{"id":812,"date":"2017-03-05T09:08:25","date_gmt":"2017-03-05T01:08:25","guid":{"rendered":"http:\/\/planckscale.info\/?p=812"},"modified":"2017-03-06T21:37:38","modified_gmt":"2017-03-06T13:37:38","slug":"notes-on-conjugate-gradient-method-2","status":"publish","type":"post","link":"https:\/\/mindspectrum.xyz\/en\/2017\/03\/05\/notes-on-conjugate-gradient-method-2\/","title":{"rendered":"Notes on Conjugate Gradient Method – 2"},"content":{"rendered":"
\u7248\u6743\u58f0\u660e\uff1a\u539f\u521b\u4f5c\u54c1\uff0c\u6b22\u8fce\u8f6c\u8f7d\uff0c\u4f46\u8f6c\u8f7d\u8bf7\u4ee5\u8d85\u94fe\u63a5\u5f62\u5f0f\u6ce8\u660e\u6587\u7ae0\u6765\u6e90(planckscale.info<\/a>)\u3001\u4f5c\u8005\u4fe1\u606f\u548c\u672c\u58f0\u660e\uff0c\u5426\u5219\u5c06\u8ffd\u7a76\u6cd5\u5f8b\u8d23\u4efb\u3002<\/strong><\/h5>\n

\u6253\u7b97\u518d\u5c06\u8fd9\u4e2a\u4e3b\u9875\u505a\u4e00\u505a\uff0c\u5148\u66f4\u4e00\u6ce2\u65e7\u7b14\u8bb0\u3002\u8fd8\u662f\u90a3\u53e5\u8bdd\uff0c\u8fd9\u4e0d\u662f\u6559\u7a0b\uff0c\u4e0d\u9002\u5408\u521d\u5b66\u3002\u672c\u6587\u4e2d\u8bb0\u53f7\u5927\u81f4\u4e0e\u524d\u4e00\u7bc7Notes on Conjugate Gradient Method<\/a>\u4e00\u81f4\u3002<\/p>\n

\u5f53\"\mat{A}\"\u5bf9\u79f0\u6b63\u5b9a\u65f6\uff0c\u5171\u8f6d\u68af\u5ea6\u6cd5\u662f\u4e00\u4e2a\u5f88\u597d\u7684\u9009\u62e9\u3002<\/p>\n

\u6b64\u65f6\u7684\u6781\u5c0f\u5316\u95ee\u9898\u5176\u5b9e\u662f\u4e00\u4e2a\"\mat{x}\"\u4e0e\"\mat{x}^{*}\"\u95f4\u7684\u8ddd\u79bb\u6781\u5c0f\u5316\u95ee\u9898<\/p>\n

(1) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u4ee5\"\mat{A}\"\u4e3a\u5ea6\u91cf\uff0c\u5728\u4e00\u4e2a\"\mat{A}\"\u6b63\u4ea4\u6807\u67b6\u4e0b\uff0c\u4e0a\u8ff0\u8ddd\u79bb\u6781\u5c0f\u5316\u95ee\u9898\u6210\u4e3a\u4e00\u4e2a\u7b80\u5355\u7684\u6b27\u5f0f\u8ddd\u79bb\u6781\u5c0f\u5316\u95ee\u9898\uff0c\u53ea\u9700\u5728\u4f9d\u6b21\u5404\"\mat{A}\"\u6b63\u4ea4\u65b9\u5411\u4e0a\u6781\u5c0f\u5316\u8ddd\u79bb\u5373\u53ef\u3002
\n\u8fd9\u9700\u8981\u89e3\u51b3\u4ee5\u4e0b\u95ee\u9898\uff1a
\n\\begin{enumerate}
\n\\item \u4e00\u4e2a\u641c\u7d22\u65b9\u5411\u4e0a\u5982\u4f55\u6781\u5c0f\u5316\"\lVert \mat{x}-\mat{x}^{*} \rVert_{\mat{A}}\"\u3002
\n\\item \u5982\u4f55\u9ad8\u6548\u7684\u6784\u9020\u51fa\u4e0e\u4e4b\u524d\u641c\u7d22\u8fc7\u7684\u5b50\u7a7a\u95f4\"\mat{A}\"\u6b63\u4ea4\u7684\u65b0\u641c\u7d22\u65b9\u5411\uff1b
\n\\end{enumerate}<\/p>\n

