.. Grid documentation .. highlight:: cpp Welcome to Grid's documentation! ================================================== .. toctree:: :maxdepth: 2 :caption: Contents: Preliminaries ==================================== .. attention:: manual version 1 (CD) Grid is primarily an *application* *development* *interface* (API) for structured Cartesian grid codes and written in C++11. In particular it is aimed at Lattice Field Theory simulations in general gauge theories, but with a particular emphasis on supporting SU(3) and U(1) gauge theories relevant to hadronic physics. Who will use this library --------------------------- As an application development interface *Grid* is primarily a programmers tool providing the building blocks and primitives for constructing lattice gauge theory programmes. Grid functionality includes: * Data parallel primitives, similar to QDP++ * gauge and fermion actions * solvers * gauge and fermion force terms * integrators and (R)HMC. * parallel field I/O * object serialisation (text, XML, JSON...) Grid is intended to enable the rapid and easy development of code with reasonably competitive performance. It is first and foremost a *library* to which people can programme, and develop new algorithms and measurements. As such, it is very much hoped that peoples principle point of contact with Grid will be in the wonderfully rich C++ language. Since import and export procedures are provided for the opaque lattice types it should be possible to call Grid from other code bases. Grid is most tightly coupled to the Hadrons package developed principally by Antonin Portelli. This package is entirely composed against the Grid data parallel interface. Interfacing to other packages is also possible. Several regression tests that combine Grid with Chroma are included in the Grid distribution. Further, Grid has been successfully interfaced to * The Columbia Physics System * The MILC code Data parallel interface ---------------------------------------------------------------------------------- Most users will wish to interact with Grid above the data parallel *Lattice* interface. At this level a programme is simply written as a series of statements, addressing entire lattice objects. Implementation details may be provided to explain how the code works, but are not strictly part of the API. **Example** For example, as an implementation detail, in a single programme multiple data (SPMD) message passing supercomputer the main programme is trivially replicated on each computing node. The data parallel operations are called *collectively* by all nodes. Any scalar values returned by the various reduction routines are the same on each node, resulting in (for example) the same decision being made by all nodes to terminate an iterative solver on the same iteration. Internal development ------------------------------------------ Internal developers may contribute to Grid at a level below the data parallel interface. Specifically, development of new lattice Dirac operators, for example, or any codes directly interacting with the * Communicators * Simd * Tensor * Stencil will make use of facilities provided by to assist the creation of high performance code. The internal data layout complexities will be exposed to some degree and the interfaces are subject to change without notice as HPC architectures change. Since some of the internal implementation details are needed to explain the design strategy of grid these will be documented, but labelled as *implementation dependent* Reasonable endeavours will be made to preserve functionality where practical but no guarantees are made. Reporting Bugs =================================== To help us tracking and solving more efficiently issues with Grid, please report problems using the issue system of GitHub rather than sending emails to Grid developers. When you file an issue, please go though the following checklist: * Check that the code is pointing to the HEAD of develop or any commit in master which is tagged with a version number. * Give a description of the target platform (CPU, network, compiler). Please give the full CPU part description, using for example cat /proc/cpuinfo | grep 'model name' | uniq (Linux) or sysctl machdep.cpu.brand_string (macOS) and the full output the --version option of your compiler. * Give the exact configure command used. * Attach config.log. * Attach grid.config.summary. * Attach the output of make V=1. * Describe the issue and any previous attempt to solve it. If relevant, show how to reproduce the issue using a minimal working example. Download, installation and build ========================================= Required libraries ---------------------------------------- * GMP, * MPFR * Eigen: bootstrapping grid downloads and uses for internal dense matrix (non-QCD operations) the Eigen library. Grid optionally uses: * HDF5 * LIME for ILDG and SciDAC file format support. * FFTW either generic version or via the Intel MKL library. * LAPACK either generic version or Intel MKL library. Compilers --------- * Intel ICPC v17 and later * Clang v3.5 and later (need 3.8 and later for OpenMP) * GCC v4.9.x * GCC v6.3 and later (recommended) **Important:** Some versions of GCC appear to have a bug under high optimisation (-O2, -O3). The safety of these compiler versions cannot be guaranteed at this time. Follow Issue 100 for details and updates. GCC v5.x GCC v6.1, v6.2 Quick start ------------ First, start by cloning the repository:: git clone https://github.com/paboyle/Grid.git Then enter the cloned directory and set up the build system:: cd Grid ./bootstrap.sh Now you can execute the `configure` script to generate makefiles (here from a build directory):: mkdir build; cd build ../configure --enable-precision=double --enable-simd=AVX --enable-comms=mpi-auto \ --prefix= where:: --enable-precision=single|double sets the **default precision**. Since this is largely a benchmarking convenience, it is anticipated that the default precision may be removed in future implementations, and that explicit type selection be made at all points. Naturally, most code will be type templated in any case.:: --enable-simd=GEN|SSE4|AVX|AVXFMA|AVXFMA4|AVX2|AVX512|NEONv8|QPX sets the **SIMD architecture**, :: --enable-comms=mpi|none selects whether to use MPI communication (mpi) or no communication (none). :: --prefix= should be passed the prefix path where you want to install Grid. Other options are detailed in the next section, you can also use :: configure --help to display them. Like with any other program using GNU autotool, the :: CXX, CXXFLAGS, LDFLAGS, ... environment variables can be modified to customise the build. Finally, you can build, check, and install Grid:: make; make check; make install If you want to build all the tests just use `make tests`. Detailed build configuration options ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ======================================== ============================================================================================================================== Option usage ======================================== ============================================================================================================================== ``--prefix=path`` installation prefix for Grid. ``--with-gmp=path`` look for GMP in the UNIX prefix `` ``--with-mpfr=path`` look for MPFR in the UNIX prefix `` ``--with-fftw=path`` look for FFTW in the UNIX prefix `` ``--with-lime=path`` look for c-lime in the UNIX prefix `` ``--enable-lapack[=path]`` enable LAPACK support in Lanczos eigensolver. A UNIX prefix containing the library can be specified (optional). --enable-mkl[=path] use Intel MKL for FFT (and LAPACK if enabled) routines. A UNIX prefix containing the library can be specified (optional). --enable-simd=code setup Grid for the SIMD target ``(default: `GEN`). A list of possible SIMD targets is detailed in a section below. --enable-gen-simd-width=size select the size (in bytes) of the generic SIMD vector type (default: 32 bytes). E.g. SSE 128 bit corresponds to 16 bytes. --enable-precision=single|double set the default precision (default: `double`). --enable-comms=mpi|none use `` for message passing (default: `none`). --enable-rng=sitmo|ranlux48|mt19937 choose the RNG (default: `sitmo`). --disable-timers disable system dependent high-resolution timers. --enable-chroma enable Chroma regression tests. --enable-doxygen-doc enable the Doxygen documentation generation (build with `make doxygen-doc`) ======================================== ============================================================================================================================== Possible communication interfaces ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The following options can be use with the `-\\-enable-comms=` option to target different communication interfaces: =============== ========================================================================================== Description =============== ========================================================================================== `none` no communications `mpi` MPI communications with compiler CXX `mpi-auto` MPI communications with compiler CXX but clone flags from MPICXX =============== ========================================================================================== For the MPI interfaces the optional `-auto` suffix instructs the `configure` scripts to determine all the necessary compilation and linking flags. This is done by extracting the informations from the MPI wrapper specified in the environment variable `MPICXX` (if not specified `configure` will scan though a list of default names). The `-auto` suffix is not supported by the Cray environment wrapper scripts. Use the standard wrappers ( `CXX=CC` ) set up by Cray `PrgEnv` modules instead. Possible SIMD types ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The following options can be use with the `-\\-enable-simd=` option to target different SIMD instruction sets: ============ ===================================================================================================================== `` Description ============ ===================================================================================================================== `GEN` generic portable vector code `SSE4` SSE 4.2 (128 bit) `AVX` AVX (256 bit) `AVXFMA` AVX (256 bit) + FMA `AVXFMA4` AVX (256 bit) + FMA4 `AVX2` AVX 2 (256 bit) `AVX512` AVX 512 bit `NEONv8` [ARM NEON](http://infocenter.arm.com/help/index.jsp?topic=/com.arm.doc.den0024a/ch07s03.html) (128 bit) `QPX` IBM QPX (256 bit) ============ ===================================================================================================================== Alternatively, some CPU codenames can be directly used: ============ ===================================================================================================================== `` Description ============ ===================================================================================================================== `KNL` [Intel Xeon Phi codename Knights Landing](http://ark.intel.com/products/codename/48999/Knights-Landing) `SKL` [Intel Skylake with AVX512 extensions](https://ark.intel.com/products/codename/37572/Skylake#@server) `BGQ` Blue Gene/Q ============ ===================================================================================================================== Notes ^^^^^^^ * We currently support AVX512 for the Intel compiler and GCC (KNL and SKL target). Support for clang will appear in future versions of Grid when the AVX512 support in the compiler is more advanced. * For BG/Q only [bgclang](http://trac.alcf.anl.gov/projects/llvm-bgq) is supported. We do not presently plan to support more compilers for this platform. * BG/Q performances are currently rather poor. This is being investigated for future versions. * The vector size for the `GEN` target can be specified with the `configure` script option `-\\-enable-gen-simd-width`. Build setup for Intel Knights Landing platform --------------------------------------------------------------------------------------- The following configuration is recommended for the Intel Knights Landing platform:: ../configure --enable-precision=double\ --enable-simd=KNL \ --enable-comms=mpi-auto \ --enable-mkl \ CXX=icpc MPICXX=mpiicpc The MKL flag enables use of BLAS and FFTW from the Intel Math Kernels Library. If you are working on a Cray machine that does not use the `mpiicpc` wrapper, please use:: ../configure --enable-precision=double\ --enable-simd=KNL \ --enable-comms=mpi \ --enable-mkl \ CXX=CC CC=cc If gmp and