1. Introduction
Many of the large dense matrices that appear in scientific computing, machine learning, integral equations, and other applications are in fact
data sparse. They may be approximated, to arbitrary accuracy, with a memory footprint that is much smaller that the prohibitive cost of the dense representation. Data sparsity is a consequence of the fact that certain blocks of the matrix may be represented by factorizations whose ranks are smaller than the block dimensions. These low rank blocks of the matrix may be of different sizes and may appear in general position in the matrix. The resulting representations are known as hierarchical matrices.Hierarchical matrices are often used to compress kernel matrices, where a general point set and an explicit kernel function are given. This is a natural setting for Nbody problems, boundary and volume integral methods, spatial statistics, regression and discretizations of fractional and nonlocal operators among others. Hierarchical matrices are also effective in settings where explicit kernels do not exist. Inverses of discretization of differential operators, Schur complements, and Hessians of PDEconstrained optimization problems also result in matrices that have a structure with many low rank blocks. General linear algebraic operations are possible in the compressed representations, presenting opportunities for tackling challenging computations and provide scalable solutions for operations that would be otherwise computationally infeasible with the dense format.
Hierarchical matrices come in different variants. A basic version uses a fixed blocking of the matrix (socalled weak admissibility partitioning), where every block touches the diagonal and is stored as a separate low rank factorization. The asymptotic memory footprint of this representation is , where is a representative rank of the blocks. Unfortunately, this basic representation requires substantial memory in practice due to both the factor and the large ranks
generally needed to reach reasonable target accuracy, as a result of the rigid partitioning of the matrix. This can be improved on in two ways. The first eliminates the logarithmic factor in the complexity estimates by expressing the block factorizations in a nested basis. The second improvement, which allows a substantial reduction in the effective
, allows both dense and low rank blocks of different granularity to appear in general position in the matrix, essentially allowing the matrix structure itself to be adapted to a given problem (strong admissibility partitioning). The hierarchical matrix variant with these two improvements is referred to as the format, and reaches the optimal asymptotic memory footprint . Given that memory is often the primary constraining resource in high performance computing environments, the representation of choice for large scale problems is a general, nested basis, strong admissibility, representation. matrices were originally motivated primarily by the needs of integral equations [32], but their algebraic nature has allowed their application in a variety of other contexts, including, for example, the representation of DirichlettoNeumann operators [27, 26], particulate flows in periodic geometries of arbitrary shapes [39], Hessians of PDEconstrained problems [9].There is tremendous interest in the development of highperformance software libraries for constructing and operating on hierarchical matrices, driven naturally by their compelling small memory footprint and lower asymptotic operational complexity. Much like quality libraries such as FMM3D [1] have made it possible to spread the use of FMM technology to broad classes of problems, similar libraries for matrices—which are algebraic generalizations of FMM—would also allow a broader use of this technology for tackling dense matrix problems that might otherwise be prohibitively expensive. Software packages that are not performanceoriented but more oriented towards readability and conciseness include [35, 40, 11, 2]. A sharedmemory highperformance package for matrices is presented in [36]. A high quality software for distributedmemory environments that targets large scale problems is STRUMPACK [5]. Distributed memory algorithms for constructing and operating on these matrices were proposed in [37, 17, 50]. Dynamic runtime systems to manage the scheduling of the operations of the irregular hierarchical matrix structure in a more convenient fashion through explicit task graphs are presented in [7, 18]. Manycore algorithms were presented in [44, 55], and BiCG solvers on GPU clusters were demonstrated in [49]. Multicore and manycore methods for operating on hierarchical matrices including matrixvector multiplication are discussed in [53, 51, 24]. Algorithms for high performance computing environments suitable for large scale problems are presented in [46, 25, 54, 45, 52].
