<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"></head><body style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">Furthermore, I tried to perform the Galerkin projection in two steps by substituting<div><br></div><div><blockquote type="cite"><div style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;"><div style="color: rgb(29, 33, 38); background-color: rgb(255, 255, 255); font-family: Menlo, Monaco, "Courier New", monospace; line-height: 18px; white-space: pre;">A_prime <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">A</span>.ptap(Phi)</div></div></blockquote><div><br></div><div>With </div><div><br></div><div><div style="color: rgb(29, 33, 38); background-color: rgb(255, 255, 255); font-family: Menlo, Monaco, "Courier New", monospace; line-height: 18px; white-space: pre;"><div><span style="color: rgb(13, 92, 181);">AL</span> <span style="color: rgb(168, 3, 20);">=</span> Phi.transposeMatMult(<span style="color: rgb(13, 92, 181);">A</span>)</div><div>A_prime <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">AL</span>.matMult(Phi)</div></div></div><div><br></div><div>And running this with 3 procs, results to the false creation of a matrix AL that has 3 times bigger dimensions that it should (A is of size 100x100 and Phi of size 100x7):</div><div><br></div></div><blockquote style="margin: 0 0 0 40px; border: none; padding: 0px;"><div><div><div>MATRIX mpiaij AL [21x300]</div></div></div><div><div><div>Assembled</div></div></div><div><div><div><br></div></div></div><div><div><div>Partitioning for AL:</div></div></div><div><div><div> Rank 0: Rows [0, 7)</div></div></div><div><div><div> Rank 1: Rows [7, 14)</div></div></div><div><div><div> Rank 2: Rows [14, 21)</div></div></div></blockquote><div><div><br></div><div>And naturally, in another dimension incompatibility:</div><div><br></div></div><blockquote style="margin: 0 0 0 40px; border: none; padding: 0px;"><div><div><div>Traceback (most recent call last):</div></div></div><div><div><div> File "/Users/boutsitron/petsc-experiments/mat_vec_multiplication2.py", line 85, in <module></div></div></div><div><div><div>Traceback (most recent call last):</div></div></div><div><div><div> File "/Users/boutsitron/petsc-experiments/mat_vec_multiplication2.py", line 85, in <module></div></div></div><div><div><div> A_prime = AL.matMult(Phi)</div></div></div><div><div><div> A_prime = AL.matMult(Phi)</div></div></div><div><div><div> ^^^^^^^^^^^^^^^</div></div></div><div><div><div> File "petsc4py/PETSc/Mat.pyx", line 1492, in petsc4py.PETSc.Mat.matMult</div></div></div><div><div><div> ^^^^^^^^^^^^^^^</div></div></div><div><div><div> File "petsc4py/PETSc/Mat.pyx", line 1492, in petsc4py.PETSc.Mat.matMult</div></div></div><div><div><div>Traceback (most recent call last):</div></div></div><div><div><div> File "/Users/boutsitron/petsc-experiments/mat_vec_multiplication2.py", line 85, in <module></div></div></div><div><div><div>petsc4py.PETSc.Error: error code 60</div></div></div><div><div><div>[2] MatMatMult() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matrix.c:10053</div></div></div><div><div><div>[2] MatProduct_Private() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matrix.c:9976</div></div></div><div><div><div>[2] MatProductSetFromOptions() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matproduct.c:541</div></div></div><div><div><div>[2] MatProductSetFromOptions_Private() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matproduct.c:421</div></div></div><div><div><div>[2] Nonconforming object sizes</div></div></div><div><div><div>[2] Matrix dimensions of A and B are incompatible for MatProductType AB: A 21x300, B 100x7</div></div></div><div><div><div>Abort(1) on node 2 (rank 2 in comm 496): application called MPI_Abort(PYOP2_COMM_WORLD, 1) - process 2</div></div></div><div><div><div>petsc4py.PETSc.Error: error code 60</div></div></div><div><div><div>[1] MatMatMult() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matrix.c:10053</div></div></div><div><div><div>[1] MatProduct_Private() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matrix.c:9976</div></div></div><div><div><div>[1] MatProductSetFromOptions() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matproduct.c:541</div></div></div><div><div><div>[1] MatProductSetFromOptions_Private() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matproduct.c:421</div></div></div><div><div><div>[1] Nonconforming object sizes</div></div></div><div><div><div>[1] Matrix dimensions of A and B are incompatible for MatProductType AB: A 21x300, B 100x7</div></div></div><div><div><div>Abort(1) on node 1 (rank 1 in comm 496): application called MPI_Abort(PYOP2_COMM_WORLD, 1) - process 1</div></div></div><div><div><div> A_prime = AL.matMult(Phi)</div></div></div><div><div><div> ^^^^^^^^^^^^^^^</div></div></div><div><div><div> File "petsc4py/PETSc/Mat.pyx", line 1492, in petsc4py.PETSc.Mat.matMult</div></div></div><div><div><div>petsc4py.PETSc.Error: error code 