需要用擬插值逼近高階導數。這將降低數值解的逼近階。因此，為了得到具有高階逼近階的數值解，就需要構造這樣的一個擬插僮格式：它可以由方程右端項的離散值及邊界條件(初始條件或者其它的條件)直接給出方程的數值解，而不需要逼近高階導數。
基於以上兩點，本論文重點研究針對線性泛函信息的擬插值的構造及其套用。
我們構造一個針對線性泛函信息的擬插值格式並給出它的誤差估計。根據誤差估計，可以找到一個擬插值運算元，使得它能夠提供與微分方程右端項的光滑階一致的最佳逼近階。
作為套用，我們用它分別去求解微分方程數值和構造動力系統中的Lyapunov函式。
理論和數值結果都表明這個格式能夠克服無格線配置法的缺陷：求解大型線性方程組、計算不穩定等。
在一些場合，採樣信息往往具有周期性，例如信號處理、醫學圖像處理等。這時，顯然用周期函式來擬合這種信息將更加合理。
三角B-樣條擬插值可以擬合周期採樣信息。但是，由於三角B-樣條擬插值以三角B-樣條基為核函式，它的光滑階低。因此，如果用三角B-樣條擬插值來逼近高階導數，就需要用高階的三角B-樣條基為核函式。這即意味著需要計算高階的廣義差商，從而導致計算不穩定。
為了克服三角B-樣條擬插值的缺陷，本論文最後構造了一個無窮次光滑的周期擬插值格式．這個格式不僅可以擬合周期採樣信息，而且還可以逼近高階導數。另外，由於這個格式的核函式的構造只需要二階廣義差商，它避免了高階廣義差商的不穩定性。
作為套用，我們用它分別逼近一個函式自身、一階導數、二階導數，以及求解與時間有關的偏微分方程。數值結果表明我們構造的擬插值格式不僅能夠很好地逼近函式，而且對導數也有較好的逼近性質。
關鍵字：擬插值；Strang-Fix條件；徑向基函式；無格線配置法；MQ三角樣條擬插值；微分方程數值解；動力系統；Lyapunov函式。
外文摘要
Quasi-interpolation plays a vital role in approximation theory and its applications. One major advantage of quasi-interpolation is that it can yield an approximant directly without the need to solve any large-scale linear system of equations. Moreover, some quasi-interpolation can even possess shape-preserving properties (such as the MQ quasi-interpolation, the spline quasi-interpolation, etc). Besides, compared with interpolation, quasi-interpolation has some other advantages such as stabilit...>> 詳細
Quasi-interpolation plays a vital role in approximation theory and its applications. One major advantage of quasi-interpolation is that it can yield an approximant directly without the need to solve any large-scale linear system of equations. Moreover, some quasi-interpolation can even possess shape-preserving properties (such as the MQ quasi-interpolation, the spline quasi-interpolation, etc). Besides, compared with interpolation, quasi-interpolation has some other advantages such as stability of computation, a small amount of computation, etc. Particularly, when the sampling data are noised, quasi-interpolation can also filter the noise. Thus, quasi-interpolation has been studied extensively both in theory and in practical applications.
However, most studies of quasi-interpolation are usually for the case when the sampling data are discrete function values (or a finite linear combination of discrete function values). Note that in practical applications, more commonly, the sampling data are linear functional data (the discrete values of the right-hand side of some differential equation) rather than the discrete function values (such as solving differential equations, remote sensing, seismic data, etc). Therefore, to make quasi-interpolation more available for practical applications, it is more meaningful to discuss quasi-interpolation for the linear functional data.
On the other hand, when solving the numerical solution of a differential equation with quasi-interpolation, one usually use quasi-interpolation to approximate high-order derivatives. This will reduce the approximation order of the numerical solution. Thus, to get a numerical solution with high approximation order, one needs to construct such a quasi-interpolation scheme that can provide the numerical solution directly from the discrete values of the right-hand side of the equation coupled with the boundary conditions (initial conditions or others) without the need to approximate high-order derivatives.
Based on the above two points, this dissertation mainly studies the construction of quasiinterpolation for the linear functional data and its applications.
We construct a quasi-interpolation scheme for the linear functional data and give its error estimate. Based on the error estimate, one can find a quasi-interpolant that provides an optimal approximation order with respect to the smoothness of the right-hand side of the differential equation.
As applications, we apply it in solving the numerical solutions of differential equations and constructing the Lyapunov function in dynamical systems, respectively.