H2Opus [3]
is an open source, GPUaccelerated, package for hierarchical matrix computations in the
format. It supports a number of core operations (see Figure 1) for constructing these matrices from a kernel and admissibility conditions, performing BLASlike operations including matrixvector multiplication, basis orthogonalization, and recompression. It also provides facilities for adding a (globally) low rank matrix to an matrix, as well as the ability to construct an matrix from randomized sampling operations, essentially generalizing the construction of globally low rank matrix decomposition [33] to the hierarchically low rank case. These core operations form the building blocks for higher level operations that perform matrixmatrix multiplication, generate approximate inverses iteratively, and compute Schur complements in various applications [20, 9]. The package runs on CPUs (with explicit optimizations for Intel and AMD CPUs) as well as on GPUs with the portability enabled by a lower level layer that either uses batch BLAS operations specialized to run directly on the GPUs when manycore hardware is available, or alternatively uses OpenMP to obtain batch parallelism in multicore environments. In addition, the library also supports running on the NECSX Vector Engine. H2Opus has interfaces to the PETSc [13, 14] package, allowing the use of the extensive facilities of the latter for manipulating distributedmemory (sparse) matrices arising from the discretization of PDEs. Efficient solvers are also available in the so called Tile Low Rank format [22].In this paper, we extend the H2Opus package with distributedmemory multiGPU operations for two of its core capabilities: matrixvector multiplication (and the related multiplication of a hierarchical matrix by multiple vectors), and matrix recompression. SingleGPU versions of these algorithms were presented in [19]
. Matrixvector multiplication is a key building block of Krylovtype iterative solvers, appearing in the inner loops of nonlinear and eigenvalue solvers. Therefore improving its performance can substantially reduce overall time to solution. The multivector case is also a performance critical kernel in many contexts such as randomized hierarchical matrixmatrix multiplication, and blockKrylov methods, such as those exposed by PETSc
[38]. The core matrixvector multiplication can benefit from processing multiple vectors concurrently; the additional arithmetic intensity made available, relative to looping over the bandwidthlimited operation of single vector multiplication, allows substantially higher absolute performance to be achieved. Matrix recompression is also another key building block for matrix operations, closely resembling the truncation operations on dense matrices and matrix blocks. Recompression is needed when an initialmatrix approximation is generated using a polynomial interpolation or other nonoptimal bases, and algebraic recompression step is relied on to produce a storageoptimal representation for the desired accuracy. It is also needed when BLAS3 operations are performed on
matrices. These operations generally increase the apparent rank of various blocks, which would then need to be recompressed to maintain the optimal complexity.We present high performance and scalable implementations of these algorithms that demonstrate nearideal scalability on up to 1024 NVIDIA V100 GPUs, with performance exceeding 2.3 Tflop/s per GPU. At this scale, a dense kernel matrix of size 536M536M, represented as a hierarchical matrix to an accuracy of can be multiplied by a single vector in ms and by 64 vectors concurrently in ms, achieving 85% of the peak efficiency of batched GEMM operations. This pushes the state of the art beyond the results of [54, 52, 46]. The compression operation also achieves nearoptimal scalability with number of GPUs, with matrices of size 67M67M compressed by a factor of 6 in their low rank memory (from an accuracy of to an accuracy of ) in around ms on 64 GPUs.
The algorithms are supported by distributed data structures for representing, accessing, and operating on hierarchical matrices with nested bases. These data structures have elements similar to those found in parallel multigrid representations, with analogous restriction and prolongation transfer operators, since the low rank blocks and their bases appear at different levels of granularity and are naturally stored at multiple levels of a hierarchy. The data structures also have patterns similar to those of parallel sparse matrix representations because at every level of the hierarchy, the low rank blocks appear in general locations forming essentially a block sparse matrix whose block sizes are on the order of the block ranks, . The matrix structure has a bounded sparsity constant [28, 16] which is exploited to optimize overall interprocess communication volume and to hide much of the MPI communication cost with concurrent local compute phases of the algorithms. Single node performance is obtained by marshaling the tree data structures in a way to allow optimized batched dense linear algebra kernels to be invoked.
We finally demonstrate the use of the algorithms in the scalable solution of a 2D variablecoefficient integral fractional diffusion equation. We use the distributedmemory algorithms to construct and compress the dense operator to an accuracy of . The solution uses a Krylov method with the matrixvector multiplication done by our distributed algorithm and the preconditioning done by a classical (nonfractional) diffusion equation which is solved by a distributedmemory algebraic multigrid method in PETSc. The results show that all aspects of the solver, including the setup of the hierarchical representation of the dense operators, the computation of the volume “Dirichlet” conditions, the setup of the preconditioner, and the work per iteration, scale linearly with problem size. The solver also exhibits a dimensionindependent number of iterations and results in nearideal weak scalability up to grids of size on 64 GPUs.
The rest of this paper is organized as follows. Section 2
describes the distributed representation of the data structures of an
matrix. Section 3 describes the basic distributed matrixvector multiplication operations and Section 4 presents the key optimizations to minimize interprocess communication volume and hide latencies. Section 5 describes the recompression algorithms which shares many of the algorithmic structures of the matrixvector multiplication algorithms but relies on batch QR and SVD instead of batch GEMM in lower level linear algebra. Section 6 presents the distributed multiGPU scalability results for the matrixvector multiplication and recompression algorithms as well as the integral fractional diffusion solver. We conclude with future potential directions in Section 7.2. Distributed Data Structures for Matrices
2.1. Matrix Structure
The representation of an matrix exploits two kinds of hierarchies: a hierarchy of blocks at different levels of granularity and a hierarchy of bases that is used to represent the data in the blocks. At every level of the hierarchy the low rank blocks essentially form a block sparse matrix. In addition, there is a block sparse matrix that stores the dense blocks of the matrix that are not amenable to low rank representations.

Hierarchy of Blocks. Let and be the sets of indices of the rows and columns of a matrix, and and
be hierarchical clusterings of these index sets, respectively. All blocks within the matrix can then be defined by cluster pairs
. Define an admissible block as a block that is either small enough to be stored in dense form of size , with denoting the socalled leaf size, or can be well approximated by a low rank matrix. A matrix in the variant of hierarchical matrices can be partitioned into admissible blocks of various sizes that are organized hierarchically as the leaves of a matrix tree. A low rank matrix on level of the tree and defined by the cluster pair is represented as the outer product of two tall matrices and of rank : .The left panel of Figure 1(a) shows the leaves of this matrix tree which define a complete partitioning of the matrix into blocks, with dense leaves colored red and low rank leaves colored green. Figure 1(b) shows the various levels of the tree, where inner nodes are in cyan. The structure of this matrix tree depends on the application. For example, in problems involving a spatial discretization, we can leverage data from the particles or nodes generated by the discretization to generate it with the aid of a geometric admissibility condition [31]
. Other applications might rely on the available algebraic data or heuristics to determine which blocks of the matrix are admissible as low rank blocks.
