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Kripke is a simple, scalable, 3D Sn deterministic particle transport code. Its primary purpose is to research how data layout, programming paradigms and architectures effect the implementation and performance of Sn transport. A main goal of Kripke is to investigate how different data-layouts affect instruction, thread and task level parallelism, and what the implications are on overall solver performance.

Kripke supports storage of angular fluxes (Psi) using all six striding orders (or "nestings") of Directions (D), Groups (G), and Zones (Z), and provides computational kernels specifically written for each of these nestings. Most Sn transport codes are designed around one of these nestings, which is an inflexibility that leads to software engineering compromises when porting to new architectures and programming paradigms.

Early research has found that the problem dimensions (zones, groups, directions, scattering order) and the scaling (number of threads and MPI tasks), can make a profound difference in the performance of each of these nestings. To our knowledge, this is a capability unique to Kripke, and should provide key insight into how data-layout affects Sn solver performance. An asynchronous MPI-based parallel sweep algorithm is provided, which employs the concepts of Group Sets (GS), Zone Sets (ZS), and Direction Sets (DS), borrowed from the Texas A&M code PDT.

As we explore new architectures and programming paradigms with Kripke, we will be able to incorporate these findings and ideas into our larger codes. The main advantages of using Kripke for this exploration is that it's light-weight (i.e. easily refactored and modified), and it gets us closer to the real question we want answered: "What is the best way to layout and traverse data in parallel in an Sn code on a given architecture+programming-model?" instead of the more commonly asked question "What is the best way to map my existing Sn code to a given architecture+programming-model?".

Mini App or Proxy App?

Kripke is a Mini-App since it has a very small code base consisting of about 5000 lines of C++ code (using cloc v1.67).

Kripke is also a Proxy-App since it is a proxy for the LLNL transport code ARDRA.


A major challenge of achieving high-performance in an Sn transport (or any physics) code is choosing a data-layout and a parallel decomposition that lends itself to the targeted architecture. Often the data-layout determines the most efficient nesting of loops in computational kernels, which then determines how well your inner-most-loop SIMD vectorizes, how you add threading (pthreads, OpenMP, etc.), and the efficiency and design of your parallel algorithms. Therefore, each nesting produces a different loop nesting orders with substantially different performance characteristics. We want to explore how easily and efficiently these different nestings map to different architectures. In particular, we are interested in exploring how we can achieve good parallel efficiency while also achieving efficient use of node resources (such as SIMD units, memory systems, and accelerators).

Parallel sweep algorithms can be explored with Kripke in multiple ways. The core MPI algorithm could be modified or rewritten to explore other approaches, domain overloading, or alternate programming models (such as Charm++). The effect of load-imbalance is an understudied aspect of Sn transport sweeps, and could easily be studied with Kripke by artificially adding more work (i.e. unknowns) to a subset of MPI tasks. Block-AMR could be added to Kripke, which would be a useful way to explore the cost-benefit analysis of adding AMR to an Sn code, and would be a way to further study load imbalances and AMR effects on sweeps.

The coupling of the on-node sweep kernel, the parallel sweep algorithm, and the choices of decomposing the problem phase space into GS's, ZS's and DS's impact the performance of the overall sweep. The trade off between large and small "units of work" can be studied. Larger "units of work" provide more opportunity for on-node parallelism, while creating larger messages, less "sends", and less efficient parallel sweeps. Smaller "units of work" make for less efficient on-node kernels, but more efficient parallel sweeps.

We can also study trading MPI tasks for threads, and the effects this has on our programming models and cache efficiency.

A simple timer infrastructure is provided that measure each compute kernels total time.

Physical Models

Kripke solves the Discrete Ordinance and Diamond Difference discretized steady-state linear Boltzmann equation.

    H * Psi = (LPlus * S * L) * Psi + Q


  • Psi is the unknown angular flux discretized over zones, directions, and energy groups

  • H is the "streaming-collision" operator. (Couples zones)

  • L is the "discrete-to-moments operator. (Couples directions and moments)

  • LPlus is the "moment-to-discrete" operator. (Couples directions and moments)

  • S is the (arbitrary) order scattering operator. (Couples groups)

  • Q is an external source. In Kripke it is represented in moment space, so really "LPlus*Q"

Kripke is hard-coded to setup and solve the 3D Kobayashi radiation benchmark, problem 3i. Since Kripke does not have reflecting boundary conditions, the full-space model is solved. Command line arguments allow the user to modify the total and scattering cross-sections. Since Kripke is a multi-group transport code and the Kobayashi problem is single-group, each energy group is setup to solve the same problem with no group-to-group coupling in the data.

The steady-state solution method uses the source-iteration technique, where each iteration is as follows:

  1. Phi = LTimes(Psi)
  2. PhiOut = Scattering(Phi)
  3. PhiOut = PhiOut + Source()
  4. Rhs = LPlusTimes(PhiOut)
  5. Psi = Sweep(Rhs, Psi) which is solving Psi=(Hinverse * Rhs) a.k.a "Inverting H"

Building and Running

Kripke comes with a BLT CMake-based build system based.


Basic requirements:

  • CMake 3.20 or later for CUDA, and 3.23 or later for HIP support (minimum respective versions required by RAJA)

  • C++14 Compiler (g++, icpc, etc.)

  • (Optional) MPI 1.0 or later

  • (Optional) OpenMP 3 or later

  • (Optional) Caliper: a performance profiling/analysis library.