\u7b2c\u4e00\u4e2a\u95ee\u9898\u3002\u7531\u7b80\u5355\u7684\u51e0\u4f55\u76f4\u89c2\uff0c\u8ddd\u79bb\u7684\u6781\u5c0f\u5316\u7b49\u4ef7\u4e8e
\n<\/a><\/p>\n

(2) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5176\u4e2d\"D_{n}=\mathrm{span}\{ \mat{d}^{(0)},\dots,\mat{d}^{(n-1)} \}\"\u8868\u793a\"\mat{x}=\mat{x}^{(n)}\"\u65f6\u5df2\u7ecf\u641c\u7d22\u8fc7\u7684\u5b50\u7a7a\u95f4\u3002<\/p>\n

\u672c\u5f0f\"D_{n}\"\u6362\u6210\"\mat{d}^{(n)}\"\u540c\u6837\u6210\u7acb\u3002\u6240\u4ee5\uff0c\u53ea\u9700\u641c\u7d22\u5230\"\mat{r}^{(n)}\"\u6b63\u4ea4\u4e8e\u641c\u7d22\u65b9\u5411\"\mat{d}^{(n)}\"\u65f6\u4e3a\u6b62\u5373\u53ef\u8fbe\u5230\u6781\u5c0f\u5316\u8ddd\u79bb\u7684\u76ee\u7684\u3002<\/p>\n

\u7b2c\u4e8c\u4e2a\u95ee\u9898\u3002\u6211\u4eec\u4ee4\"D_{n}=\mathcal{K}_{m}(\mat{A},\mat{r}^{(0)})\",\u5176\u4e2d\"\mat{r}^{(0)}\"\u4e3a\u521d\u59cb\u6b8b\u5dee\uff0c\u4e8e\u662f\u6709\"D_{n}=\mathrm{span}\{ \mat{r}^{(0)},\mat{A}D_{n-1} \}\".\u8fd9\u6837\u5f0f2<\/a>\u5c31\u610f\u5473\u7740<\/p>\n

(3) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u8fd9\u5e26\u6765\u4e00\u4e2a\u6781\u5927\u7684\u4fbf\u5229\uff0c\u5373\u7b97\u6cd5\u53ef\u4ee5\u901a\u8fc7\"\mat{r}^{(n)}\"\u5feb\u901f\u6784\u9020\u51fa\u4e0e\"D_n\"\u5b50\u7a7a\u95f4\"\mat{A}\"\u6b63\u4ea4\u7684\"\mat{d}^{(n)}\",\u4e3a\u6b64\u53ea\u9700\u8981\u5c06\"\mat{r}^{(n)}\"\u4e2d\"\mat{d}^{(n-1)}\"\u65b9\u5411\u7684\u5206\u91cf\uff08\"\mat{A}\"\u5185\u79ef\u610f\u4e49\u4e0a\uff09\u6263\u9664\u5373\u53ef\u3002<\/p>\n

\u8fd9\u6837\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5171\u8f6d\u68af\u5ea6\u6cd5\u7684\u7b97\u6cd5\u6d41\u7a0b\uff1a\u6bcf\u6b21\u641c\u7d22\u90fd\u641c\u7d22\u5230\u6b8b\u5dee\u65b9\u5411\u4e0e\u641c\u7d22\u65b9\u5411\u6b63\u4ea4\u4e3a\u6b62\uff0c\u65b0\u7684\u641c\u7d22\u65b9\u5411\u7531\u5f53\u524d\u6b8b\u5dee\u7528\"\mat{A}\"\u5185\u79ef\u4e0e\"\{ \mat{d}^{(0)},\dots,\mat{d}^{(n-1)} \}\"\u505a\u6b63\u4ea4\u5316\u5f97\u5230\uff0c\u5b9e\u9645\u4e0a\u53ea\u9700\u8981\u6263\u9664\"\mat{d}^{(n-1)}\"\u65b9\u5411\u7684\u5206\u91cf\u5373\u53ef\u3002
\n\u7b97\u6cd5\u63cf\u8ff0\u4e3a<\/p>\n

(4) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5176\u4e2d\"\alpha\"\u7531\"\mat{r}^{(n+1)}\"\u548c\"\mat{d}^{(n)}\"\u7684\u6b63\u4ea4\u6027\u7ed9\u51fa<\/p>\n

