Solving sparse systems - Revision history

Mitjajancic at 07:42, 15 February 2023

2023-02-15T07:42:37Z

Mitjajancic: Added some observations.

2023-02-14T12:56:57Z

Added some observations.

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Mitjajancic at 12:27, 14 February 2023

2023-02-14T12:27:34Z

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Mdepolli at 10:57, 19 July 2017

2017-07-19T10:57:19Z

Gkosec: /* Herzian contact */

2017-05-15T08:15:09Z

‎Herzian contact

Gkosec: /* Herzian contact */

2017-05-15T08:14:57Z

‎Herzian contact

Gkosec: /* ILUT factorization */

2017-05-15T08:14:45Z

‎ILUT factorization

Gkosec: /* Herzian contact */

2017-05-15T08:14:29Z

‎Herzian contact

Gkosec at 10:24, 20 April 2017

2017-04-20T10:24:58Z

Gkosec at 10:23, 20 April 2017

2017-04-20T10:23:36Z

@@ Line 156: / Line 156: @@
 * [https://eigen.tuxfamily.org/dox/classEigen_1_1PardisoLU.html PardisoLU].
-The performance tests are focused on the accuracy of the numerical solution, number of iterations and time required to solve the final sparse system. We also comment on the amount of RAM required by a specific solver. If required, the [https://gitlab.com/e62Lab/eigen_solvers_analysis gitlab repository] with the implementation is also provided.
+The performance tests are focused on the accuracy of the numerical solution, number of iterations and time required to solve the final sparse system. We also comment on the amount of RAM required by a specific solver. If required, the [https://gitlab.com/e62Lab/Medusa-disucssions/-/tree/master/eigen_solvers gitlab repository] with the implementation is also provided.
 <syntaxhighlight lang="text">
 NOTE: Although the SparseQR has been included in our tests, significantly larger solve times compared to other solvers have been observed. For that reason, the SparseQR is discarded from further analyses even though it was able to obtain a solution of comparable accuracy.
 </syntaxhighlight>
@@ Line 178: / Line 182: @@
        \boldsymbol e_i$ with unit vectors $\boldsymbol e_i$ on the Neumann boundary.
 * $f(\boldsymbol x) = 2a e^{-a \left \| \boldsymbol x - \boldsymbol x_s \right \|^2}(2a\left \| \boldsymbol x -
-   \boldsymbol x_s \right \| - d)$ yielding
+   \boldsymbol x_s \right \| - d)$ for width $a=10^2$ of the exponential source positioned at $\boldsymbol x_s=\boldsymbol 0$ yielding
 ** $f_d(\boldsymbol x) = e^{-a \left \| \boldsymbol x - \boldsymbol x_s \right \|^2}$ yielding on Dirichlet boundary and
 ** $\boldsymbol f_n (\boldsymbol x) = -2a(\boldsymbol x - \boldsymbol x_s)e^{-a \left \| \boldsymbol x - \boldsymbol x_s \right \|^2}$ on Neumann boundary.
-Example solutions to both scenarios are shown in figure below.
+Example numerical solutions to both scenarios are shown in figure below.
 [[File:solution_poisson.png|800px]]
@@ Line 191: / Line 194: @@
 Below, we show the performance of the selected solvers in case of a two-dimensional Poisson problem. The first column shows the accuracy of the numerical solution in terms of the infinity norm, the second column shows the number of iterations a given solver required (direct solvers were assigned a value 1) and the third column shows the total time required to solve the system and obtain the numerical solution $\widehat u$.
 [[File:convergence_poisson_sin.png|1200px]]

← Older revision		Revision as of 10:57, 19 July 2017
Line 123:		Line 123:

	We can see that low BiCGSTAB error estimate coincides with the range of actual small error compared to the analytical solution as well as with the range of lesser number of iterations.		We can see that low BiCGSTAB error estimate coincides with the range of actual small error compared to the analytical solution as well as with the range of lesser number of iterations.
		+
		+	==Pardiso library==
		+	[[Pardiso]]

	==Herzian contact ==		==Herzian contact ==

← Older revision		Revision as of 08:15, 15 May 2017
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	The condition number iz arther large and grows superlinearly with size in the example below.		The condition number iz arther large and grows superlinearly with size in the example below.
−	[[File:cond.png\|~~500px~~]]	+	[[File:cond.png\|600px]]

← Older revision		Revision as of 08:14, 15 May 2017
Line 129:		Line 129:

	The condition number iz arther large and grows superlinearly with size in the example below.		The condition number iz arther large and grows superlinearly with size in the example below.
−	[[File:cond.png\|~~800px~~]]	+	[[File:cond.png\|500px]]

@@ Line 120: / Line 120: @@
 such a convergence graph for the diffusion equation is presented in figures below:
-[[File:bicgstab_err.png|600px]][[File:bicgstab_conv.png|600px]][[File:bicgstab_iter.png|600px]]
+[[File:bicgstab_err.png|400px]][[File:bicgstab_conv.png|400px]][[File:bicgstab_iter.png|400px]]
 We can see that low BiCGSTAB error estimate coincides with the range of actual small error compared to the analytical solution as well as with the range of lesser number of iterations.

@@ Line 126: / Line 126: @@
 ==Herzian contact ==
 The matrix of the [[Hertzian contact]] problem is shown below:
-  [[File:matrix_hertzian.png|1000px]]
+  [[File:matrix_hertzian.png|500px]]
 The condition number iz arther large and grows superlinearly with size in the example below.
   [[File:cond.png|800px]]

@@ Line 17: / Line 17: @@
 Solving a simple sparse system $A x = b$ for steady space of heat equation in 1d with $n$ nodes, results in a matrix shown in <xr id="fig:matrix1"/>.
-<!--
 The following timings of solvers are given in seconds:
 {| class="wikitable"
@@ Line 36: / Line 36: @@
 | 0.82
 |-
 |}
--->
 Incomplete LU preconditioner was used for BICGStab.
 Without the preconditioner BICGStab does not converge.