Both theory and numerical results show that the scheme can overcome the drawbacks of the meshless collocation method: solving large-scale linear system of equations, unstability of computation, etc.
In some cases, the sampling data usually possess periodicity, such as signal processing, medical image processing, etc. Then, obviously, it is more reasonable to fit the data with a periodic function.
Trigonometric B-splines quasi-interpolation can fit the periodic sampling data. However, the order of smoothness of trigonometric B-splines quasi-interpolation is low, since it uses trigonometric B-splines as its kernel functions. Thus, if approximating high-order derivatives with trigonometric B-splines quasi-interpolation, one should use high-order trigonometric Bsplines as the kernel functions. This implies computation of high-order generalized divided differences, which leads to the unstability of computation.
To overcome the drawbacks of trigonometric B-splines quasi-interpolation, a periodic quasi-interpolation scheme with infinite smoothness is constructed in the last part of this dissertation. The scheme can not only fit the periodic sampling data, but also approximate highorder derivatives. Moreover, since the construction of the kernel functions of the scheme only requires second-order generalized divided differences, it avoids the unstability of high-order generalized divided differences.
As applications, we apply it in approximating a function, the first-order derivative, the second-order derivative of the function, and solving the numerical solution of time-dependent partial differential equations, respectively. Numerical results show that the scheme can not only provide an excellent approximation to the function, but also give an approximation to derivatives of the function.
Key Words: Quasi-interpolation; Strang-Fix Condition; Radial Basis Function; Meshless Collocation Method; MQ Trigonometric Spline Quasi-interpolation; Numerical Solution of Differential Equations; Dynamical System; Lyapunov Function.
However, most studies of quasi-interpolation are usually for the case when the sampling data are discrete function values (or a finite linear combination of discrete function values). Note that in practical applications, more commonly, the sampling data are linear functional data (the discrete values of the right-hand side of some differential equation) rather than the discrete function values (such as solving differential equations, remote sensing, seismic data, etc). Therefore, to make quasi-interpolation more available for practical applications, it is more meaningful to discuss quasi-interpolation for the linear functional data.
On the other hand, when solving the numerical solution of a differential equation with quasi-interpolation, one usually use quasi-interpolation to approximate high-order derivatives. This will reduce the approximation order of the numerical solution. Thus, to get a numerical solution with high approximation order, one needs to construct such a quasi-interpolation scheme that can provide the numerical solution directly from the discrete values of the right-hand side of the equation coupled with the boundary conditions (initial conditions or others) without the need to approximate high-order derivatives.
Based on the above two points, this dissertation mainly studies the construction of quasiinterpolation for the linear functional data and its applications.
We construct a quasi-interpolation scheme for the linear functional data and give its error estimate. Based on the error estimate, one can find a quasi-interpolant that provides an optimal approximation order with respect to the smoothness of the right-hand side of the differential equation.
As applications, we apply it in solving the numerical solutions of differential equations and constructing the Lyapunov function in dynamical systems, respectively.
Both theory and numerical results show that the scheme can overcome the drawbacks of the meshless collocation method: solving large-scale linear system of equations, unstability of computation, etc.
In some cases, the sampling data usually possess periodicity, such as signal processing, medical image processing, etc. Then, obviously, it is more reasonable to fit the data with a periodic function.
Trigonometric B-splines quasi-interpolation can fit the periodic sampling data. However, the order of smoothness of trigonometric B-splines quasi-interpolation is low, since it uses trigonometric B-splines as its kernel functions. Thus, if approximating high-order derivatives with trigonometric B-splines quasi-interpolation, one should use high-order trigonometric Bsplines as the kernel functions. This implies computation of high-order generalized divided differences, which leads to the unstability of computation.
To overcome the drawbacks of trigonometric B-splines quasi-interpolation, a periodic quasi-interpolation scheme with infinite smoothness is constructed in the last part of this dissertation. The scheme can not only fit the periodic sampling data, but also approximate highorder derivatives. Moreover, since the construction of the kernel functions of the scheme only requires second-order generalized divided differences, it avoids the unstability of high-order generalized divided differences.
As applications, we apply it in approximating a function, the first-order derivative, the second-order derivative of the function, and solving the numerical solution of time-dependent partial differential equations, respectively. Numerical results show that the scheme can not only provide an excellent approximation to the function, but also give an approximation to derivatives of the function.
Key Words: Quasi-interpolation; Strang-Fix Condition; Radial Basis Function; Meshless Collocation Method; MQ Trigonometric Spline Quasi-interpolation; Numerical Solution of Differential Equations; Dynamical System; Lyapunov Function.