Hierarchy of Bases. The representation uses block low rank factorizations of the form for every block at every level , where and are bases for the block rows and block column of level , respectively. is a small coupling matrix representing the interaction between the row and column cluster pairs. These coupling matrices are organized in the matrix tree defined by . In our implementation, we currently use a fixed rank per level that we refer to as , or even simply when the context is clear, which allows us to use higherperforming fixedsize batch linear algebra kernels when executing basis operations.
An additional hierarchy is introduced in the row and column basis trees, where a basis node is expressed in terms of the bases of its children. Basis nodes are only stored explicitly at the leaves of the tree, and inner nodes can be computed from their children using small interlevel transfer matrices. For example, one can compute an inner node from its two children, and , and their corresponding transfer matrices and as
The explicit bases at the leaves and the interlevel transfer matrices are organized in a basis tree. Figure 3 shows an example of the nested basis , with the implicitly represented inner nodes shaded.
Putting it all together, the matrix representation may be succinctly written as
where is a block sparse matrix with dense blocks of size , shown as the red leaves at the finest level of the matrix tree in Figure 1(a). is a matrix tree of coupling matrices, whose leaves, which appear at different levels, are shown as the cyan blocks in the right panel of Figure 1(a), and and are the block row and block column basis trees, each consisting of explicitly stored bases at the leaf level, and interlevel transfer matrices and , as shown in Figure 3 for basis tree . The tripletree product notation is defined as the assembly of all blocks of all levels where every block may be computed as .
2.2. Distributed Representation and Construction
Symbol  Description 

,  Total number of GPUs, and index of the local GPU 
Complete basis trees  
Complete matrix tree  
Local branch of the basis tree on GPU  
Root branch of the basis tree on the master process  
Local branch of the matrix tree on GPU  
Root branch of the matrix tree on the master process  
Matrix tree branch, from dual tree traversal of and  
, , etc.  Marshaled , , etc. arrays for batched execution 
Input and output vectors  
Local branches of vector trees and on GPU  
Local subvectors of the input and output on GPU 
The description of the parallel distribution of the matrix and the algorithms operating on it uses the notation in Table 1. We use a left subscript to refer to local data owned by a GPU, and a triple bar subscript to refer to data that has been marshaled to allow batched GPU kernels to be invoked on it.
Treating each level of the matrix tree of the hierarchical matrix as a block sparse matrix, we decompose the levels into block rows, with each block row stored on a single GPU, as illustrated in Figure 4. The basis trees are similarly decomposed by level, assigning the nodes corresponding to the stored block rows/columns to the same GPU. This decomposition allows much of the upsweep and downsweep to be carried out independently as described in Section 3 for matrixvector multiplication and in Section 5 for matrix compression. Above a critical Clevel, the decomposition stops and a single root GPU owns all top levels. The interlevel transfer operators at the roots of each of the local branch basis trees are duplicated at the leaf level of (see Figure 4) to allow upsweep and downsweep operations to begin and end at the Clevel.
We construct the distributed matrix by first splitting the row cluster tree into independent branches at level and a local branch of the basis tree on GPU is generated for each . The top levels of the basis tree are kept on a master process and could potentially be split further if becomes large enough. The root branch of the matrix tree can be generated on the master process in any number of ways, such as general admissibility dual tree traversal [31]. Once this matrix tree is constructed, a list of the basis nodes from can be extracted for each GPU corresponding to the inadmissible nodes of the block row at level of the matrix tree. As an example, consider the distribution on GPUs of the fourth level of the matrix tree in Figure 1(b). The column indices corresponding to the cyan nodes are extracted for GPU , setting from the second block row. Once the list for each process has been compiled, they are scattered from the master process to all other processes.
Since the nodes in were generated from the inadmissible nodes of the matrix tree that have to be subdivided further, the structure of the matrix tree on each GPU can then be generated independently using multiple dual tree traversals of the root node of with the nodes in , where each traversal generates a local branch . Once the structure has been determined, the entries of the transfer and leaf matrices of the basis trees, together with the entries of the matrix tree can be populated independently on each GPU using established techniques [16].
3. Distributedmemory MatrixVector Multiplication
This section describes the overall structure of the distributed multiplication operation with vectors concurrently. We use this structure to highlight the interprocess communication bottlenecks, that are then optimized in Section 4.