(5) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u800c\"\beta\"\u5219\u8fd9\u6837\u8ba1\u7b97\u51fa<\/p>\n

(6) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5171\u8f6d\u68af\u5ea6\u6cd5\u5728\u52a0\u5165precondition\u65f6\u4f1a\u9047\u5230\u4e00\u4e2a\u95ee\u9898\uff0c\u5373\u5982\u4f55\u4fdd\u8bc1\"\mat{M}^{-1}\mat{A}\"\u4ecd\u662f\u5bf9\u79f0\u77e9\u9635\u3002
\n\u5bf9\u4e8e\u5bf9\u79f0\u6b63\u5b9a\u77e9\u9635\"\mat{M}\",\u5b58\u5728\u4e00\u4e2a\u5206\u89e3\"\mat{M}=\mat{E}\mat{E}^T\".\u5bb9\u6613\u8bc1\u660e\"\mat{M}^{-1}\mat{A}\"\u548c\"\mat{E}^{-1}\mat{A}\mat{E}^{-T}\"\u5177\u6709\u76f8\u540c\u7684\u672c\u5f81\u503c\uff0c\u56e0\u800c\u5177\u6709\u540c\u6837\u7684precondition\u6548\u679c\u3002\u4e8e\u662f\u53ef\u4ee5\u5c06\u95ee\u9898\u8f6c\u5316\u4e3a<\/p>\n

(7) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u518d\u7528CG\u6c42\u89e3\u3002\"\mat{E}^{-1}\mat{A}\mat{E}^{-T}\"\u7684\u51e0\u4f55\u610f\u4e49\u662f\u5c06\u95ee\u9898\u8f6c\u5165\u4e00\u4e2a\u65b0\u7684\u5750\u6807\u7cfb\u4e0b\u6c42\u89e3\uff0c\u8fd9\u4e2a\u5750\u6807\u7cfb\u4e0b\u4e8c\u6b21\u578b\u7684\u8d85\u692d\u7403\u88ab\u53d8\u6362\u7684\u66f4\u63a5\u8fd1\u8d85\u7403\u4e86\u4e00\u4e9b\u3002\u8fd9\u6837\u8981\u6c42\u8ba1\u7b97\u51fa\"\mat{E}\",\u4f46\u5b9e\u9645\u4e0a\u6211\u4eec\u53ef\u4ee5\u4e0d\u7528\u8ba1\u7b97\"\mat{E}\"\u800c\u76f4\u63a5\u7528\"\mat{M}\"\u5b8c\u6210\u8ba1\u7b97\u3002\u4e3a\u6b64\u53ea\u9700\u8981\u5c06\u539f\u6765\u7684\u8fed\u4ee3\u516c\u5f0f\u505a\u5750\u6807\u53d8\u6362\uff0c\u5199\u51fa\u65b0\u5750\u6807\u7cfb\u4e0b\u7684\u8fed\u4ee3\u516c\u5f0f\u5373\u53ef\u3002\u8981\u6ce8\u610f\u8fed\u4ee3\u516c\u5f0f\u5728\u8be5\u53d8\u6362\u4e0b\u5e76\u4e0d\u5b8c\u5168\u4fdd\u6301\u5f62\u5f0f\u4e00\u81f4\uff0c\u5177\u4f53\u7684\u5bf9\u5e94\u5173\u7cfb\u5982\u4e0b\u6240\u793a<\/p>\n

\"Rendered
\n\u6709\u7a0d\u8bb8\u5fae\u5206\u51e0\u4f55\u6216\u76f8\u5bf9\u8bba\u80cc\u666f\u7684\u540c\u5b66\u4e0d\u96be\u770b\u51fa\u8fd9\u5176\u5b9e\u662f\u4e00\u7ec4\u5728\u5750\u6807\u7cfb\u53d8\u6362\u4e0b\u5177\u6709\u4e0d\u540c\u53d8\u6362\u6027\u8d28\u7684\u91cf\u4e4b\u53d8\u6362\u5f0f\uff0c\u5305\u542b\u4e86\u534f\/\u9006\u53d8\u77e2\u91cf\uff0c\u6807\u91cf\u7684\u53d8\u6362\u6027\u8d28\uff0c\u4ee5\u53ca\u5982\u4f55\u5c06\u5f53\u524d\u5750\u6807\u7cfb\u4e0b\u5b9a\u4e49\uff0c\u975e\u5f62\u5f0f\u4e0d\u53d8\u7684\u91cf\"\langle \mat{r},\mat{r}^{\prime} \rangle\"\u63d0\u5347\u4e3a\u4efb\u610f\u5750\u6807\u7cfb\u4e0b\u5747\u4fdd\u6301\u5f62\u5f0f\u4e0d\u53d8\u7684\u91cf\"\langle \mat{r},\mat{M}^{-1}\mat{r}^{\prime} \rangle\"\u7684\u64cd\u4f5c.<\/p>\n