60</div></div></div><div><div><div>[0] MatMatMult() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matrix.c:10053</div></div></div><div><div><div>[0] MatProduct_Private() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matrix.c:9976</div></div></div><div><div><div>[0] MatProductSetFromOptions() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matproduct.c:541</div></div></div><div><div><div>[0] MatProductSetFromOptions_Private() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matproduct.c:421</div></div></div><div><div><div>[0] Nonconforming object sizes</div></div></div><div><div><div>[0] Matrix dimensions of A and B are incompatible for MatProductType AB: A 21x300, B 100x7</div></div></div><div><div><div>Abort(1) on node 0 (rank 0 in comm 496): application called MPI_Abort(PYOP2_COMM_WORLD, 1) - process 0</div></div></div></blockquote><div><div><br><blockquote type="cite"><div>On 10 Oct 2023, at 23:33, Thanasis Boutsikakis <thanasis.boutsikakis@corintis.com> wrote:</div><br class="Apple-interchange-newline"><div><meta http-equiv="content-type" content="text/html; charset=utf-8"><div style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">Hi all,<div><br></div><div>Revisiting my code and the proposed solution from Pierre, I realized this works only in sequential. The reason is that PETSc partitions those matrices only row-wise, which leads to an error due to the mismatch between number of columns of A (non-partitioned) and the number of rows of Phi (partitioned).<div><br></div><div><div style="color: rgb(29, 33, 38); background-color: rgb(255, 255, 255); font-family: Menlo, Monaco, "Courier New", monospace; line-height: 18px; white-space: pre;"><div><span style="color: rgb(10, 66, 131);">"""Experimenting with PETSc mat-mat multiplication"""</span></div><br><div><span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">time</span></div><br><div><span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">numpy</span> <span style="color: rgb(168, 3, 20);">as</span> <span style="color: rgb(72, 31, 148);">np</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">colorama</span> <span style="color: rgb(168, 3, 20);">import</span> Fore</div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">firedrake</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(13, 92, 181);">COMM_SELF</span>, <span style="color: rgb(13, 92, 181);">COMM_WORLD</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">firedrake</span>.<span style="color: rgb(72, 31, 148);">petsc</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">PETSc</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">mpi4py</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">MPI</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">numpy</span>.<span style="color: rgb(72, 31, 148);">testing</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">assert_array_almost_equal</span></div><br><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">utilities</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">Print</span></div><br><div>nproc <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">COMM_WORLD</span>.size</div><div>rank <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">COMM_WORLD</span>.rank</div><div><br></div><div><div style="line-height: 18px;"><div><span style="color: rgb(168, 3, 20);">def</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(<span style="color: rgb(166, 61, 0);">input_array,</span> <span style="color: rgb(166, 61, 0);">sparse</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">True</span>):</div><div> <span style="color: rgb(10, 66, 131);">"""Create a PETSc matrix from an input_array</span></div><br><div><span style="color: rgb(10, 66, 131);"> Args:</span></div><div><span style="color: rgb(10, 66, 131);"> input_array (np array): Input array</span></div><div><span style="color: rgb(10, 66, 131);"> partition_like (PETSc mat, optional): Petsc matrix. Defaults to None.</span></div><div><span style="color: rgb(10, 66, 131);"> sparse (bool, optional): Toggle for sparese or dense. Defaults to True.</span></div><br><div><span style="color: rgb(10, 66, 131);"> Returns:</span></div><div><span style="color: rgb(10, 66, 131);"> PETSc mat: PETSc mpi matrix</span></div><div><span style="color: rgb(10, 66, 131);"> """</span></div><div> <span style="color: rgb(164, 176, 188);"># Check if input_array is 1D and reshape if necessary</span></div><div> <span style="color: rgb(168, 3, 20);">assert</span> <span style="color: rgb(72, 31, 148);">len</span>(<span style="color: rgb(166, 61, 0);">input_array</span>.shape) <span style="color: rgb(168, 3, 20);">==</span> <span style="color: rgb(13, 92, 181);">2</span>, <span style="color: rgb(10, 66, 131);">"Input array should be 2-dimensional"</span></div><div> global_rows, global_cols <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(166, 61, 0);">input_array</span>.shape</div><div> size <span style="color: rgb(168, 3, 20);">=</span> ((<span style="color: rgb(13, 92, 181);">None</span>, global_rows), (global_cols, global_cols))</div><br><div> <span style="color: rgb(164, 176, 188);"># Create a sparse or dense