mpfr are NOT in standard places (/usr/) these flags may be needed:: --with-gmp= \ --with-mpfr= where `` is the UNIX prefix where GMP and MPFR are installed. Knight's Landing with Intel Omnipath adapters with two adapters per node presently performs better with use of more than one rank per node, using shared memory for interior communication. We recommend four ranks per node for best performance, but optimum is local volume dependent. :: ../configure --enable-precision=double\ --enable-simd=KNL \ --enable-comms=mpi-auto \ --enable-mkl \ CC=icpc MPICXX=mpiicpc Build setup for Intel Haswell Xeon platform --------------------------------------------------------------------------------------- The following configuration is recommended for the Intel Haswell platform:: ../configure --enable-precision=double\ --enable-simd=AVX2 \ --enable-comms=mpi-auto \ --enable-mkl \ CXX=icpc MPICXX=mpiicpc The MKL flag enables use of BLAS and FFTW from the Intel Math Kernels Library. If gmp and mpfr are NOT in standard places (/usr/) these flags may be needed:: --with-gmp= \ --with-mpfr= where `` is the UNIX prefix where GMP and MPFR are installed. If you are working on a Cray machine that does not use the `mpiicpc` wrapper, please use:: ../configure --enable-precision=double\ --enable-simd=AVX2 \ --enable-comms=mpi \ --enable-mkl \ CXX=CC CC=cc Since Dual socket nodes are commonplace, we recommend MPI-3 as the default with the use of one rank per socket. If using the Intel MPI library, threads should be pinned to NUMA domains using:: export I_MPI_PIN=1 This is the default. Build setup for Intel Skylake Xeon platform ---------------------------------------------------------------------------- The following configuration is recommended for the Intel Skylake platform:: ../configure --enable-precision=double\ --enable-simd=AVX512 \ --enable-comms=mpi \ --enable-mkl \ CXX=mpiicpc The MKL flag enables use of BLAS and FFTW from the Intel Math Kernels Library. If gmp and mpfr are NOT in standard places (/usr/) these flags may be needed:: --with-gmp= \ --with-mpfr= \ where `` is the UNIX prefix where GMP and MPFR are installed. If you are working on a Cray machine that does not use the `mpiicpc` wrapper, please use:: ../configure --enable-precision=double\ --enable-simd=AVX512 \ --enable-comms=mpi \ --enable-mkl \ CXX=CC CC=cc Since Dual socket nodes are commonplace, we recommend MPI-3 as the default with the use of one rank per socket. If using the Intel MPI library, threads should be pinned to NUMA domains using:: export I_MPI_PIN=1 This is the default. Build setup for AMD EPYC / RYZEN ---------------------------------------------------------------------------- The AMD EPYC is a multichip module comprising 32 cores spread over four distinct chips each with 8 cores. So, even with a single socket node there is a quad-chip module. Dual socket nodes with 64 cores total are common. Each chip within the module exposes a separate NUMA domain. There are four NUMA domains per socket and we recommend one MPI rank per NUMA domain. MPI-3 is recommended with the use of four ranks per socket, and 8 threads per rank. The following configuration is recommended for the AMD EPYC platform:: ../configure --enable-precision=double\ --enable-simd=AVX2 \ --enable-comms=mpi \ CXX=mpicxx If gmp and mpfr are NOT in standard places (/usr/) these flags may be needed:: --with-gmp= \ --with-mpfr= where `` is the UNIX prefix where GMP and MPFR are installed. Using MPICH and g++ v4.9.2, best performance can be obtained using explicit GOMP_CPU_AFFINITY flags for each MPI rank. This can be done by invoking MPI on a wrapper script omp_bind.sh to handle this. It is recommended to run 8 MPI ranks on a single dual socket AMD EPYC, with 8 threads per rank using MPI and shared memory to communicate within this .. describe:: command line mpirun -np 8 ./omp_bind.sh ./Benchmark_dwf --mpi 2.2.2.1 --dslash-unroll --threads 8 --grid 16.16.16.16 --cacheblocking 4.4.4.4 Where omp_bind.sh does the following:: #!/bin/bash numanode=` expr $PMI_RANK % 8 ` basecore=`expr $numanode \* 16` core0=`expr $basecore + 0 ` core1=`expr $basecore + 2 ` core2=`expr $basecore + 4 ` core3=`expr $basecore + 6 ` core4=`expr $basecore + 8 ` core5=`expr $basecore + 10 ` core6=`expr $basecore + 12 ` core7=`expr $basecore + 14 ` export GOMP_CPU_AFFINITY="$core0 $core1 $core2 $core3 $core4 $core5 $core6 $core7" echo GOMP_CUP_AFFINITY $GOMP_CPU_AFFINITY $@ Build setup for laptops, other compilers, non-cluster builds ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Many versions of g++ and clang++ work with Grid, and involve merely replacing CXX (and MPICXX), and omit the enable-mkl flag. Single node builds are enabled with:: --enable-comms=none FFTW support that is not in the default search path may then enabled with:: --with-fftw= BLAS will not be compiled in by default, and Lanczos will default to Eigen diagonalisation. Execution model ============================================ Grid is intended to support performance portability across a many of platforms ranging from single processors to message passing CPU clusters and accelerated computing nodes. The library provides data parallel C++ container classes with internal memory layout that is transformed to map efficiently to SIMD architectures. CSHIFT facilities are provided, similar to HPF and cmfortran, and user control is given over the mapping of array indices to both MPI tasks and SIMD processing elements. Identically shaped arrays then be processed with perfect data parallelisation. Such identically shaped arrays are called conformable arrays. The transformation is based on the observation that Cartesian array processing involves identical processing to be performed on different regions of the Cartesian array. The library will both geometrically decompose into MPI tasks and across SIMD lanes. Local vector loops are parallelised with OpenMP pragmas. Data parallel array operations can then be specified with a SINGLE data parallel paradigm, but optimally use MPI, OpenMP and SIMD parallelism under the hood. This is a significant simplification for most programmers. The two broad optimisation targets are: * MPI, OpenMP, and SIMD parallelism Presently SSE4, ARM NEON (128 bits) AVX, AVX2, QPX (256 bits), and AVX512 (512 bits) targets are supported with aggressive use of