Tiled Matrix Multiplication

The tiled matrix multiplication implementation demonstrates efficient handling of matrices \((9 \times 9)\) using small tiles \((3 \times 3)\). Here’s how it works:

1. Shared memory allocation

   Input matrices (9×9) - Perfect fit for (3×3) tiling:
   A = [0  1  2  3  4  5  6  7  8 ]    B = [0  2  4  6  8  10 12 14 16]
       [9  10 11 12 13 14 15 16 17]        [18 20 22 24 26 28 30 32 34]
       [18 19 20 21 22 23 24 25 26]        [36 38 40 42 44 46 48 50 52]
       [27 28 29 30 31 32 33 34 35]        [54 56 58 60 62 64 66 68 70]
       [36 37 38 39 40 41 42 43 44]        [72 74 76 78 80 82 84 86 88]
       [45 46 47 48 49 50 51 52 53]        [90 92 94 96 98 100 102 104 106]
       [54 55 56 57 58 59 60 61 62]        [108 110 112 114 116 118 120 122 124]
       [63 64 65 66 67 68 69 70 71]        [126 128 130 132 134 136 138 140 142]
       [72 73 74 75 76 77 78 79 80]        [144 146 148 150 152 154 156 158 160]
   
   Shared memory per block (3×3):
   a_shared[TPB, TPB]  b_shared[TPB, TPB]
   

2. Tile processing loop

   Number of tiles = 9 // 3 = 3 tiles (perfect division!)
   
   For each tile:
   1. Load tile from A and B
   2. Compute partial products
   3. Accumulate in register
   

3. Memory loading pattern

   • With perfect \((9 \times 9)\) tiling, bounds check is technically unnecessary but included for defensive programming and consistency with other matrix sizes.

        # Load A tile - global row stays the same, col determined by tile
        if tiled_row < size and (tile * TPB + local_col) < size:
            a_shared[local_row, local_col] = a[
                tiled_row, tile * TPB + local_col
            ]
     
        # Load B tile - row determined by tile, global col stays the same
        if (tile * TPB + local_row) < size and tiled_col < size:
            b_shared[local_row, local_col] = b[
                tile * TPB + local_row, tiled_col
            ]
     

4. Computation within tile

   for k in range(min(TPB, size - tile * TPB)):
       acc += a_shared[local_row, k] * b_shared[k, local_col]
   

   • Avoids shared memory bank conflicts:

     Bank Conflict Free (Good):        Bank Conflicts (Bad):
     Thread0: a_shared[0,k] b_shared[k,0]  Thread0: a_shared[k,0] b_shared[0,k]
     Thread1: a_shared[0,k] b_shared[k,1]  Thread1: a_shared[k,0] b_shared[1,k]
     Thread2: a_shared[0,k] b_shared[k,2]  Thread2: a_shared[k,0] b_shared[2,k]
     ↓                                     ↓
     Parallel access to different banks    Serialized access to same bank of b_shared
     (or broadcast for a_shared)           if shared memory was col_major (transposed)
     

     Left: When threads from the same warp access different banks of shared memory during the same memory transaction (b_shared[k,threadIdx.x]), or all threads in a warp access the same shared memory address (a_shared[0,k]), accesses can proceed in parallel. Right: When threads from the same warp access the same bank simultaneously (bank k in b_shared), accesses are serialized, reducing performance. Note: Shared memory has 32 banks (on modern GPUs). Bank conflicts occur when multiple threads in a warp access different addresses in the same bank simultaneously. Broadcasts (same address) are handled efficiently.

5. Synchronization points

   barrier() after:
   1. Tile loading
   2. Tile computation
   

Key performance features:

• Processes \((9 \times 9)\) matrix using \((3 \times 3)\) tiles (perfect fit!)
• Uses shared memory for fast tile access
• Minimizes global memory transactions with coalesced memory access
• Optimized shared memory layout and access pattern to avoid shared memory bank conflicts

6. Result writing:

   if tiled_row < size and tiled_col < size:
      output[tiled_row, tiled_col] = acc
   

   • Defensive bounds checking included for other matrix sizes and tiling strategies
   • Direct assignment to output matrix
   • All threads write valid results

Key optimizations

1. Layout optimization:

   • Row-major layout for all tensors
   • Efficient 2D indexing

2. Memory access:

   • Coalesced global memory loads
   • Efficient shared memory usage

3. Computation:

   • Register-based accumulation i.e. var acc: output.element_type = 0
   • Compile-time loop unrolling via @parameter

This implementation achieves high performance through:

• Efficient use of LayoutTensor for memory access
• Optimal tiling strategy
• Proper thread synchronization
• Careful boundary handling

The idiomatic tiled matrix multiplication leverages Mojo’s LayoutTensor API and asynchronous memory operations for a beautifully clean implementation. With the \((9 \times 9)\) matrix size, we get perfect tiling that eliminates all boundary checks:

1. LayoutTensor tile API

   out_tile = output.tile[TPB, TPB](block_idx.y, block_idx.x)
   a_tile = a.tile[TPB, TPB](block_idx.y, idx)
   b_tile = b.tile[TPB, TPB](idx, block_idx.x)
   

   This directly expresses “get the tile at position (block_idx.y, block_idx.x)” without manual coordinate calculation. See the documentation for more details.

2. Asynchronous memory operations

   copy_dram_to_sram_async[thread_layout=load_a_layout](a_shared, a_tile)
   copy_dram_to_sram_async[thread_layout=load_b_layout](b_shared, b_tile)
   async_copy_wait_all()
   

   These operations:

   • Use dedicated copy engines that bypass registers and enable compute-memory overlap via copy_dram_to_sram_async
   • Use specialized thread layouts for optimal memory access patterns
   • Eliminate the need for manual memory initialization
   • Note: Standard GPU loads are already asynchronous; these provide better resource utilization

3. Specialized compile-time load layouts

   alias load_a_layout = Layout.row_major(1, TPB)
   alias load_b_layout = Layout.row_major(TPB, 1)
   

   These layouts optimize how threads cooperate during memory transfers:

   • load_a_layout: Each thread loads a slice of a row (coalesced access)
   • load_b_layout: Each thread loads a slice of a column (transposed access)

4. Perfect tiling eliminates boundary checks

   @parameter
   for idx in range(size // TPB):  # Perfect division: 9 // 3 = 3
   

   With \((9 \times 9)\) matrices and \((3 \times 3)\) tiles, every tile is exactly full-sized. No boundary checking needed!

5. Clean tile processing with defensive bounds checking

   # Defensive bounds checking included even with perfect tiling
   if tiled_row < size and tiled_col < size:
       out_tile[local_row, local_col] = acc
   

   With perfect \((9 \times 9)\) tiling, this bounds check is technically unnecessary but included for defensive programming and consistency with other matrix sizes.

Performance considerations

The idiomatic implementation maintains the performance benefits of tiling while providing cleaner abstractions:

1. Memory locality: Exploits spatial and temporal locality through tiling
2. Coalesced access: Specialized load layouts ensure coalesced memory access patterns
3. Compute-memory overlap: Potential overlap through asynchronous memory operations
4. Shared memory efficiency: No redundant initialization of shared memory
5. Register pressure: Uses accumulation registers for optimal compute throughput

This implementation shows how high-level abstractions can express complex GPU algorithms without sacrificing performance. It’s a prime example of Mojo’s philosophy: combining high-level expressiveness with low-level performance control.

Key differences from manual tiling

The idiomatic approach is not just cleaner but also potentially more performant due to the use of specialized memory layouts and asynchronous operations.