The input multivector of size is distributed among GPUs, where each subvector is extracted from the index set defined by the root node of . The hierarchical matrixvector product operation requires a standard block sparse multiplication for the dense blocks of the matrix, which can be overlapped with the low rank multiplication . The low rank portion of the product is a generalization to the hierarchical tree setting of a regular dense low rank matrix by a vector, and is illustrated in Figure 5. In a first phase, we perform an upsweep of the basis tree and compute , which contains the products of the matrices of all block columns at all levels with the corresponding pieces of . In a second phase, a tree is computed. also consists of multilevel data corresponding to the products of for every level . In the third phase, a downsweep phase through the basis tree accumulates the multilevel tree data into the output vector , an operation we may symbolically write as .
3.1. Distributed Upsweep
The upsweep phase is illustrated in Figure 6 and summarized in Algorithm 2. It proceeds from the leaf level where the explicitly stored block row bases, each of size , are simply multiplied by . At every higher level, can be computed from the children at level using the transfer operators [16]. For a parent node with children and we have
In the distributed setting, the upsweep on each GPU can proceed in parallel on each of the local branches independently, using the kernel shown in Algorithm 1. Once the upsweep reaches the roots of each branch, the data from all root nodes of the trees are gathered on the master process to populate the leaf level of the root branch , allowing the upsweep to complete. The regular upsweep Algorithm 1 can be used for the branch upsweep as well as the root upsweep by simply omitting the first batched operation on the leaves of the root branch, since that step is replaced by the gathering of the data.
From an efficiency perspective, the primary challenge in the upsweep comes from the fact that the individual operations in the basis tree nodes involve small matrix operators that do not possess a sufficient amount of data parallelism for effective GPU utilization. We overcome this problem by marshaling the appropriate level data to allow batched GPU kernels to be invoked, allowing to be computed with only batched kernel executions. Except for the top few levels of the root process , these executions happen concurrently on the GPUs. The marshaling GPU kernel is shown for a step of the upsweep operation in Algorithm 3. The marshaling kernel essentially plays the role of a fast static scheduler for all operations performed on a given level of the tree. It prepares for the execution of the operations by efficiently gathering data from the basis tree. Marshaling involves no data movement and therefore has very little overhead and in practice consumes only a tiny fraction of total execution time.
3.2. Distributed Intermediate Multiplication
The second phase of the operation builds a vector tree , where each node in a level is the product of the block row of level of the coupling matrix tree with the corresponding nodes in . This operation can be expressed as
where is the set of column indices of the matrix blocks in the block row . This is a block sparse matrixvector multiplication at every level, and is illustrated in the middle portion of Figure 5, where the blockrow, multiGPU decomposition of each level is also highlighted. The scalar version of this problem is a wellstudied computational kernel that has many possible high quality solutions [30, 41, 15, 23] which may be adapted to the blocksparse version. While we could rely on vendor kernels, their relatively low performance and lack of support for nonuniform blocks necessitates a different approach. Our solution relies on a key result regarding hierarchical matrices which puts a bound on the sparsity constant , the maximum number of blocks in any block row at any level of the matrix tree. Such constant is bounded by a dimensionindependent value that only depends on the specific structure of the matrix and the admissibility criterion[28, 16].
During the construction of the matrix tree, we generate conflictfree batches of matrix products that can then be executed by a series of nonuniform batched matrixvector and matrixmatrix kernels. A conflictfree ordering of the batch indices can be obtained by assigning a batch index based on the position within the block row or column that increases from left to right, allowing us to marshal all the batches in a single marshaling kernel. While there will potentially be some kernels that perform little work, the bounded sparsity constant guarantees that there will be few of them, and they will thus represent a small portion of the total runtime. This could be optimized further by running those small batches on a separate low priority stream and then combining the results with the main execution stream.
The boundedness of the sparsity constants also guarantees that the block row local to a GPU will require input from nodes belonging to a limited number of remote GPUs: we therefore consider a standard approach in distributed memory sparse matrix vector products, and split the block row into diagonal and offdiagonal parts. The contributions of these two parts can be overlapped with the needed communication as described in Section 4.1. Once the needed parts of are assembled, the multiplication phase can proceed on each GPU independently on the coupling matrices of using the treeMultiply routine of Algorithm 4. Processing the multiplication of the root branch on the master process to produce the root branch finalizes the phase as shown in Algorithm 5. The selectiveGather routine communicates and assembles the needed portion of the tree from the remote branches to use in the local treeMultiply. This is described in Section 4.1.
3.3. Distributed Downsweep
The vector tree now contains, at every level , a partial contribution to the output vector expressed in terms of the bases , and it could be computed as if the bases at level were explicitly available. However, since we only store interlevel transfer operators, we express in terms of bases of the children of node and accumulate them in a downsweep phase through the tree from root to leaves. This partial accumulation takes the form
between a parent node at level and children and at the finer level .
The downsweep procedure, pictured in the left part of Figure 5, is summarized in Algorithm 7. A downsweep through the root subtree is first performed. Once the leaf level of has been updated, it can be scattered from the master process to all other GPUs and added to the roots of local trees containing data from the previous phase of the multiplication. When the roots of the local branches on the GPUs have their proper accumulation, obtained from the leaves of the tree , we can sweep down independently on all GPUs setting each node to ; the level at each step now also contains the partial sum of for all levels above expressed in the basis of . The leaf level will then contain the complete sum which is finally expanded into through a multiplication by the explicitly stored leaf level bases. We follow the same approach used for the upsweep, where each level is also processed in parallel by first marshaling the operations and then executing using a batch matrix vector product. This leads us to Algorithm 6 for computing , i.e., the distributed vector . The downsweep marshaling algorithm is structurally similar to the one described in Algorithm 3 and is omitted for brevity.