Submodule dependencies:

  • BLT v0.5.3: a CMake based build system (required)

  • RAJA v2023.06.1: a loop abstraction library (required)

  • CHAI v2023.06.0: a copy hiding abstraction for moving data between memory spaces (optional)

  • Umpire v2023.06.0: a memory management abstraction (required if using CHAI)

  • Cub: algorithm primitives library for CUDA (required by RAJA if using CUDA)

Getting Kripke

Two options are available:

The following are the instruction for cloning the tarball, and setting up your clone repository.

Clone the latest released version from github:

    git clone

Clone all of the submodules. The Kripke build system, BLT, resides in another repository on github so one must initialize and update the "git submodules"

    cd Kripke
    git submodule update --init --recursive

The released source tarball on github is created with all of the submodules included already.

Quick Start

The easiest way to get Kripke running, is to directly invoke CMake and take whatever system defaults you have for compilers and let CMake find MPI for you.

  • Step 1: Create a build space (assuming you are starting in the Kripke root directory)

     mkdir build
  • Step 2: Run CMake in that build space

     cd build
     cmake ..
     For a number of platforms, we have CMake cache files that make things easier:
     cd build
     cmake .. -C ../host-configs/llnl-bgqos-clang.cmake -DCMAKE_BUILD_TYPE=Release
  • Step 3: Now make Kripke:

     make -j8
  • Step 5: Run Kripke's default problem:


There are a number of cache init files for LLNL machines and operating systems.
These might not meet your needs, but can be a very good starting point for developing your own. The current list of cache init files (located in the ./host-configs/ directory) are:

  • llnl-bgqos-clang.cmake

  • llnl-toss3-clang4.cmake

  • llnl-toss3-intel18.cmake

  • llnl-toss3-gcc7.1.cmake

  • llnl-toss3-gcc8.1.cmake

  • llnl-blueos-P100-nvcc-clang.cmake

  • llnl-blueos-V100-nvcc-clang.cmake

  • llnl-toss4-MI250X-rocm5-vernal.cmake

  • llnl-toss4-intel22.cmake

Running Kripke

Environment Variables

If Kripke is built with OpenMP support, then the environment variable OMP_NUM_THREADS is used to control the number of OpenMP threads. Kripke does not attempt to modify the OpenMP runtime in any way, so other OMP_* environment variables should also work as well.

If Kripke is built with Caliper support, Caliper performance measurements can be configured through Caliper environment variables. For example,

CALI_CONFIG_PROFILE=runtime-report ./kripke ...

will print a time profile of annotated code regions in Kripke. For more information, see

If building Kripke on RZVernal, then the rocm and cmake/3.24.2 modules need to be loaded.

Command Line Options

Command line option help can also be viewed by running "./kripke --help"

Problem Size Options:

  • --groups <ngroups>

    Number of energy groups. (Default: --groups 32)

  • --legendre <lorder>

    Scattering Legendre Expansion Order (0, 1, ...). (Default: --legendre 4)

  • --quad <ndirs>, or --quad <polar>:<azim>

    Define the quadrature set to use either a fake S2 with points, OR Gauss-Legendre with by points. (Default: --quad 96)

  • --zones <x>,<y>,<z>

    Number of zones in x,y,z. (Default: --zones 16,16,16)

Physics Parameters:

  • --sigt <sigt0,sigt1,sigt2>

    Total material cross-sections. (Default: --sigt 0.1,0.0001,0.1)

  • --sigs <sigs0,sigs1,sigs2>

    Total material cross-sections. (Default: --sigs 0.05,0.00005,0.05)

On-Node Options:

  • --arch <ARCH>

    Architecture selection. Selects the back-end used for computation, available are Sequential, OpenMP and CUDA. The default depends on capabilities selected by the build system and is selected from list of increasing precedence: Sequential, OpenMP and CUDA.

  • --layout <LAYOUT>

    Data layout selection. This determines the data layout and kernel implementation details (such as loop nesting order). The layouts are determined by the order of unknowns in the angular flux: Direction, Group, and Zone. Available layouts are DGZ, DZG, GDZ, GZD, ZDG, and ZGD. The order is specified left-to-right in longest-to-shortest stride. For example: DGZ means that Directions are the longest stride, and Zones are stride-1. (Default: --nest DGZ)

Parallel Decomposition Options:

  • --pdist <lout>

    Layout of spatial subdomains over MPI ranks. 0 for "Blocked" where local zone sets represent adjacent regions of space. 1 for "Scattered" where adjacent regions of space are distributed to adjacent MPI ranks. (Default: --layout 0)

  • --procs <npx,npy,npz>

    Number of MPI ranks in each spatial dimension. (Default: --procs 1,1,1)

  • --dset <ds>

    Number of direction-sets. Must be a factor of 8, and divide evenly the number of quadrature points. (Default: --dset 8)

  • --gset <gs>

    Number of energy group-sets. Must divide evenly the number energy groups. (Default: --gset 1)

  • --zset <zx>,<zy>,<zz>

    Number of zone-sets in x, y, and z. (Default: --zset 1:1:1)

Solver Options:

  • --niter <NITER>

    Number of solver iterations to run. (Default: --niter 10)

  • --pmethod <method>

    Parallel solver method. "sweep" for full up-wind sweep (wavefront algorithm). "bj" for Block Jacobi. (Default: --pmethod sweep)

Future Plans

Some ideas for future study:

  • More tuning of CUDA implementation

  • Block AMR

  • More FLOP intensive spatial discretizations such as DFEM's



Copyright (c) 2014-23, Lawrence Livermore National Security, LLC.

Produced at the Lawrence Livermore National Laboratory.

All rights reserved.


Unlimited Open Source - BSD Distribution

For release details and restrictions, please read the COPYRIGHT, LICENSE, and NOTICE files, also linked here: