本研究應用灰色系統理論之灰預測方法(Grey Prediction Method)來改善逆向矩陣法(Reversed Matrix Method)於分析熱傳逆問題(Inverse Heat Transfer Problems, IHTP)時，因溫度量測誤差所衍生的逆算誤差問題。
於分析熱傳逆問題，本文採用逆向矩陣法配合線性最小均方根誤差法(Linear Least-squares Error Method)來建構一線性逆模型。以有限差分法將欲求解之熱傳逆問題統御方程式離散化(Discretization)建構一線性矩陣方程式。藉由重新排列矩陣方程式，使得未知條件(如初始條件、邊界條件、熱物性或幾何形狀等)能以獨立明確地表示出來；再以少量的溫度量測點數代入此線性逆模型中，利用線性最小均方根誤差法求得其解。
於逆解過程中只需量取少量溫度即可推算出熱傳逆問題之解，但是在實際量測溫度時，必定會產生不可避免的溫度量測誤差。此量測誤差將會影響逆解運算估計值的準確度，甚至會導致錯誤的結果。其改善方式之一為增加溫度量測點數，當然若要得到更準確的逆算結果其所需要增加的量測點數也就要越多。因此本文引用灰預測方法來嘗試改善此缺點，以期能大量的減少實際溫度量測的點數，也同樣能獲得準確的逆算結果。
灰預測方法可以把少量的直接量測溫度(通常只要數點即可)擴增到較多的溫度點數，而且這些擴增點的溫度仍然保有先前直接量測溫度的關聯性。用這些擴增的溫度點數即可取代逆解運算所需要增加的溫度點數，以達到改善因溫度量測誤差所造成的逆算結果錯誤。換句話說，灰預測方法可以使實際溫度量測點數大量的減少，一樣能夠達到原有使用大量直接量測溫度逆算結果的準確度。

This article applies Grey Prediction Method of Grey System Theory to improve the problem of errors in inverse operation due to the error of temperature measurement when analyze Inverse Heat Transfer Problems (IHTP) with Reversed Matrix Method.
For IHTP, this research adopted Revered Matrix Method with Linear Least-squares Error Method to construct a linear inverse model. With finite difference method, we discretized governing equation that is designed to solve IHTP to construct a linear matrix equation. Through the re-arrangement of matrix equation, the unknown conditions (such as initial conditions, boundary conditions, thermal property or geometrical shape) could be demonstrated clearly and independently. Then substitute a small amount of successive measuring points temperature into the linear inverse model and solve the problems by Linear Least-squares Error Method.
The process of inverse operation only need to measure a small amount points temperature to estimate the solution of IHTP, but in practical measurement of temperature, the errors of measurement of temperature are inevitable. Such errors will affect the accuracy of estimation value of inverse operation or even lead to an erroneous results. One of improvement method is to increase the number of temperature measurement points. Certainly, more accurate results of inverse operation we want to obtain, the number of measurement points we should increase. Therefore, this research uses the Grey Prediction Method to improve the defect with a hope that significant reduction of the number of practical temperature measurement points could also obtain the same accurate results of inverse operation.
The small amount of direct temperature measurement points can increase to more amount of temperature points by Grey Prediction Method, and the temperatures of those increased points could still keep the correlations with previous temperatures from direct measurement. The increased number of temperature points could replace the number of temperature points that is necessary to increase for inverse operation. In other words, Grey Prediction Method could significantly reduce the number of practical temperature measurement points while keep the same accuracy as the results of inverse operation using a great number of direct temperature measurement points.

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