4. Optimizing Communication
4.1. Optimizing Communication Volume
We first discuss a communicationoptimized parallel block sparse matrixvector multiplication for a given level matrix of , the low rank portion of the hierarchical matrix; the same logic applies to and it is omitted for brevity.
The matrix tree is first split into two distinct trees: a diagonal matrix tree and an offdiagonal matrix tree . While the offdiagonal portion could simply be represented as a set of trees, with one tree for every interaction with a GPU , it is far more efficient to have them all in a single flattened one to allow marshaling operations on the offdiagonal blocks to be completed with a single kernel call.
Once the trees are split, the list of basis nodes that interact with can be generated by iterating through its pairs and determining all unique values. Given the boundedness of the sparsity constant , on a given level , a GPU will receive data only from a few other GPUs; we thus determine the list of GPUs that need to send data to , as well as the list of nodes that should be received, and store such information in a compressedstorage format as representatively shown in Figure 7. The data needed from process pid, corresponding to global indices listed in nodes from nodes_ptr to nodes_ptr, is communicated among GPUs during the setup phase. A list of unique entries of nodes represents the compressed storage for the offiagonal block.
pid  1  3  

nodes_ptr  0  2  5  
nodes  5  6  12  13  14 
The diagonal multiplication phase can proceed without any communication while the offdiagonal phase needs input from other GPUs. The nodes at a level of the local branch which are needed by other GPUs are packed into a single buffer using a marshaling kernel and a single batched copy kernel populates a send buffer , which is then used to issue nonblocking sends to the neighboring GPUs. Nonblocking receives are issued to populate a receive buffer , which can then be directly used for the offdiagonal level of the tree, since the column basis indices have already been updated to use the compressed node format.
Algorithm 8 summarizes the overall communicationoptimized multiplication phase. The exchangeData routine executes the nonblocking MPI sends and receives using the compressed offdiagonal data that is marshaled using the marshalOffdiag routine and copied using a batch kernel.
4.2. Achieving Communication and Computation Overlap
While the communication volume has been reduced and a portion of the communication can be hidden by overlapping it with the diagonal multiplication, there are still a few challenges left that hold back performance. Assuming the lack of hardware support for more advanced memory transfer features such as GPUDirect RDMA, transferring the data from the GPU to the host adds some overhead to the execution, as well as synchronization points that impact GPU usage. Moreover, not all MPI distributions guarantee that communication can progress in the background without explicit intervention from the process.
The effectiveness of overlapping the diagonal multiplication phase with the needed communication depends on the structure of the tree, the capabilities of the communication network, and the compute capabilities of the GPU. For a tailored, finegrained control of the communications involved in the algorithm at hand, we explicitly create communication threads that queue up asynchronous copies on separate streams to overlap the transfers with the processing of each level of the local branch upsweep. The transfers can then be carried out on the same thread without interrupting or forcing synchronizations with the main execution stream, allowing communication and computation to overlap. When the diagonal block kernels have been queued up on the stream, the communication thread can then join the main thread. The multiplication of the root branch on the master process can also be hidden, to some extent, by overlapping it with the main stream of work, and scheduling it on a low priority stream. This will allow the work to be completed during phases of low GPU utilization, such as the smaller top levels of the basis and matrix tree. Finally, by adding the dense block multiplication phase to a low priority stream, GPU utilization on all GPUs can be increased as well.
Figure 8 shows the effect of overlapping communication with computation on the timeline of the overall distributed matrixvector multiplication. As expected, the gaps in the timeline due to communication are significantly smaller when it is overlapped with the computation.
5. Algebraic Matrix Compression
algebraic matrix compression is an operation that takes as input an matrix and produces another matrix of lower rank that approximates the input to a desired target accuracy. Compression is akin to the truncation operation that is commonly used to approximate dense matrices or dense matrix blocks by low rank approximations.
Recompression is a core operation when working with matrices. In particular, it is common in the discretization of integral equations to generate an initial matrix from a kernel and an admissibility condition by approximating the kernel with Chebyshev polynomials for well separated clusters. The ranks produced by this approximation are not optimal however, resulting in increased storage and increased arithmetic costs for operations such as the matrixvector multiplication. A recompression step, purely algebraic, is then performed to generate an optimal basis in which the ranks are as small as possible. Another context where recompression is essential arises when adding matrices, performing low rank updates, or implementing BLAS3like operations. When matrix blocks get added there is an increase in the apparent rank of the resulting blocks. The matrix would then need to be recompressed in order to maintain the linear asymptotic rate for storage and operations. The key task of recompression is to generate a new nested compressed basis in which the matrix blocks are expressed. This may be done by a pair of downsweep and upsweep operations that we describe next.
