Using BLIS for tensor computations in Q-Chem
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1 Using BLIS for tensor computations in Q-Chem Evgeny Epifanovsky Q-Chem BLIS Retreat, September 19 20, 2016
2 Q-Chem is an integrated software suite for modeling the properties of molecular systems from first principles. Density functional theory Pure and hybrid functionals Range-separated functionals Double hybrid functionals Time-dependent DFT Wavefunction theory Hartree Fock theory Møller Plesset perturbation theory Coupled-cluster theory Methods for excited states: CIS, ADC, EOM,... Sustainability model: Commercial licenses for research in academia and industry (includes technical support) Free developer licenses for using Q-Chem as a development platform (200+ contributors)
3 Outline Libtensor as a programming tool for tensor-based electronic structure methods Block tensor format for storing and manipulating tensors Block tensor contraction algorithm Interface with BLIS and results
4 Coupled cluster theory methods in Q-Chem Ground state Excited state Properties MP2 QCISD CCD, CCSD CCSD(T), (dt), (ft) CISD EOM-CCSD (EA, EE, IP, SF, DIP, DSF) IP-CISD, EA-CISD OPDM, TPDM Properties (all methods) Gradient (CC, EOM) 2013 RI-CCSD RI-EOM-CCSD (EA, EE, IP, SF) RI-OPDM, RI-TPDM RI properties CS/CX-MP2 CS/CX-CCSD CS/CX-CISD CS/CX-EOM-CCSD (EA, EE, IP, SF) CS/CX-OPDM Real, complex Dyson orbitals Two-photon absorption Spin-orbit coupling 2016 CCVB-SD (RI-)EOM-MP2 (EA, EE, IP, SF) Polarizabilities for CC and EOM wavefunctions Over 1000 programmable expressions implemented 17 contributors to date
5 Overview of libtensor Declarative internal domain-specific language for programming tensor expressions in C++ Multiple back-ends for tensor computations: built-in disk/memory-based disk-based contraction algorithm via libxm (I. Kaliman) distributed memory via CTF library (E. Solomonik)
6 Coupled cluster programmable expressions ( ) ( ) Tij ab Dij ab = ij ab + P (ab) f bctij ac 1 kl cd tkl bd tij ac P (ij) f jktik ab + 1 kl cd tjl cd tik ab 2 2 c klcd k klcd ( ) + 1 ij kl tkl ab + 1 kl cd tij cd tkl ab + 1 ab cd tij cd P (ij)p (ab) kb jc tik ac 1 kl cd tlj db tik ac kl klcd cd kc klcd P (ij)a ij = A ij A ji Dij ab = ɛ i + ɛ j ɛ a ɛ b f1_oo(i j) = f_oo(i j) * contract(k a b, i_oovv(j k a b), t2(i k a b)); f1_vv(b c) = f_vv(b c) * contract(k l d, i_oovv(k l c d), t2(k l b d)); ii_oooo(i j k l) = i_oooo(i j k l) * contract(a b, i_oovv(k l a b), t2(i j a b)); ii_ovov(i a j b) = i_ovov(i a j b) * contract(k c, i_oovv(i k b c), t2(k j c a)); t2new(i j a b) = div( i_oovv(i j a b) + asymm(a, b, contract(c, t2(i j a c), f1_vv(b c))) - asymm(i, j, contract(k, t2(i k a b), f1_oo(j k))) * contract(k l, ii_oooo(i j k l), t2(k l a b)) * contract(c d, i_vvvv(a b c d), t2(i j c d)) - asymm(a, b, asymm(i, j, contract(k c, ii_ovov(k b j c), t2(i k a c)))), d_oovv(i j a b));
7 Evaluation of expressions t ab ij = P(ij)P(ab) kc I (1) kbic tac jk + P(ij) c jc ba t c i t2(i j a b) = asymm(i, j, asymm(a, b, contract(k c, i1(k b i c), t2(j k a c)))) + asymm(i, j, contract(c, ovvv(j c b a), t2(i c))); { = t2(i,j,a,b) { + { asym(i,j; a,b) { * i1(k,b,i,c) t2(j,k,a,c) } } { asym(i,j) { * ovvv(j,c,b,a) t1(i,c) } } } } Factorization and optimization Rendering as basic tensor operations and execution
8 Block tensor format Three components: 1. Block tensor space: dimensions + tiling pattern. 2. Symmetry relations between blocks. 3. Non-zero canonical data blocks. Natural for block-sparse tensors. Imbalance in diagonal blocks with permutational symmetry. B A A B 1 B 2 B 3 α α B β B 1 B 2 β B 3 Permutational Point group Spin
9 Block tensor contraction algorithm C ijba = kc A kbic B jkac C ibja = kc à ibkc B jakc C IJ = K à IK BJK 1. Compute the symmetry of C analytically 2. Compute unfolding transformations for A, B, C 3. Perform block tensor contraction of Ã, B into C
10 Block tensor contraction algorithm C IJ = K A IK B JK 1. Compute the list of non-zero canonical blocks in C: For each K, take sparse column - row outer products symbolically, reduce into a list of non-zero block indices Filter out non-canonical block indices 2. Compute non-zero canonical blocks of C in parallel (one task per block): Given (I, J), compute list of K that give non-zero products Using symmetry of A and B, convert (I, K) and (J, K) into canonical (IK, Ta), (JK, Tb) Optimize list of blockwise contractions to avoid duplication Evaluate CIJ = (T aa IK )(T b B JK ) using the dense tensor contraction algorithm
11 Dense tensor contraction algorithm Dense tensor contractions via loop merge followed by loop-over-gemm: C [ib][ja] = [kc] A [ib][kc] B [ja][kc] 1. Reduce the number of nested loops by merging tensor indices 2. Match inner loops against available kernels (BLAS primitives) 3. Evaluate loops-over-kernel Integration with BLIS: enable use of BLIS routines as linear algebra primitives
12 libtensor+blis vs. libtensor+mkl Timings (CPU sec.) for a UCCSD T amplitude update in water clusters, cc-pvdz. NERSC Cori: 2x16-core 2.3 GHz Haswell Xeon E5-2698, 128 GB DDR MHz BS=16 W8 W10 W12 MKL BLIS MKL BLIS MKL BLIS dgemm pack kernel dcopy BS=32 dgemm pack kernel dcopy
13 Note on BLAS primitives Given two vectors, it is possible to perform: Inner product (dot): c = i a ib i Outer product (ger): c ij = a i b j Pointwise product: c i = a i b i axpy = inner product + translation: c i = k a ikb k gemm = inner product + outer product c ij = k a ikb jk inner product + pointwise product? c i = k a ikb ik all three? c ijk = m a ikmb jkm
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