matrix based on the 'sparse' argument</span></div><div> <span style="color: rgb(168, 3, 20);">if</span> <span style="color: rgb(166, 61, 0);">sparse</span>:</div><div> matrix <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">PETSc</span>.Mat().createAIJ(<span style="color: rgb(166, 61, 0);">size</span><span style="color: rgb(168, 3, 20);">=</span>size, <span style="color: rgb(166, 61, 0);">comm</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">COMM_WORLD</span>)</div><div> <span style="color: rgb(168, 3, 20);">else</span>:</div><div> matrix <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">PETSc</span>.Mat().createDense(<span style="color: rgb(166, 61, 0);">size</span><span style="color: rgb(168, 3, 20);">=</span>size, <span style="color: rgb(166, 61, 0);">comm</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">COMM_WORLD</span>)</div><div> matrix.setUp()</div><br><div> local_rows_start, local_rows_end <span style="color: rgb(168, 3, 20);">=</span> matrix.getOwnershipRange()</div><br><div> <span style="color: rgb(168, 3, 20);">for</span> counter, i <span style="color: rgb(168, 3, 20);">in</span> <span style="color: rgb(72, 31, 148);">enumerate</span>(<span style="color: rgb(72, 31, 148);">range</span>(local_rows_start, local_rows_end)):</div><div> <span style="color: rgb(164, 176, 188);"># Calculate the correct row in the array for the current process</span></div><div> row_in_array <span style="color: rgb(168, 3, 20);">=</span> counter <span style="color: rgb(168, 3, 20);">+</span> local_rows_start</div><div> matrix.setValues(</div><div> i, <span style="color: rgb(72, 31, 148);">range</span>(global_cols), <span style="color: rgb(166, 61, 0);">input_array</span>[row_in_array, :], <span style="color: rgb(166, 61, 0);">addv</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">False</span></div><div> )</div><br><div> <span style="color: rgb(164, 176, 188);"># Assembly the matrix to compute the final structure</span></div><div> matrix.assemblyBegin()</div><div> matrix.assemblyEnd()</div><br><div> <span style="color: rgb(168, 3, 20);">return</span> matrix</div></div></div><br><div><span style="color: rgb(164, 176, 188);"># --------------------------------------------</span></div><div><span style="color: rgb(164, 176, 188);"># EXP: Galerkin projection of an mpi PETSc matrix A with an mpi PETSc matrix Phi</span></div><div><span style="color: rgb(164, 176, 188);"># A' = Phi.T * A * Phi</span></div><div><span style="color: rgb(164, 176, 188);"># [k x k] <- [k x m] x [m x m] x [m x k]</span></div><div><span style="color: rgb(164, 176, 188);"># --------------------------------------------</span></div><br><div>m, k <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">100</span>, <span style="color: rgb(13, 92, 181);">7</span></div><div><span style="color: rgb(164, 176, 188);"># Generate the random numpy matrices</span></div><div><span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">random</span>.seed(<span style="color: rgb(13, 92, 181);">0</span>) <span style="color: rgb(164, 176, 188);"># sets the seed to 0</span></div><div>A_np <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">random</span>.randint(<span style="color: rgb(166, 61, 0);">low</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">0</span>, <span style="color: rgb(166, 61, 0);">high</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">6</span>, <span style="color: rgb(166, 61, 0);">size</span><span style="color: rgb(168, 3, 20);">=</span>(m, m))</div><div>Phi_np <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">random</span>.randint(<span style="color: rgb(166, 61, 0);">low</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">0</span>, <span style="color: rgb(166, 61, 0);">high</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">6</span>, <span style="color: rgb(166, 61, 0);">size</span><span style="color: rgb(168, 3, 20);">=</span>(m, k))</div><br><div><span style="color: rgb(164, 176, 188);"># --------------------------------------------</span></div><div><span style="color: rgb(164, 176, 188);"># TEST: Galerking projection of numpy matrices A_np and Phi_np</span></div><div><span style="color: rgb(164, 176, 188);"># --------------------------------------------</span></div><div>Aprime_np <span style="color: rgb(168, 3, 20);">=</span> Phi_np.T <span style="color: rgb(13, 92, 181);">@</span> A_np <span style="color: rgb(13, 92, 181);">@</span> Phi_np</div><div><span style="color: rgb(72, 31, 148);">Print</span>(<span style="color: rgb(168, 3, 20);">f</span><span style="color: rgb(10, 66, 131);">"MATRIX Aprime_np [</span><span style="color: rgb(13, 92, 181);">{</span>Aprime_np.shape[<span style="color: rgb(13, 92, 181);">0</span>]<span style="color: rgb(13, 92, 181);">}</span><span style="color: rgb(10, 66, 131);">x</span><span style="color: rgb(13, 92, 181);">{</span>Aprime_np.shape[<span style="color: rgb(13, 92, 181);">1</span>]<span style="color: rgb(13, 92, 181);">}</span><span style="color: rgb(10, 66, 131);">]"</span>)</div><div><span style="color: rgb(72, 31, 