architecture vectorisation intrinsic functions. * MPI between nodes with and data parallel offload to GPU's. For the latter generic C++ code is used both on the host and on the GPU, with a common vectorisation granularity. Accelerator memory model ---------------------------------------------- For accelerator targets it is assumed that heap allocations can be shared between the CPU and the accelerator. This corresponds to lattice fields having their memory allocated with *cudaMallocManaged* with Nvidia GPU's. Grid does not assume that stack or data segments share a common address space with an accelerator. * This constraint presently rules out porting Grid to AMD GPU's which do not support managed memory. * At some point in the future a cacheing strategy may be implemented to enable running on AMD GPU's Data parallel API ===================================== Data parallel array indices are divided into two types. * Internal indices, such as complex, colour, spin degrees of freedom * spatial (space-time) indices. The ranges of all internal degrees are determined by template parameters, and known at compile time. The ranges of spatial indices are dynamic, run time values and the Cartesian structure information is contained and accessed via `Grid` objects. Grid objects are the controlling entity for the decomposition of a distributed `Lattice` array across MPI tasks, nodes, SIMD lanes, accelerators. Threaded loops are used as appropriate on host code. (binary) Data parallel operations can only be performed between Lattice objects constructed from the same Grid pointer. These are called `conformable` operations. We will focus initially on the internal indices as these are the building blocks assembled in Lattice container classes. Every Lattice container class constructor requires a Grid object pointer. Tensor classes ---------------------------- The Tensor data structures are built up from fundamental scalar matrix and vector classes:: template class iScalar { private: vobj _internal ; } template class iVector { private: vobj _internal[N] ; } template class iMatrix { private: vobj _internal[N][N] ; } These are template classes and can be passed a fundamental scalar or vector type, or nested to form arbitrarily complicated tensor products of indices. All mathematical expressions are defined to operate recursively, index by index. Presently the constants * Nc * Nd are globally predefined. However, this is planned for changed in future and policy classes for different theories (e.g. QCD, QED, SU2 etc...) will contain these constants and enable multiple theories to coexist more naturally. Arbitrary tensor products of fundamental scalar, vector and matrix objects may be formed in principle by the basic Grid code. For Lattice field theory, we define types according to the following tensor product structure ordering. The suffix "D" indicates either double types, and replacing with "F" gives the corresponding single precision type. The test cases have R, which takes the compiled default precision (either F or D). This is for convenience only and may be deprecated in future forcing code external to Grid to choose the specific word size. Type definitions are provided in qcd/QCD.h to give the internal index structures of QCD codes. For example:: template using iSinglet = iScalar > >; using iSpinMatrix = iScalar, Ns> >; using iColourMatrix = iScalar > > ; using iSpinColourMatrix = iScalar, Ns> >; using iLorentzColourMatrix = iVector >, Nd > ; using iDoubleStoredColourMatrix = iVector >, Nds > ; using iSpinVector = iScalar, Ns> >; using iColourVector = iScalar > >; using iSpinColourVector = iScalar, Ns> >; using iHalfSpinVector = iScalar, Nhs> >; using iHalfSpinColourVector = iScalar, Nhs> >; Giving the type table: ======= ======= ====== ====== =========== ======================= Lattice Lorentz Spin Colour scalar_type Field ======= ======= ====== ====== =========== ======================= Scalar Scalar Scalar Scalar RealD RealD Scalar Scalar Scalar Scalar ComplexD ComplexD Scalar Scalar Scalar Matrix ComplexD ColourMatrixD Scalar Vector Scalar Matrix ComplexD LorentzColourMatrixD Scalar Scalar Vector Vector ComplexD SpinColourVectorD Scalar Scalar Vector Vector ComplexD HalfSpinColourVectorD Scalar Scalar Matrix Matrix ComplexD SpinColourMatrixD ======= ======= ====== ====== =========== ======================= The types are implemented via a recursive tensor nesting system. **Example** Here, the prefix "i" indicates for internal use, preserving the template nature of the class. Final types are declared with vtype selected to be both scalar and vector, appropriate to a single datum, or stored in a partial SoA transformed lattice object:: // LorentzColour typedef iLorentzColourMatrix LorentzColourMatrix; typedef iLorentzColourMatrix LorentzColourMatrixF; typedef iLorentzColourMatrix LorentzColourMatrixD; typedef iLorentzColourMatrix vLorentzColourMatrix; typedef iLorentzColourMatrix vLorentzColourMatrixF; typedef iLorentzColourMatrix vLorentzColourMatrixD; Arbitrarily deep tensor nests may be formed. Grid uses a positional and numerical rule to associate indices for contraction in the Einstein summation sense. ============= ======= ======== Symbolic name Number Position ============= ======= ======== LorentzIndex 0 left SpinIndex 1 middle ColourIndex 2 right ============= ======= ======== The conventions are that the index ordering left to right are: Lorentz, Spin, Colour. A scalar type (either real or complex, single or double precision) is be provided to the innermost structure. Tensor arithmetic rules (lib/tensors/Tensor_arith.h) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Arithmetic rules are defined on these types The multiplication operator follows the natural multiplication table for each index, index level by index level. `Operator *` == == == == x S V M == == == == S S V M V S S V M M V M == == == == The addition and subtraction rules disallow a scalar to be added to a vector, and vector to be added to matrix. A scalar adds to a matrix on the diagonal. *Operator* + and *Operator* - === == == == +/- S V M === == == == S S M V V M M M === == == == The rules for a nested objects are recursively inferred level by level from basic rules of multiplication addition and subtraction for scalar/vector/matrix. Legal expressions can only be formed