5.1. Downsweep for Generating a New Basis
Consider how a new basis for a block row at the finest level would be generated. here denotes only the low rank portions of the hierarchical matrix, since the dense blocks are not compressed. consists of low rank blocks expressed at level as with , and additional pieces representing the restriction of blocks from coarser levels to their “i” rows.
(1) 
A new more efficient basis can be generated by computing the SVD of , and using the left singular vectors as the new basis. This would however require an expensive SVD of the sized
. In order to avoid it, we first compute the QR decomposition of
and then perform the SVD on the small factor.(2) 
The new basis for level , is effectively a reweighing of the columns of the previous basis .
The task of computing of the QR decomposition of can be done efficiently by exploiting the nestedness of the bases. Let us assume that the QR decomposition of , the parent block at level , is available as . Then,
(3) 
with conveniently expressible as:
(4) 
Assuming the bases are orthogonal, the block diagonal matrix in Eq. 4 is orthogonal and the QR of simply reduces to the QR of the small stack at the end of Eq. 4 which involves only blocks, each being a small coupling/transfer matrix. Since this QR uses the matrix from level , the overall computation starts from the root and goes down the tree computing all the matrices for all levels in a downsweep traversal.
As with previous operations, all blocks at a given level can be processed in parallel, and importantly for the distributed setting, all the processes owning the subtrees below the Clevel can proceed independently. Therefore, the computational pattern is identical to the distributed downsweep of the matrixvector multiplication. Above the Clevel, a single GPU is responsible for processing the top levels of the basis tree. The leaves of the top subtree, which hold the factors at level , are then scattered to all GPUs and seed the roots of the individual subtrees, which continue the downseep independently all the way to the leaf level of the basis. The computational work at every level being, for every block row, a QR decomposition of the small stack at the end of Eq. 4. Batched QR operations [21] are used within every GPU to achieve high performance.
5.2. Upsweep for Truncating the New Basis
Once the new reweighed basis is generated for all levels, it can then be truncated to the desired accuracy to generate the optimal basis . The coupling blocks are then compressed by projecting them on the new truncated basis.
The truncation operation has to preserve the nestedness property. This can be accomplished by starting the truncation at the leaf level and sweeping up the basis tree. Given the reweighed basis ( at the leaves and at nodes in the tree), we seek a basis ( at leaves and at nodes in the tree) that spans the same subspace as to the desired accuracy.
At the leaf level, this may be done through an SVD of the explicitly available basis
is a subset of columns of the left singular vectors . is a transformation matrix (which may be computed as ) used to compute the new transfer matrix from a node to its parent as described next.
For nonleaves, the truncated basis is expressed in terms of new compressed transfer operators:
and are the new transfer matrices for the children of a nonleaf node , and is a transformation that will be used to compute the new transfer matrix from to its parent, in a similar fashion. are computed by first computing an SVD of the matrix
. The left singular vectors corresponding to singular values below the target compression threshold are truncated, and the remaining subset is partitioned to generate the new transfer matrices
and . is computed as .The structure of the truncation algorithm is identical to that of the upsweep phase in the matrixvector multiplication. All GPUs start the truncation operations concurrently, each on its subtree, with no interprocess communication required. The computational work at every level being, for every block row, an SVD involving the leaf basis at the leaf level, or the stacked transfer operators for nonleaf levels. Within a GPU, batched SVDs are used [19]. Once all GPUs reach the Clevel, a gather operation communicates the new transfer operators from the roots of the subtrees to the leaves of the root tree that is stored on a single GPU. This bootstraps the last phase of the upweep which proceeds on the root GPU.
Two details remain to be discussed. First, in the downsweep phase, the algorithm relied on the basis being orthogonal. When this is not the case, a preprocessing step is needed to orthogonalize the basis tree. A basis is orthogonal if
is the identity matrix for all levels
. Orthogonalizing a basis involves performing QR on the finest level basis and then going up the tree to compute new transfer matrices that allow higher level nodes to satisfy the orthogonality condition. This is also done in an upsweep pass that is very similar to the one described above for truncation, but replacing the SVD operations by QR operations. The distributed implementation therefore also proceeds independently on all GPUs up until the Clevel. A gather operation is then performed to bootstrap the orthogonalization of the top levels of the basis tree which reside on a single GPU.Finally, once a new compressed basis is computed, it remains to approximate the coupling blocks in it. Since is an orthogonal basis, the best approximation of a block is obtained by an orthogonal projection. Therefore, we can obtain the approximation by projecting every low rank block on the new basis to obtain:
The new coupling blocks are computed via batched matrix multiplication operations. In this step, there is parallelism across all GPUs and across all levels. In addition, The GPUs are particularly efficient in GEMM operations, and in practice this projection step consumes much less time that the other operations, particularly those involving batched SVDs.
6. Performance Results
In this section, we report performance results for the distributedmemory matrixvector multiplication and algebraic compression operations, as well as performance on a complete application involving the setup and solution of an integral equation.