148);">Print</span>(<span style="color: rgb(168, 3, 20);">f</span><span style="color: rgb(10, 66, 131);">"</span><span style="color: rgb(13, 92, 181);">{</span>Aprime_np<span style="color: rgb(13, 92, 181);">}</span><span style="color: rgb(10, 66, 131);">"</span>)</div><br><div><span style="color: rgb(164, 176, 188);"># Create A as an mpi matrix distributed on each process</span></div><div><span style="color: rgb(13, 92, 181);">A</span> <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(A_np, <span style="color: rgb(166, 61, 0);">sparse</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">False</span>)</div><br><div><span style="color: rgb(164, 176, 188);"># Create Phi as an mpi matrix distributed on each process</span></div><div>Phi <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(Phi_np, <span style="color: rgb(166, 61, 0);">sparse</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">False</span>)</div><br><div><span style="color: rgb(164, 176, 188);"># Create an empty PETSc matrix object to store the result of the PtAP operation.</span></div><div><span style="color: rgb(164, 176, 188);"># This will hold the result A' = Phi.T * A * Phi after the computation.</span></div><div>A_prime <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(<span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">zeros</span>((k, k)), <span style="color: rgb(166, 61, 0);">sparse</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">False</span>)</div><br><div><span style="color: rgb(164, 176, 188);"># Perform the PtAP (Phi Transpose times A times Phi) operation.</span></div><div><span style="color: rgb(164, 176, 188);"># In mathematical terms, this operation is A' = Phi.T * A * Phi.</span></div><div><span style="color: rgb(164, 176, 188);"># A_prime will store the result of the operation.</span></div><div>A_prime <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">A</span>.ptap(Phi)</div></div><div><br></div><div>Here is the error</div><div><br></div></div><blockquote style="margin: 0 0 0 40px; border: none; padding: 0px;"><div><div><div>MATRIX mpiaij A [100x100]</div></div></div><div><div><div>Assembled</div></div></div><div><div><div><br></div></div></div><div><div><div>Partitioning for A:</div></div></div><div><div><div> Rank 0: Rows [0, 34)</div></div></div><div><div><div> Rank 1: Rows [34, 67)</div></div></div><div><div><div> Rank 2: Rows [67, 100)</div></div></div><div><div><div><br></div></div></div><div><div><div>MATRIX mpiaij Phi [100x7]</div></div></div><div><div><div>Assembled</div></div></div><div><div><div><br></div></div></div><div><div><div>Partitioning for Phi:</div></div></div><div><div><div> Rank 0: Rows [0, 34)</div></div></div><div><div><div> Rank 1: Rows [34, 67)</div></div></div><div><div><div> Rank 2: Rows [67, 100)</div></div></div><div><div><div><br></div></div></div><div><div><div>Traceback (most recent call last):</div></div></div><div><div><div> File "/Users/boutsitron/work/galerkin_projection.py", line 87, in <module></div></div></div><div><div><div> A_prime = A.ptap(Phi)</div></div></div><div><div><div> ^^^^^^^^^^^</div></div></div><div><div><div> File "petsc4py/PETSc/Mat.pyx", line 1525, in petsc4py.PETSc.Mat.ptap</div></div></div><div><div><div>petsc4py.PETSc.Error: error code 60</div></div></div><div><div><div>[0] MatPtAP() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matrix.c:9896</div></div></div><div><div><div>[0] MatProductSetFromOptions() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matproduct.c:541</div></div></div><div><div><div>[0] MatProductSetFromOptions_Private() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matproduct.c:435</div></div></div><div><div><div>[0] MatProductSetFromOptions_MPIAIJ() at /Users/boutsitron/firedrake/src/petsc/src/mat/impls/aij/mpi/mpimatmatmult.c:2372</div></div></div><div><div><div>[0] MatProductSetFromOptions_MPIAIJ_PtAP() at /Users/boutsitron/firedrake/src/petsc/src/mat/impls/aij/mpi/mpimatmatmult.c:2266</div></div></div><div><div><div>[0] Nonconforming object sizes</div></div></div><div><div><div>[0] Matrix local dimensions are incompatible, Acol (0, 100) != Prow (0,34)</div></div></div><div><div><div>Abort(1) on node 0 (rank 0 in comm 496): application called MPI_Abort(PYOP2_COMM_WORLD, 1) - process 0</div></div></div></blockquote><div><div><div><br></div><div>Any thoughts?</div><div><br></div><div>Thanks,</div><div>Thanos</div></div><div><br><blockquote type="cite"><div>On 5 Oct 2023, at 14:23, Thanasis Boutsikakis <thanasis.boutsikakis@corintis.com> wrote:</div><br class="Apple-interchange-newline"><div><meta http-equiv="content-type" content="text/html; charset=utf-8"><div style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">This works Pierre. Amazing input, thanks a lot!