作者llrabel (不屈不撓)
看板Math
標題Re: 是變分法嗎？
時間Wed Jan 15 01:27:31 2014
※ 引述《physmd (smd)》之銘言：
: 我變分法是很多年前學的，而且只懂皮毛......
: 底下這個問題不好意思我用英文輸入啦 :P
: 有請版友指點，多謝～～
: Consider real valued functions in the closed domain [0, 1]
: Find f(x) that minimizes: Integrate{ g(x)^2 / f(x) },
: with respect to a given g(x), integrate over x from 0 to 1,
: subject to the constraints
: (1) f > 0 for all x in [0, 1]
: (2) |f| = 1, that is, Integrate{ f(x) } over x = 0 to 1 is unity.
: Suppose everything you need to know about g(x) is completely specified.
: (the derivative or anti-derivative of g(x) etc)
: In general g(x) can be just some real valued integrable function (that doesn't
: even have to be piece-wise continuous), but if needed we can focus on the solution
: f(x) for some smooth g(x).
: p.s.
: Some of you might notice that f is a probability density.
: In fact, this question is simplified from a question regarding minimizing
: the variance of some stuff.
是變分沒錯, 不要太要求嚴謹, 就像下面這樣做.
Let f(x) be a function which satisfies (1), (2) and minimizes
Int{ g(x)^2 / f(x) }. Let h(x) be any smooth function satisfying
Int{ h(x) }=0. Then f(x)+εh(x) also satisfies (1) and (2)
when ε is very close to zero.
Therefore, the function I(ε) := Int{ g(x)^2 / (f(x)+εh(x)) } has a
local maximum at ε=0, which implies I'(0)=0.
By direct computation, we find this means
Int{ [g(x)^2/f(x)^2]．h(x) } = 0.
Note that this is true for arbitrary smooth function h(x)
satisfying Int{h(x)}=0. The remaining problem is to ask what f(x) makes
this possible.
By observation, we find it's true when
g(x)^2/f(x)^2 is a constant. Hence f(x)=c|g(x)|,
where c is the constant so that (2) holds.
Precisely, c = 1/Int{|g(x)|}.
--
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◆ From: 111.243.114.220
→ wohtp :第五行，I的定義裡面，分母多了個平方 01/15 04:42
已修正, 謝謝.
→ wohtp :最後的答案是沒錯 01/15 04:43
推 yw1002 :是沒誠意還是小看？意思就是原本比較難，把簡化的 01/15 05:31
→ yw1002 :來問大家 以免大家不會 01/15 05:32
推 suhorng :原原PO應該是說把原本問題的其中一步抽取出來吧... 01/15 11:05
※ 編輯: llrabel 來自: 140.109.105.70 (01/15 15:16)
※ 編輯: llrabel 來自: 140.109.105.70 (01/15 15:17)
推 jacky7987 :... 01/15 22:39
推 physmd :喔多謝啦，我還想說要找什麼公式檢查條件來代一代 01/16 04:11
→ physmd :其實還是回歸基本定義最好了. 01/16 04:11
→ physmd :喔那個 p.s. 要講的是如果把我po的問題轉化成機率 01/16 04:13
→ physmd :問題來看的話其實是走回頭路....除非有另外的妙解. 01/16 04:14
→ physmd :Anyway, thanks guys. 01/16 04:14
→ JASS0213 :其實這個寫法已經非常嚴謹了。硬要多說一句就是最後 01/16 05:29
→ JASS0213 :一步，把h取成在兩個相異點附近rescale的那種appox. 01/16 05:31
→ JASS0213 :dirac family。然後你就可以證明f=c|g| a.e 01/16 05:33
→ willydp :小問題: 1.最大化的f存在 2.g在一些地方=0怎麼辦? 01/16 08:29
→ willydp : 小 01/16 08:30
→ willydp :根據內文的論證, 頂多只能說f是個critical pt而已 01/16 08:32
→ willydp :我想“不嚴謹”這幾個字是這個意思吧? 嚴謹的變分法 01/16 08:37
→ willydp :還需要多一點的分析. 原本的作法已能讓很多人信服了. 01/16 08:39
→ JASS0213 :我覺得各位把問題搞複雜了 01/16 19:42
→ JASS0213 :這只是柯西不等式罷了 01/16 19:42