6.1. Experimental Setup
The H2Opus library as well as the code to generate and run all examples below is open source and is available at https://github.com/ecrc/h2opus. Batched kernels for GEMM operations are executed using MAGMA [12, 4], while SVD and QR batches are performed using the algorithms shipped with the opensource library KBLAS, available at https://github.com/ecrc/kblasgpu. The marshaling GPU kernel and various other utility routines use Thrust [6].
All tests are conducted on Summit, the IBM Power System AC922 supercomputer installed at Oak Ridge National Laboratory. Individual nodes on Summit have 6 NVIDIA V100SXM2 GPUs with 16GB of HBM2 memory each, but we only used 4 GPUs per node (2 per socket) in our runs. Summit has a fast hosttodevice bandwidth which can deliver up to 50 GB/s over PCIe 4.0; in our application, we were able to use a significant portion of it, 40 GB/s, as measured by nvprof. It also has a fattree topology network for internode communication that delivers 200Gb/s of bandwidth. To assess the efficiency attained by our algorithms, we measure the performance of the single GPU batched GEMM implementation from MAGMA, with batch elements of size . All computations are done in doubleprecision and every point in every plot has been generated as the average of 10 runs after discarding the fastest and slowest timings.
To test the performance and scalability of the matrixvector multiplication and matrix compression implementations, we performed numerical experiments on two sample matrix sets with different structural characteristics. The first matrix set comes from a spatial statistics application using a point set placed on a 2D grid of side length , and an exponential kernel with a correlation length of . The hierarchical matrix representation of this covariance matrix uses as the finest block size and size of the dense leaves. A geometric admissibility condition is used, where and refer to the center and diagonal size of a bounding box of the corresponding point set. We use a value of and set a rank in the low rank blocks, resulting in an approximation with relative accuracy better than for all problem sizes. This accuracy is computed by sampling of the rows and computing
with randomly generated vectors with entries from a uniform distribution. The resulting sparsity constant of the matrix, which is a proxy for how finely refined the matrix is in its offdiagonal portions, is
. At the largest size, the matrix has 23 levels, with the top 10 levels on a single master GPU, and the bottom 13 levels on separate 1024 GPUs for a matrix size of 536M.The second matrix set comes from a 3D Gaussian process and is intended to show the effect of memory pressure—due to a finer refinement in the offdiagonal blocks—on scalability. The matrices are constructed using a set of points on a 3D grid of side length and uses an exponential kernel with correlation length [10], a similar admissibility condition to the previous case, and a rank for the low rank blocks. The resulting relative accuracy is now for all problem sizes considered in this section, and the resulting matrix tree has many more leaves than in the 2D case with a larger sparsity constant .
6.2. MatrixVector Multiplication
6.2.1. Weak Scalability
We first report on the weak scalability of the implementation, using a number of vectors ranging from 1 to 64. Both 2D and 3D test sets were scaled using a local matrix size of per GPU. Results are summarized in Figure 9, with the top and bottom rows showing the results from the 2D and 3D test sets, respectively. We observed no significant variability in the timings, with the highs and lows within 1–3% of reported average. There is a slight jitter in the plots as the problems are scaled up, due to small changes in the structure of the matrix tree affecting the amount of actual computations performed, which do not grow exactly at the same rate as the problem size. The relative efficiency was computed as , the ratio of the relative flops performed and the scaling factor relative to a base case of GPUs.
For the 2D tests, the scalability is near ideal up to 512 GPUs across all multiple vector sizes, . For the single vector case, a bandwidthlimited operation, the throughput is about 150 Gflop/s per GPU. With , the additional arithmetic intensity pushes the computation into a computebound regime and achieves 2.6 Tflop/s per GPU, more than 95% of the sustained peak of the batched GEMM operation measured at 2.7 Tflop/s. Even with the larger data volume being communicated, corresponding to various parts of 64 vectors, the communication is essentially hidden by the local computations. Only with 1024 GPUs, the plots show a slight degradation in performance and a deviation from ideal scalability, particularly for the case, with performance dropping to 2.3 Tflop/s per GPU. The primary reason for this is that the root tree on the master GPU is now 10 levels deep and is of size . The computations on this sizeable top tree become a bottleneck. In order to scale up to this number of GPUs or larger, we will need to coarsen to a single top GPU more smoothly than the to ratio used in the current implementation. For example, the trees above a first Clevel can be distributed on multiple GPUs, and back to a single GPU after a second Clevel, so that the top level work is small enough to be overlapped with other parts of the computation.
The results of the 3D problem test set display a similar behavior, but the communication overhead appears earlier. For the single vector case, results scale reasonably well with problem size and the relative efficiency is about 90% with 1024 GPUs, with a performance of about 130 Gflop/s per GPU. As the number of vectors increases however, the communication needed for transferring all the relevant portions of the vectors to the GPUs that need those data becomes substantial. This is in addition to the larger root tree difficulty mentioned above. The compute part of the operation grows sublinearly with problem size because of the favorable increase in arithmetic intensity; however, the communication volume grows linearly and can no longer be hidden by the now relatively faster compute phases. While a performance of 2.6 Tflop/s per GPU is reached for 2 GPUs, the performance at 1024 GPUs reaches only 1.1 Tflop/s, an efficiency less than 45%.