<br id="lineBreakAtBeginningOfMessage"><div><br><blockquote type="cite"><div>On 5 Oct 2023, at 14:17, Pierre Jolivet <pierre@joliv.et> wrote:</div><br class="Apple-interchange-newline"><div><meta http-equiv="content-type" content="text/html; charset=utf-8"><div style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;"><div>Not a petsc4py expert here, but you may to try instead:</div><div>A_prime = A.ptap(Phi)</div><div><br></div><div>Thanks,</div><div>Pierre</div><div><br><blockquote type="cite"><div>On 5 Oct 2023, at 2:02 PM, Thanasis Boutsikakis <thanasis.boutsikakis@corintis.com> wrote:</div><br class="Apple-interchange-newline"><div><meta http-equiv="content-type" content="text/html; charset=utf-8"><div style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">Thanks Pierre! So I tried this and got a segmentation fault. Is this supposed to work right off the bat or am I missing sth?<div><br></div><div><div>[0]PETSC ERROR: ------------------------------------------------------------------------</div><div>[0]PETSC ERROR: Caught signal number 11 SEGV: Segmentation Violation, probably memory access out of range</div><div>[0]PETSC ERROR: Try option -start_in_debugger or -on_error_attach_debugger</div><div>[0]PETSC ERROR: or see https://petsc.org/release/faq/#valgrind and https://petsc.org/release/faq/</div><div>[0]PETSC ERROR: configure using --with-debugging=yes, recompile, link, and run</div><div>[0]PETSC ERROR: to get more information on the crash.</div><div>[0]PETSC ERROR: Run with -malloc_debug to check if memory corruption is causing the crash.</div><div>Abort(59) on node 0 (rank 0 in comm 0): application called MPI_Abort(MPI_COMM_WORLD, 59) - process 0</div><div><div><br></div><div><div style="color: rgb(29, 33, 38); background-color: rgb(255, 255, 255); font-family: Menlo, Monaco, "Courier New", monospace; line-height: 18px; white-space: pre;"><div><span style="color: rgb(10, 66, 131);">"""Experimenting with PETSc mat-mat multiplication"""</span></div><br><div><span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">time</span></div><br><div><span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">numpy</span> <span style="color: rgb(168, 3, 20);">as</span> <span style="color: rgb(72, 31, 148);">np</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">colorama</span> <span style="color: rgb(168, 3, 20);">import</span> Fore</div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">firedrake</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(13, 92, 181);">COMM_SELF</span>, <span style="color: rgb(13, 92, 181);">COMM_WORLD</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">firedrake</span>.<span style="color: rgb(72, 31, 148);">petsc</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">PETSc</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">mpi4py</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">MPI</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">numpy</span>.<span style="color: rgb(72, 31, 148);">testing</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">assert_array_almost_equal</span></div><br><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">utilities</span> <span style="color: rgb(168, 3, 20);">import</span> (</div><div> <span style="color: rgb(72, 31, 148);">Print</span>,</div><div> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>,</div><div> <span style="color: rgb(72, 31, 148);">print_matrix_partitioning</span>,</div><div>)</div><br><div>nproc <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">COMM_WORLD</span>.size</div><div>rank <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">COMM_WORLD</span>.rank</div><br><div><span style="color: rgb(164, 176, 188);"># --------------------------------------------</span></div><div><span style="color: rgb(164, 176, 188);"># EXP: Galerkin projection of an mpi PETSc matrix A with an mpi PETSc matrix Phi</span></div><div><span style="color: rgb(164, 176, 188);"># A' = Phi.T * A * Phi</span></div><div><span style="color: rgb(164, 176, 188);"># [k x k] <- [k x m] x [m x m] x [m x k]</span></div><div><span style="color: rgb(164, 176, 188);"># --------------------------------------------</span></div><br><div>m, k <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">11</span>, <span style="color: rgb(13, 92, 181);">7</span></div><div><span style="color: rgb(164, 176, 188);"># Generate the random numpy matrices</span></div><div><span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">random</span>.seed(<span style="color: rgb(13, 92, 181);">0</span>) <span style="color: rgb(164, 176, 188);"># sets the seed to 0</span></div><div>A_np <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">random</span>.randint(<span style="color: rgb(166, 61, 0);">low</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">0</span>, <span style="color: rgb(166, 61, 0);">high</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">6</span>, <span style="color: rgb(166, 61, 0);">size</span><span style="color: rgb(168, 3, 20);">=</span>(m, m))</div><div>Phi_np <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">random</span>.randint(<span style="color: rgb(166, 61, 0);">low</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">0</span>, <span style="color: rgb(166, 61, 0);">high</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">6</span>, <span style="color: rgb(166, 61, 0);">size</span><span style="color: rgb(168, 3, 20);">=</span>(m, k))</div><br><div><span style="color: rgb(164, 176, 188);"># --------------------------------------------</span></div><div><span style="color: rgb(164, 176, 188);"># TEST: Galerking projection of numpy matrices A_np and Phi_np</span></div><div><span style="color: rgb(164, 176, 188);"># --------------------------------------------</span></div><div>Aprime_np <span style="color: rgb(168, 3, 20);">=</span> Phi_np.T <span style="color: rgb(13, 92, 181);">@</span> A_np <span style="color: rgb(13, 92, 181);">@</span> Phi_np</div><div><span style="color: rgb(72, 31, 148);">Print</span>(<span style="color: rgb(168, 3, 20);">f</span><span style="color: rgb(10, 66, 131);">"MATRIX Aprime_np [</span><span style="color: rgb(13, 92, 181);">{</span>Aprime_np.shape[<span style="color: rgb(13, 92, 181);">0</span>]<span style="color: rgb(13, 92, 181);">}</span><span style="color: rgb(10, 66, 131);">x</span><span style="color: rgb(13, 92, 181);">{</span>Aprime_np.shape[<span style="color: rgb(13, 92, 181);">1</span>]<span style="color: rgb(13, 92, 181);">}</span><span style="color: rgb(10, 66, 131);">]"</span>)</div><div><span style="color: rgb(72, 31, 148);">Print</span>(<span style="color: rgb(168, 3, 20);">f</span><span style="color: rgb(10, 66, 131);">"</span><span style="color: rgb(13, 92, 181);">{</span>Aprime_np<span style="color: rgb(13, 92, 181);">}</span><span style="color: rgb(10, 66, 131);">"</span>)</div><br><div><span style="color: rgb(164, 176, 188);"># Create A as an mpi matrix distributed on each process</span></div><div><span style="color: rgb(13, 92, 181);">A</span> <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(A_np, <span style="color: rgb(166, 61, 0);">sparse</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">False</span>)</div><div><br></div><div><span style="color: rgb(164, 176, 188);"># Create Phi as an mpi matrix distributed on each process</span></div><div>Phi <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(Phi_np, <span style="color: rgb(166, 61, 0);">sparse</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">False</span>)</div><div><br></div><div><span style="color: rgb(164, 176, 188);"># Create an empty PETSc matrix object to store the result of the PtAP operation.</span></div><div><span style="color: rgb(164, 176, 188);"># This will hold the result A' = Phi.T * A * Phi after the computation.</span></div><div>A_prime <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(<span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">zeros</span>((k, k)), <span style="color: rgb(166, 61, 0);">sparse</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">False</span>)</div><div><br></div><div><span style="color: rgb(164, 176, 188);"># Perform the PtAP (Phi Transpose times A times Phi) operation.</span></div><div><span style="color: rgb(164, 176, 188);"># In mathematical terms, this operation is A' = Phi.T * A * Phi.</span></div><div><span style="color: rgb(164, 176, 188);"># A_prime will store the result of the operation.</span></div><div>Phi.PtAP(<span style="color: rgb(13, 92, 181);">A</span>, A_prime)</div></div><div><br><blockquote type="cite"><div>On 5 Oct 2023, at 13:22, Pierre Jolivet <pierre@joliv.et> wrote:</div><br class="Apple-interchange-newline"><div><meta http-equiv="content-type" content="text/html; charset=utf-8"><div style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">How about using ptap which will use MatPtAP?<div>It will be more efficient (and it will help you bypass the issue).</div><div><br></div><div>Thanks,</div><div>Pierre<br id="lineBreakAtBeginningOfMessage"><div><br><blockquote type="cite"><div>On 5 Oct 2023, at 1:18 PM, Thanasis Boutsikakis <thanasis.boutsikakis@corintis.com> wrote:</div><br class="Apple-interchange-newline"><div><meta http-equiv="content-type" content="text/html; charset=us-ascii"><div style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">Sorry, forgot function <span style="caret-color: rgb(72, 31, 148); color: rgb(72, 31, 148); font-family: Menlo, Monaco, "Courier New", monospace; white-space: pre; background-color: rgb(255, 255, 255);">create_petsc_matrix()</span><div><br></div><div><div style="color: rgb(29, 33, 38); background-color: rgb(255, 255, 255); font-family: Menlo, Monaco, "Courier New", monospace; line-height: 18px; white-space: pre;"><div><span style="color: rgb(168, 3, 20);">def</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(<span style="color: rgb(166, 61, 0);">input_array</span> <span style="color: rgb(166, 61, 0);">sparse</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">True</span>):</div><div> <span style="color: rgb(10, 66, 131);">"""Create a PETSc matrix from an input_array</span></div><br><div><span style="color: rgb(10, 66, 131);"> Args:</span></div><div><span style="color: rgb(10, 66, 131);"> input_array (np array): Input array</span></div><div><span style="color: rgb(10, 66, 131);"> partition_like (PETSc mat, optional): Petsc matrix. Defaults to None.