between objects with the same number of nested internal indices. All the Grid QCD datatypes have precisely three internal indices, some of which may be trivial scalar to enable expressions to be formed. Arithmetic operations are possible where the left or right operand is a scalar type. **Example**:: LatticeColourMatrixD U(grid); LatticeColourMatrixD Udag(grid); Udag = adj(U); RealD unitary_err = norm2(U*adj(U) - 1.0); Will provide a measure of how discrepant from unitarity the matrix U is. Internal index manipulation (lib/tensors/Tensor_index.h) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ General code can access any specific index by number with a peek/poke semantic:: // peek index number "Level" of a vector index template auto peekIndex (const vtype &arg,int i); // peek index number "Level" of a vector index template auto peekIndex (const vtype &arg,int i,int j); // poke index number "Level" of a vector index template void pokeIndex (vtype &pokeme,arg,int i) // poke index number "Level" of a matrix index template void pokeIndex (vtype &pokeme,arg,int i,int j) **Example**:: for (int mu = 0; mu < Nd; mu++) { U[mu] = PeekIndex(Umu, mu); } Similar to the QDP++ package convenience routines are provided to access specific elements of vector and matrix internal index types by physics name or meaning aliases for the above routines with the appropriate index constant. * peekColour * peekSpin * peekLorentz and * pokeColour * pokeSpin * pokeLorentz For example, we often store Gauge Fields with a Lorentz index, but can split them into polarisations in relevant pieces of code. **Example**:: for (int mu = 0; mu < Nd; mu++) { U[mu] = peekLorentz(Umu, mu); } For convenience, direct access as both an l-value and an r-value is provided by the parenthesis operator () on each of the Scalar, Vector and Matrix classes. For example one may write **Example**:: ColourMatrix A, B; A()()(i,j) = B()()(j,i); bearing in mind that empty parentheses are need to address a scalar entry in the tensor index nest. The first (left) empty parentheses move past the (scalar) Lorentz level in the tensor nest, and the second (middle) empty parantheses move past the (scalar) spin level. The (i,j) index the colour matrix. Other examples are easy to form for the many cases, and should be obvious to the reader. This form of addressing is convenient and saves peek, modifying, poke multiple temporary objects when both spin and colour indices are being accessed. There are many cases where multiple lines of code are required with a peek/poke semantic which are easier with direct l-value and r-value addressing. Matrix operations ^^^^^^^^^^^^^^^^^^^^^^^^^^ Transposition and tracing specific internal indices are possible using:: template auto traceIndex (const vtype &arg) template auto transposeIndex (const vtype &arg) These may be used as **Example**:: LatticeColourMatrixD Link(grid); ComplexD link_trace = traceIndex (Link); Again, convenience aliases for QCD naming schemes are provided via * traceColour * traceSpin * transposeColour * transposeSpin **Example**:: ComplexD link_trace = traceColour (Link); The operations only makes sense for matrix and scalar internal indices. The trace and transpose over all indices is also defined for matrix and scalar types:: template auto trace(const iMatrix &arg) -> iScalar template auto transpose(const iMatrix &arg ) -> iMatrix Similar functions are: * conjugate * adjoint The traceless anti-Hermitian part is taken with:: template iMatrix Ta(const iMatrix &arg) SU(N) Reunitarisation (or reorthogonalisation) is enabled by:: template iMatrix ProjectOnGroup(const iMatrix &arg) **Example**:: LatticeColourMatrixD Mom(grid); LatticeColourMatrixD TaMom(grid); TaMom = Ta(Mom); Querying internal index structure ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Templated code may find it useful to use query functions on the Grid datatypes they are provided. For example general Serialisation and I/O code can inspect the nature of a type a routine has been asked to read from disk, or even generate descriptive type strings:: //////////////////////////////////////////////////// // Support type queries on template params: //////////////////////////////////////////////////// // int _ColourScalar = isScalar(); // int _ColourVector = isVector(); // int _ColourMatrix = isMatrix(); template int isScalar(void) template int isVector(void) template int isMatrix(void) **Example** (lib/parallelIO/IldgIO.h):: template std::string ScidacRecordTypeString(int &colors, int &spins, int & typesize,int &datacount) { ///////////////////////////////////////// // Encode a generic tensor as a string ///////////////////////////////////////// typedef typename getPrecision::real_scalar_type stype; int _ColourN = indexRank(); int _ColourScalar = isScalar(); int _ColourVector = isVector(); int _ColourMatrix = isMatrix(); int _SpinN = indexRank(); int _SpinScalar = isScalar(); int _SpinVector = isVector(); int _SpinMatrix = isMatrix(); int _LorentzN = indexRank(); int _LorentzScalar = isScalar(); int _LorentzVector = isVector(); int _LorentzMatrix = isMatrix(); std::stringstream stream; stream << "GRID_"; stream << ScidacWordMnemonic(); if ( _LorentzVector ) stream << "_LorentzVector"<<_LorentzN; if ( _LorentzMatrix ) stream << "_LorentzMatrix"<<_LorentzN; if ( _SpinVector ) stream << "_SpinVector"<<_SpinN; if ( _SpinMatrix ) stream << "_SpinMatrix"<<_SpinN; if ( _ColourVector ) stream << "_ColourVector"<<_ColourN; if ( _ColourMatrix ) stream << "_ColourMatrix"<<_ColourN; if ( _ColourScalar && _LorentzScalar && _SpinScalar ) stream << "_Complex"; typesize = sizeof(typename vobj::scalar_type); if ( _ColourMatrix ) typesize*= _ColourN*_ColourN; else typesize*= _ColourN; if ( _SpinMatrix ) typesize*= _SpinN*_SpinN; else typesize*= _SpinN; }; Inner and outer products ^^^^^^^^^^^^^^^^^^^^^^^^^ We recursively define (tensors/Tensor_inner.h), ultimately returning scalar in all indices:: ///////////////////////////////////////////////////////////////////////// // innerProduct Scalar x Scalar -> Scalar // innerProduct Vector x Vector -> Scalar // innerProduct Matrix x Matrix -> Scalar ///////////////////////////////////////////////////////////////////////// template auto innerProductD (const iScalar& lhs,const iScalar& rhs) template auto innerProductD (const iVector& lhs,const