6.2.2. Strong Scalability
We report, in Figure 10, strong scalability results of the matrixvector multiplication for the 2D and 3D test data described above, for different numbers of test vectors. In all cases, the problem size is to fit on a single GPU of Summit and is executed on an increasingly larger number of GPUs. As expected, the more arithmetically intensive multiplication involving a larger set of vectors scales better than the single vector operation. In all cases however, the limits of strong scalability is reached around 32 GPUs. This is not unexpected, since at this scale the local problem size is now only . There is very little local work to do to hide communication, and the whole computation takes only few milliseconds.
6.3. Matrix Compression
To test the performance and scalability of the algebraic compression routines, we used the same sets of matrices described earlier. For the 2D tests, we start with the matrix defined as above, with point clusters of size , and admissibility parameter . Its low rank blocks are initially constructed using a Chebyshev polynomial approximation of the kernel in the bounding boxes of the point clusters. A grid is used, resulting in all low rank blocks having a uniform rank . Compression seeks to reduce these ranks to maintain an accuracy of . The 2D test set was scaled using a local matrix size , reaching a matrix size of 67M on 64 GPUs.
For the 3D tests, a point cluster size and an admissibility parameter are used in the matrix construction. The low rank blocks are initially computed using a tricubic Chebyshev polynomial approximation of the kernel in the bounding boxes of the point clusters, resulting in all low rank blocks having a uniform rank . Compression seeks to reduce these ranks to maintain an accuracy of . The test test was scaled using a local matrix size , reaching a matrix size of 16M on 64 GPUs.
6.3.1. Weak Scalability
Figure 11 shows the weak scalability and effectiveness of the compression operation. Compression starts with an initial phase to orthogonalize the (nonorthogonal) bases constructed by Chebyshev interpolation without modifying their ranks. We time and report the orthogonalization phase separately. This is followed by a pair of downsweep/upsweep passes to construct a new reweighed basis and truncate it, finalized with a projection of the low rank blocks on the new bases. These steps are reported together under the compression phase label. The orthogonalization phase involves fewer floating point operations and as a result it executes faster than the compression phase. In 2D, the scaling is near ideal for up to 64 GPUs, with a slight bump when moving beyond a single GPU, because of interGPU communication. However for , the execution time is essentially flat indicating that all communication has been hidden by the computational steps. In 3D, we note a slight deviation from ideal scalability when , because the communication volume is larger than the 2D case and it can no longer be totally hidden by local computations. The orthogonalization phase executes at around 2.1 Gflop/s/GPU (near the practicallyachievable peak of 2.7 Gflop/s per GPU, as measured by the batched GEMM operation). The compression phase, featuring QR on stacks of blocks and SVD kernels, is not able to reach that level of performance, but this limitation comes purely form the single GPU performance of the corresponding batched routines for these operations, and would be directly improved when more performant batch kernels are implemented. Nonetheless, we should note that even with our current batch kernels, the 67M and 16M matrices in 2D and 3D respectively, are compressed in just a fraction of a second, as reported in the central column of Figure 11.
In the rightmost column of Figure 11, we report on the effectiveness of the compression operation in reducing the memory footprint of the low rank blocks. In the 2D case, there is a factor of reduction between precompression low rank memory (with all blocks having a uniform rank ) and postcompression to an accuracy of . In the 3D case, the compression is a factor of for the low rank data, primarily because the starting matrix was generated with a relatively small footprint (all blocks have rank 64, generated from tricubic polynomials) and not much reduction is possible to maintain an accuracy of . In all cases however, we note the ideal growth in memory.
6.3.2. Strong Scalability
Strong scalability results are shown in Figure 12. Deviation from ideal scalability becomes noticeable as soon as the local problem size is small enough where there are not enough computations to perform locally and communication time dominates. For the 2D tests, the problem size is . With GPUs, the local problem size is only and results in an efficiency reduction to near . On 32 GPUs, the local problem size is and the limit of strong scalability is essentially reached: there is very little local work to do and the whole operation takes a few ms, spent mostly in communications. For the 3D tests, the problem size is . On GPUs, the local problem size is already only , and the resulting efficiency drops below . With 16 GPUs, the strong scalability limit is essentially reached as the problem size is now only , with very little local work available for each GPU.
6.4. Integral Fractional Diffusion Equation
We consider the performance of a Krylov solver for the solution of the integral equation where is the fractional diffusion operator defined as:
(5) 
The spatially varying diffusivity
is defined as the geometric mean of the usual diffusion coefficient,
, is the fractional order () and we assume it to be constant in this experiment, is the region in which the solution is sought, and is a surrounding region in which volume constraints, somewhat analogous to the Dirichlet boundary conditions in the classical diffusion equations, are imposed on the solution, for . We solve the problem with , , and constant fractional order . We use a diffusivity field of the form(6) 
with the bump function defined as:
(7) 
and a right hand side in .
The singularity of the kernel in (5) requires that the discretization of the integral be done carefully to allow standard quadrature rules to attain their theoretical convergence rate. To this end, we rewrite the integral as:
(8) 
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