</span></div><div><span style="color: rgb(10, 66, 131);"> sparse (bool, optional): Toggle for sparese or dense. Defaults to True.</span></div><br><div><span style="color: rgb(10, 66, 131);"> Returns:</span></div><div><span style="color: rgb(10, 66, 131);"> PETSc mat: PETSc matrix</span></div><div><span style="color: rgb(10, 66, 131);"> """</span></div><div> <span style="color: rgb(164, 176, 188);"># Check if input_array is 1D and reshape if necessary</span></div><div> <span style="color: rgb(168, 3, 20);">assert</span> <span style="color: rgb(72, 31, 148);">len</span>(<span style="color: rgb(166, 61, 0);">input_array</span>.shape) <span style="color: rgb(168, 3, 20);">==</span> <span style="color: rgb(13, 92, 181);">2</span>, <span style="color: rgb(10, 66, 131);">"Input array should be 2-dimensional"</span></div><div> global_rows, global_cols <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(166, 61, 0);">input_array</span>.shape</div><div><br></div><div> size <span style="color: rgb(168, 3, 20);">=</span> ((<span style="color: rgb(13, 92, 181);">None</span>, global_rows), (global_cols, global_cols))</div><br><div> <span style="color: rgb(164, 176, 188);"># Create a sparse or dense matrix based on the 'sparse' argument</span></div><div> <span style="color: rgb(168, 3, 20);">if</span> <span style="color: rgb(166, 61, 0);">sparse</span>:</div><div> matrix <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">PETSc</span>.Mat().createAIJ(<span style="color: rgb(166, 61, 0);">size</span><span style="color: rgb(168, 3, 20);">=</span>size, <span style="color: rgb(166, 61, 0);">comm</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">COMM_WORLD</span>)</div><div> <span style="color: rgb(168, 3, 20);">else</span>:</div><div> matrix <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">PETSc</span>.Mat().createDense(<span style="color: rgb(166, 61, 0);">size</span><span style="color: rgb(168, 3, 20);">=</span>size, <span style="color: rgb(166, 61, 0);">comm</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">COMM_WORLD</span>)</div><div> matrix.setUp()</div><br><div> local_rows_start, local_rows_end <span style="color: rgb(168, 3, 20);">=</span> matrix.getOwnershipRange()</div><br><div> <span style="color: rgb(168, 3, 20);">for</span> counter, i <span style="color: rgb(168, 3, 20);">in</span> <span style="color: rgb(72, 31, 148);">enumerate</span>(<span style="color: rgb(72, 31, 148);">range</span>(local_rows_start, local_rows_end)):</div><div> <span style="color: rgb(164, 176, 188);"># Calculate the correct row in the array for the current process</span></div><div> row_in_array <span style="color: rgb(168, 3, 20);">=</span> counter <span style="color: rgb(168, 3, 20);">+</span> local_rows_start</div><div> matrix.setValues(</div><div> i, <span style="color: rgb(72, 31, 148);">range</span>(global_cols), <span style="color: rgb(166, 61, 0);">input_array</span>[row_in_array, :], <span style="color: rgb(166, 61, 0);">addv</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">False</span></div><div> )</div><br><div> <span style="color: rgb(164, 176, 188);"># Assembly the matrix to compute the final structure</span></div><div> matrix.assemblyBegin()</div><div> matrix.assemblyEnd()</div><br><div> <span style="color: rgb(168, 3, 20);">return</span> matrix</div></div><div><br><blockquote type="cite"><div>On 5 Oct 2023, at 13:09, Thanasis Boutsikakis <thanasis.boutsikakis@corintis.com> wrote:</div><br class="Apple-interchange-newline"><div><meta http-equiv="content-type" content="text/html; charset=us-ascii"><div style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">Hi everyone,<div><br></div><div>I am trying a Galerkin projection (see MFE below) and I cannot get the <span style="caret-color: rgb(29, 33, 38); color: rgb(29, 33, 38); font-family: Menlo, Monaco, "Courier New", monospace; white-space: pre; background-color: rgb(255, 255, 255);">Phi</span><span style="caret-color: rgb(29, 33, 38); color: rgb(29, 33, 38); font-family: Menlo, Monaco, "Courier New", monospace; white-space: pre; background-color: rgb(255, 255, 255);">.transposeMatMult(A, A1) work. The error is</span></div><div><span style="caret-color: rgb(29, 33, 38); color: rgb(29, 33, 38); font-family: Menlo, Monaco, "Courier New", monospace; white-space: pre; background-color: rgb(255, 255, 255);"><br></span></div><div><div> Phi.transposeMatMult(A, A1)</div><div> File "petsc4py/PETSc/Mat.pyx", line 1514, in petsc4py.PETSc.Mat.transposeMatMult</div><div>petsc4py.PETSc.Error: error code 56</div><div>[0] MatTransposeMatMult() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matrix.c:10135</div><div>[0] MatProduct_Private() at /Users/boutsitron/firedrake/src/petsc/src/mat/interface/matrix.c:9989</div><div>[0] No support for this operation for this object type</div><div>[0] Call MatProductCreate() first</div></div><div><br></div><div>Do you know if these exposed to petsc4py or maybe there is another way? I cannot get the MFE to work (neither in sequential nor in parallel)</div><div><br></div><div><div style="color: rgb(29, 33, 