iVector& rhs) template auto innerProductD (const iMatrix& lhs,const iMatrix& rhs) template auto innerProduct (const iScalar& lhs,const iScalar& rhs) template auto innerProduct (const iVector& lhs,const iVector& rhs) template auto innerProduct (const iMatrix& lhs,const iMatrix& rhs) The sum is always performed in double precision for the innerProductD variant. We recursively define (tensors/Tensor_outer.h):: ///////////////////////////////////////////////////////////////////////// // outerProduct Scalar x Scalar -> Scalar // Vector x Vector -> Matrix ///////////////////////////////////////////////////////////////////////// template auto outerProduct (const iScalar& lhs,const iScalar& rhs) template auto outerProduct (const iVector& lhs,const iVector& rhs) Functions of Tensor ^^^^^^^^^^^^^^^^^^^^^^^^^ The following unary functions are defined, which operate element by element on a tensor data structure:: sqrt(); rsqrt(); sin(); cos(); asin(); acos(); log(); exp(); abs(); Not(); toReal(); toComplex(); Element wise functions are defined for:: div(tensor,Integer); mod(tensor,Integer); pow(tensor,RealD); Matrix exponentiation (as opposed to element wise exponentiation is implemented via power series in:: Exponentiate(const Tensor &r ,RealD alpha, Integer Nexp = DEFAULT_MAT_EXP) the exponentiation is distributive across vector indices (i.e. proceeds component by component for a LorentzColourMatrix). Determinant is similar:: iScalar Determinant(const Tensor &r ) Vectorisation -------------- Internally, Grid defines a portable abstraction SIMD vectorisation, via the following types: * vRealF * vRealD * vComplexF * vComplexD These have the usual range of arithmetic operators and functions acting upon them. They do not form part of the API, but are mentioned to (partially) explain the need for controlling the layout transformation in lattice objects. They contain a number consecutive elements of the appropriate Real/Complex type, where number is architecture depemendent. The number may be queried at runtime using:: vComplexF::Nsimd(); The layout transformations in indexing functions in the Grid objects as completely parameterised by this Nsimd(). They are documented further in the Internals chapter. Coordinates ------------ The Grid is define on a N-dimensional set of integer coordinates. The maximum dimension is eight, and indexes in this space make use of the Coordinate class. The coordinate class shares a similar interface to `std::vector`, but contains all data within the object, and has a fixed maximum length (template parameter). **Example**:: const int Nd=4; Coordinate point(Nd); for(int i=0;i _processors; // Which dimensions get relayed out over processors lanes. int _processor; // linear processor rank std::vector _processor_coor; // linear processor coordinate unsigned long _ndimension; Grid_MPI_Comm communicator; The final of these is potentially an MPI Cartesian communicator, mapping some total number of processors to an N-dimensional coordinate system. This is used by Grid to geometrically decompose the subvolumes of a lattice field across processing elements. Grid is aware of multiple ranks per node and attempts to ensure that the geometrical decomposition keeps as many neigbours as possible on the same node. This is done by reordering the ranks in the constructor of a Communicator object once the topology requested has been indicated, via an internal call to the method OptimalCommunicator(). The reordering is chosen by Grid to trick MPI, which makes a simple lexicographic assignment of ranks to coordinate, to ensure that the simple lexicographic assignment of the reordered ranks is the optimal choice. MPI does not do this by default and substantial improvements arise from this design choice. Lattice containers ----------------------------------------- Lattice objects may be constructed to contain the local portion of a distribued array of any tensor type. For performance reasons the tensor type uses a vector Real or Complex as the fundamental datum. Every lattice requires a GridBase object pointer to be provided in its constructor. Memory is allocated at construction time. If a Lattice is passed a RedBlack grid, it allocates half the storage of the full grid, and may either store the red or black checkerboard. The Lattice object will automatically track through assignments which checkerboard it refers to. For example, shifting a Even checkerboard by an odd distance produces an Odd result field. Struct of array objects are defined, and used in the template parameters to the lattice class. **Example** (lib/qcd/QCD.h):: template using iSpinMatrix = iScalar, Ns> >; typedef iSpinMatrix SpinMatrixF; //scalar typedef iSpinMatrix vSpinMatrixF;//vectorised typedef Lattice LatticeSpinMatrixF; The full range of QCD relevant lattice objects is given below. ======= ======= ====== ====== =========== ============================= ===================== Lattice Lorentz Spin Colour scalar_type Field Synonym ======= ======= ====== ====== =========== ============================= ===================== Vector Scalar Scalar Scalar Integer LatticeInteger N/A Vector Scalar Scalar Scalar RealD LatticeRealD N/A Vector Scalar Scalar Scalar ComplexD LatticeComplexD N/A Vector Scalar Scalar Matrix ComplexD LatticeColourMatrixD LatticeGaugeLink Vector Vector Scalar Matrix ComplexD LatticeLorentzColourMatrixD LatticeGaugeFieldD Vector Scalar Vector Vector ComplexD LatticeSpinColourVectorD LatticeFermionD Vector Scalar Vector Vector ComplexD LatticeHalfSpinColourVectorD LatticeHalfFermionD Vector Scalar Matrix Matrix ComplexD LatticeSpinColourMatrixD LatticePropagatorD ======= ======= ====== ====== =========== ============================= ===================== Additional single precison variants are defined with the suffix "F". Other lattice objects can be defined using the sort of typedef's shown above if needed. LatticeInteger is typically only used in the form of predicate fields for where statements. Opaque containers ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The layout within the container is complicated to enable maximum opportunity for vectorisation, and is opaque from the point of view of the API definition. The key implementation observation is that so long as data parallel operations are performed and adjacent SIMD lanes correspond to well separated