\u66ff\u6362\u540e\u53ef\u5c06\"\mat{E}\"\u5168\u90e8\u6d88\u9664\u6389\uff0c\u5269\u4e0b\u4ec5\u7528\"\mat{M}\"\u8868\u8fbe\u7684\u8fed\u4ee3\u516c\u5f0f<\/p>\n

(8) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n","protected":false},"excerpt":{"rendered":"

\u7248\u6743\u58f0\u660e\uff1a\u539f\u521b\u4f5c\u54c1\uff0c\u6b22\u8fce\u8f6c\u8f7d\uff0c\u4f46\u8f6c\u8f7d\u8bf7\u4ee5\u8d85\u94fe\u63a5\u5f62\u5f0f\u6ce8\u660e\u6587\u7ae0\u6765\u6e90(planckscale.info)\u3001\u4f5c\u8005\u4fe1\u606f\u548c\u672c\u58f0\u660e\uff0c\u5426\u5219\u5c06\u8ffd\u7a76\u6cd5\u5f8b\u8d23\u4efb\u3002 \u6253\u7b97\u518d\u5c06\u8fd9\u4e2a\u4e3b\u9875\u505a\u4e00\u505a\uff0c\u5148\u66f4\u4e00\u6ce2\u65e7\u7b14\u8bb0\u3002\u8fd8\u662f\u90a3\u53e5\u8bdd\uff0c\u8fd9\u4e0d\u662f\u6559\u7a0b\uff0c\u4e0d\u9002\u5408\u521d\u5b66\u3002\u672c\u6587\u4e2d\u8bb0\u53f7\u5927\u81f4\u4e0e\u524d\u4e00\u7bc7Notes on Conjugate Gradient Method\u4e00\u81f4\u3002 \u5f53\u5bf9\u79f0\u6b63\u5b9a\u65f6\uff0c\u5171\u8f6d\u68af\u5ea6\u6cd5\u662f\u4e00\u4e2a\u5f88\u597d\u7684\u9009\u62e9\u3002 \u6b64\u65f6\u7684\u6781\u5c0f\u5316\u95ee\u9898\u5176\u5b9e\u662f\u4e00\u4e2a\u4e0e\u95f4\u7684\u8ddd\u79bb\u6781\u5c0f\u5316\u95ee\u9898 (1)   \u4ee5\u4e3a\u5ea6\u91cf\uff0c\u5728\u4e00\u4e2a\u6b63\u4ea4\u6807\u67b6\u4e0b\uff0c\u4e0a\u8ff0\u8ddd\u79bb\u6781\u5c0f\u5316\u95ee\u9898\u6210\u4e3a\u4e00\u4e2a\u7b80\u5355\u7684\u6b27\u5f0f\u8ddd\u79bb\u6781\u5c0f\u5316\u95ee\u9898\uff0c\u53ea\u9700\u5728\u4f9d\u6b21\u5404\u6b63\u4ea4\u65b9\u5411\u4e0a\u6781\u5c0f\u5316\u8ddd\u79bb\u5373\u53ef\u3002 \u8fd9\u9700\u8981\u89e3\u51b3\u4ee5\u4e0b\u95ee\u9898\uff1a \\begin{enumerate} \\item \u4e00\u4e2a\u641c\u7d22\u65b9\u5411\u4e0a\u5982\u4f55\u6781\u5c0f\u5316\u3002 \\item \u5982\u4f55\u9ad8\u6548\u7684\u6784\u9020\u51fa\u4e0e\u4e4b\u524d\u641c\u7d22\u8fc7\u7684\u5b50\u7a7a\u95f4\u6b63\u4ea4\u7684\u65b0\u641c\u7d22\u65b9\u5411\uff1b \\end{enumerate} \u7b2c\u4e00\u4e2a\u95ee\u9898\u3002\u7531\u7b80\u5355\u7684\u51e0\u4f55\u76f4\u89c2\uff0c\u8ddd\u79bb\u7684\u6781\u5c0f\u5316\u7b49\u4ef7\u4e8e (2)   \u5176\u4e2d\u8868\u793a\u65f6\u5df2\u7ecf\u641c\u7d22\u8fc7\u7684\u5b50\u7a7a\u95f4\u3002 \u672c\u5f0f\u6362\u6210\u540c\u6837\u6210\u7acb\u3002\u6240\u4ee5\uff0c\u53ea\u9700\u641c\u7d22\u5230\u6b63\u4ea4\u4e8e\u641c\u7d22\u65b9\u5411\u65f6\u4e3a\u6b62\u5373\u53ef\u8fbe\u5230\u6781\u5c0f\u5316\u8ddd\u79bb\u7684\u76ee\u7684\u3002 \u7b2c\u4e8c\u4e2a\u95ee\u9898\u3002\u6211\u4eec\u4ee4,\u5176\u4e2d\u4e3a\u521d\u59cb\u6b8b\u5dee\uff0c\u4e8e\u662f\u6709.\u8fd9\u6837\u5f0f2\u5c31\u610f\u5473\u7740 (3)   \u8fd9\u5e26\u6765\u4e00\u4e2a\u6781\u5927\u7684\u4fbf\u5229\uff0c\u5373\u7b97\u6cd5\u53ef\u4ee5\u901a\u8fc7\u5feb\u901f\u6784\u9020\u51fa\u4e0e\u5b50\u7a7a\u95f4\u6b63\u4ea4\u7684,\u4e3a\u6b64\u53ea\u9700\u8981\u5c06\u4e2d\u65b9\u5411\u7684\u5206\u91cf\uff08\u5185\u79ef\u610f\u4e49\u4e0a\uff09\u6263\u9664\u5373\u53ef\u3002 \u8fd9\u6837\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5171\u8f6d\u68af\u5ea6\u6cd5\u7684\u7b97\u6cd5\u6d41\u7a0b\uff1a\u6bcf\u6b21\u641c\u7d22\u90fd\u641c\u7d22\u5230\u6b8b\u5dee\u65b9\u5411\u4e0e\u641c\u7d22\u65b9\u5411\u6b63\u4ea4\u4e3a\u6b62\uff0c\u65b0\u7684\u641c\u7d22\u65b9\u5411\u7531\u5f53\u524d\u6b8b\u5dee\u7528\u5185\u79ef\u4e0e\u505a\u6b63\u4ea4\u5316\u5f97\u5230\uff0c\u5b9e\u9645\u4e0a\u53ea\u9700\u8981\u6263\u9664\u65b9\u5411\u7684\u5206\u91cf\u5373\u53ef\u3002 \u7b97\u6cd5\u63cf\u8ff0\u4e3a (4)   \u5176\u4e2d\u7531\u548c\u7684\u6b63\u4ea4\u6027\u7ed9\u51fa (5)   \u800c\u5219\u8fd9\u6837\u8ba1\u7b97\u51fa (6)   \u5171\u8f6d\u68af\u5ea6\u6cd5\u5728\u52a0\u5165precondition\u65f6\u4f1a\u9047\u5230\u4e00\u4e2a\u95ee\u9898\uff0c\u5373\u5982\u4f55\u4fdd\u8bc1\u4ecd\u662f\u5bf9\u79f0\u77e9\u9635\u3002 \u5bf9\u4e8e\u5bf9\u79f0\u6b63\u5b9a\u77e9\u9635,\u5b58\u5728\u4e00\u4e2a\u5206\u89e3.\u5bb9\u6613\u8bc1\u660e\u548c\u5177\u6709\u76f8\u540c\u7684\u672c\u5f81\u503c\uff0c\u56e0\u800c\u5177\u6709\u540c\u6837\u7684precondition\u6548\u679c\u3002\u4e8e\u662f\u53ef\u4ee5\u5c06\u95ee\u9898\u8f6c\u5316\u4e3a (7)   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