38); background-color: rgb(255, 255, 255); font-family: Menlo, Monaco, "Courier New", monospace; line-height: 18px; white-space: pre;"><div><span style="color: rgb(10, 66, 131);">"""Experimenting with PETSc mat-mat multiplication"""</span></div><br><div><span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">time</span></div><br><div><span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">numpy</span> <span style="color: rgb(168, 3, 20);">as</span> <span style="color: rgb(72, 31, 148);">np</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">colorama</span> <span style="color: rgb(168, 3, 20);">import</span> Fore</div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">firedrake</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(13, 92, 181);">COMM_SELF</span>, <span style="color: rgb(13, 92, 181);">COMM_WORLD</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">firedrake</span>.<span style="color: rgb(72, 31, 148);">petsc</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">PETSc</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">mpi4py</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">MPI</span></div><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">numpy</span>.<span style="color: rgb(72, 31, 148);">testing</span> <span style="color: rgb(168, 3, 20);">import</span> <span style="color: rgb(72, 31, 148);">assert_array_almost_equal</span></div><br><div><span style="color: rgb(168, 3, 20);">from</span> <span style="color: rgb(72, 31, 148);">utilities</span> <span style="color: rgb(168, 3, 20);">import</span> (</div><div> <span style="color: rgb(72, 31, 148);">Print</span>,</div><div> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>,</div><div>)</div><br><div>nproc <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">COMM_WORLD</span>.size</div><div>rank <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">COMM_WORLD</span>.rank</div><br><div><span style="color: rgb(164, 176, 188);"># --------------------------------------------</span></div><div><span style="color: rgb(164, 176, 188);"># EXP: Galerkin projection of an mpi PETSc matrix A with an mpi PETSc matrix Phi</span></div><div><span style="color: rgb(164, 176, 188);"># A' = Phi.T * A * Phi</span></div><div><span style="color: rgb(164, 176, 188);"># [k x k] <- [k x m] x [m x m] x [m x k]</span></div><div><span style="color: rgb(164, 176, 188);"># --------------------------------------------</span></div><br><div>m, k <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(13, 92, 181);">11</span>, <span style="color: rgb(13, 92, 181);">7</span></div><div><span style="color: rgb(164, 176, 188);"># Generate the random numpy matrices</span></div><div><span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">random</span>.seed(<span style="color: rgb(13, 92, 181);">0</span>) <span style="color: rgb(164, 176, 188);"># sets the seed to 0</span></div><div>A_np <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">random</span>.randint(<span style="color: rgb(166, 61, 0);">low</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">0</span>, <span style="color: rgb(166, 61, 0);">high</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">6</span>, <span style="color: rgb(166, 61, 0);">size</span><span style="color: rgb(168, 3, 20);">=</span>(m, m))</div><div>Phi_np <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">random</span>.randint(<span style="color: rgb(166, 61, 0);">low</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">0</span>, <span style="color: rgb(166, 61, 0);">high</span><span style="color: rgb(168, 3, 20);">=</span><span style="color: rgb(13, 92, 181);">6</span>, <span style="color: rgb(166, 61, 0);">size</span><span style="color: rgb(168, 3, 20);">=</span>(m, k))</div><br><div><span style="color: rgb(164, 176, 188);"># Create A as an mpi matrix distributed on each process</span></div><div><span style="color: rgb(13, 92, 181);">A</span> <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(A_np)</div><div><br></div><div><span style="color: rgb(164, 176, 188);"># Create Phi as an mpi matrix distributed on each process</span></div><div>Phi <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(Phi_np)</div><div><br></div><div><span style="color: rgb(13, 92, 181);">A1</span> <span style="color: rgb(168, 3, 20);">=</span> <span style="color: rgb(72, 31, 148);">create_petsc_matrix</span>(<span style="color: rgb(72, 31, 148);">np</span>.<span style="color: rgb(72, 31, 148);">zeros</span>((k, m)))</div><div><br></div><div><span style="color: rgb(164, 176, 188);"># Now A1 contains the result of Phi^T * A</span></div><div>Phi.transposeMatMult(<span style="color: rgb(13, 92, 181);">A</span>, <span style="color: rgb(13, 92, 181);">A1</span>)</div><br></div></div></div></div></blockquote></div><br></div></div></div></blockquote></div><br></div></div></div></blockquote></div><br></div></div></div></div></div></blockquote></div><br></div></div></blockquote></div><br></div></div></blockquote></div><br></div></div></div></div></blockquote></div><br></div></body></html>