lattice sites, then identical operations are performed on all SIMD lanes and enable good vectorisation. Because the layout is opaque, import and export routines from naturally ordered x,y,z,t arrays are provided (lib/lattice/Lattice_transfer.h):: unvectorizeToLexOrdArray(std::vector &out, const Lattice &in); vectorizeFromLexOrdArray(std::vector &in , Lattice &out); sobj and vobj should be a matching pair of scalar and vector objects of the same internal structure. The compiler will let you know with a long and verbose complaint if they are not. The Lexicographic order of data in the external vector fields is defined by (lib/util/Lexicographic.h):: Lexicographic::IndexFromCoor(const Coordinate &lcoor, int &lex,Coordinate *local_dims); This ordering is :math:`x + L_x * y + L_x*L_y*z + L_x*L_y*L_z *t` Peek and poke routines are provided to perform single site operations. These operations are extremely low performance and are not intended for algorithm development or performance-critical code. The following are `collective` operations and involve communication between nodes. All nodes receive the same result by broadcast from the owning node:: void peekSite(sobj &s,const Lattice &l,const Coordinate &site); void pokeSite(const sobj &s,Lattice &l,const Coordinate &site); The following are executed independently by each node:: void peekLocalSite(sobj &s,const Lattice &l,Coordinate &site); void pokeLocalSite(const sobj &s,Lattice &l,Coordinate &site); Lattices of one tensor type may be transformed into lattices of another tensor type by peeking and poking specific indices in a data parallel manner:: template // Vector data parallel index peek auto PeekIndex(const Lattice &lhs,int i); template // Matrix data parallel index peek auto PeekIndex(const Lattice &lhs,int i,int j); template // Vector poke void PokeIndex(Lattice &lhs,const Lattice<> & rhs,int i) template // Matrix poke void PokeIndex(Lattice &lhs,const Lattice<> & rhs,int i,int j) The inconsistent capitalisation on the letter P is due to an obscure bug in g++ that has not to our knowledge been fixed in any version. The bug was reported in 2016. .. todo:: CD: Do you want to mention/expose PropToFerm and FermToProp? Are there other such convenience routines to make part of the API? Global Reduction operations ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Reduction operations for any lattice field are provided. The result is identical on each computing node that is part of the relevant Grid communicator:: template RealD norm2(const Lattice &arg); template ComplexD innerProduct(const Lattice &left,const Lattice &right); template vobj sum(const Lattice &arg) Site local reduction operations ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Internal indices may be reduced, site by site, using the following routines:: template auto localNorm2 (const Lattice &rhs) template auto localInnerProduct (const Lattice &lhs,const Lattice &rhs) Outer product ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ A site local outer product is defined:: template auto outerProduct (const Lattice &lhs,const Lattice &rhs) Slice operations ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Slice operations are defined to operate on one lower dimension than the full lattice. The omitted dimension is the parameter orthogdim:: template void sliceSum(const Lattice &Data, std::vector &result, int orthogdim); template void sliceInnerProductVector( std::vector & result, const Lattice &lhs, const Lattice &rhs, int orthogdim); template void sliceNorm (std::vector &sn, const Lattice &rhs, int orthogdim); Data parallel expression template engine ------------------------------------------ The standard arithmetic operators and some data parallel library functions are implemented site by site on lattice types. Operations may only ever combine lattice objects that have been constructed from the **same** grid pointer. **Example**:: LatticeFermionD A(&grid); LatticeFermionD B(&grid); LatticeFermionD C(&grid); A = B - C; Such operations are said to be **conformable** and are the lattice are guaranteed to have the same dimensions and both MPI and SIMD decomposition because they are based on the same grid object. The conformability check is lightweight and simply requires the same grid pointers be passed to the lattice objects. The data members of the grid objects are not compared. Conformable lattice fields may be combined with appropriate scalar types in expressions. The implemented rules follow those already documented for the tensor types. Unary operators and functions ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The following sitewise unary operations are defined: ===================== ============================================ Operation Description ===================== ============================================ operator- negate adj Hermitian conjugate conjugate complex conjugate trace sitewise trace transpose sitewise transpose Ta take traceles anti Hermitian part ProjectOnGroup reunitarise or orthogonalise real take the real part imag take the imaginary part toReal demote complex to real toComplex promote real to complex timesI elementwise +i mult (0 is not multiplied) timesMinusI elementwise -i mult (0 is not multiplied) abs elementwise absolute value sqrt elementwise square root rsqrt elementwise reciprocal square root sin elementwise sine cos elementwise cosine asin elementwise inverse sine acos elementwise inverse cosine log elementwise logarithm exp elementwise exponentiation operator! Logical negation of integer field Not Logical negation of integer field ===================== ============================================ The following sitewise applied functions with additional parameters are:: template Lattice pow(const Lattice &rhs_i,RealD y); template Lattice mod(const Lattice &rhs_i,Integer y); template Lattice div(const Lattice &rhs_i,Integer y); template Lattice expMat(const Lattice &rhs_i, RealD alpha, Integer Nexp = DEFAULT_MAT_EXP); Binary operators ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The following binary operators are defined:: operator+ operator- operator* operator/ Logical are defined on LatticeInteger types:: operator& operator| operator&& operator|| Ternary operator, logical operations and where ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Within the data parallel level of the API the only way to perform operations that are differentiated between sites is use predicated execution. The predicate takes the form of a LatticeInteger which is confromable with both the iftrue and iffalse argument:: template void where(const Lattice &pred, Lattice &iftrue, Lattice &iffalse); This plays the data parallel analogue of the C++ ternary operator:: a == b ? c : d; In order to create the predicate in a coordinate dependent fashion it is often useful to use the lattice coordinates. The LatticeCoordinate function:: template LatticeCoordinate(Lattice &coor,int dir); fills an Integer field with the coordinate in the direction specified by "dir". A usage example is given **Example**:: int dir = 3; int block = 4; LatticeInteger coor(FineGrid); LatticeCoordinate(coor,dir); result = where(mod(coor,block)==(block-1),x,z); This example takes result to be either "x" or "z" in a coordinate dependent way. When third (z) lattice coordinate lies at the boundaries of a block size (periodic arithmetic). This example is lifted and paraphrased from code that (data parallel) evaluates matrix elements for a coarse representation of the Dirac operator in multigrid. Other usage cases of LatticeCoordinate include the generation of plane wave momentum phases. Site local fused operations ------------------------------------------ The biggest limitation of expression template engines is that the optimisation visibility is a single assignment statement in the original source code. There is no scope for loop fusion between multiple statements. Multi-loop fusion gives scope for greater cache locality. Two primitives for hardware aware parallel loops are provided. These will operate directly on the site objects which are expanded by a factor of the vector length (in our struct of array datatypes). Since the mapping of sites to data lanes is opaque, these vectorised loops are *only* appropriate for optimisation of site local operations. View objects ^^^^^^^^^^^^^^ Due to an obscure aspect of the way that Nvidia handle device C++11 lambda functions, it is necessary to disable the indexing of a Lattice object. Rather, a reference to a lattice object must be first obtained. The reference is copyable to a GPU, and is able to be indexed on either accelerator code, or by host code. In order to prevent people developing code that dereferences Lattice objects in a way that works on CPU compilation, but fails on GPU compilation, we have decided to remove the ability to index a lattice object on CPU code. As a result of Nvidia's constraints, all accesses to lattice objects are required to be made through a View object. In the following, the type is LatticeView, however it is wise to use the C++11 auto keyword to avoid naming the type. See code examples below. thread_loops ^^^^^^^^^^^^^^ The first parallel primitive is the thread_loop **Example**:: LatticeField r(grid); LatticeField x(grid); LatticeField p(grid); LatticeField mmp(grid); auto r_v = r.View(); auto x_v = x.View(); auto p_v = p.View(); auto mmp_v = mmp.View(); thread_loop(s , r_v, { r_v[s] = r_v[s] - a * mmp_v[s]; x_v[s] = x_v[s] + a*p_v[s]; p_v[s] = p_v[s]*b + r_v[s]; }); accelerator_loops ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The second parallel primitive is the "accelerator_loop". The thread loop runs on host processor cores only. If the enabled architecture is VGPU, if Grid is configured with --enable-simd=VGPU, the acccelerator_loop may run on a GPU if present. On non-accelerated architectures, the accelerator_loop will simply run as a an OpenMP thread_loop. It is planned to support multiple forms of acccelerator in future, including OpenMP 5.0 offload, and possibly SyCL based offload. **Example**:: LatticeField r(grid); LatticeField x(grid); LatticeField p(grid); LatticeField mmp(grid); auto r_v = r.View(); auto x_v = x.View(); auto p_v = p.View(); auto mmp_v = mmp.View(); accelerator_loop(s , r_v, { r_v[s] = r_v[s] - a * mmp_v[s]; x_v[s] = x_v[s] + a*p_v[s]; p_v[s] = p_v[s]*b + r_v[s]; }); Cshift ^^^^^^^^^^ Site shifting operations are provided using the Cshift function:: template Lattice Cshift(const Lattice &rhs,int dimension,int shift) This shifts the whole vector by any distance shift in the appropriate dimension. For the avoidance of doubt on direction conventions,a positive shift moves the lattice site :math:`x_\mu = 1` in the rhs to :math:`x_\mu = 0` in the result. **Example** (benchmarks/Benchmark_wilson.cc):: { // Naive wilson implementation ref = Zero(); for(int mu=0;mu void pickCheckerboard(int cb,Lattice &half,const Lattice &full); template void setCheckerboard(Lattice &full,const Lattice &half); These are used to set up Schur red-black decomposed solvers, for example. Multi-grid projection between a fine and coarse grid:: template void blockProject(Lattice > &coarseData, const Lattice &fineData, const std::vector > &Basis); Multi-grid promotion to a finer grid:: template void blockPromote(const Lattice > &coarseData, Lattice &fineData, const std::vector > &Basis) Support for subblock Linear algebra:: template void blockZAXPY(Lattice &fineZ, const Lattice &coarseA, const Lattice &fineX, const Lattice &fineY) template void blockInnerProduct(Lattice &CoarseInner, const Lattice &fineX, const Lattice &fineY) template void blockNormalise(Lattice &ip,Lattice &fineX) template void blockSum(Lattice &coarseData,const Lattice &fineData) template void blockOrthogonalise(Lattice &ip,std::vector > &Basis) Conversion between different SIMD layouts:: template void localConvert(const Lattice &in,Lattice &out) Slices between grid of dimension N and grid of dimentions N+1:: template void InsertSlice(const Lattice &lowDim,Lattice & higherDim,int slice, int orthog) template void ExtractSlice(Lattice &lowDim,const Lattice & higherDim,int slice, int orthog) Growing a lattice by a multiple factor, with periodic replication:: template void Replicate(Lattice &coarse,Lattice & fine) That latter is useful to, for example, pre-thermalise a smaller volume and then grow the volume in HMC. It was written while debugging G-parity boundary conditions. Random number generators ========================================= Grid provides three configure time options for random the number generator engine. * sitmo * ranlux48 * mt19937 The selection is controlled by the --enable-rng=