Solving sparse linear systems with approximate inverse preconditioners on analog devices

Abstract

Numerical computation is essential to many areas of artificial intelligence (AI), whose computing demands continue to grow dramatically, yet their continued scaling is jeopardized by the slowdown in Moore's law. Multi-function multi-way analog (MFMWA) technology, a computing architecture comprising arrays of memristors supporting in-memory computation of matrix operations, can offer tremendous improvements in computation and energy, but at the expense of inherent unpredictability and noise. We devise novel randomized algorithms tailored to MFMWA architectures that mitigate the detrimental impact of imperfect analog computations while realizing their potential benefits across various areas of AI, such as applications in computer vision. Through analysis, measurements from analog devices, and simulations of larger systems, we demonstrate orders of magnitude reduction in both computation and energy with accuracy similar to digital computers.

In-memory sparse matrix multiplication with a low-power self-rectifying memristor array

“…Moreover, solving sparse matrix equations usually faces the challenge of ill-conditioned problems, which are di cult to determine the exact solution and are highly sensitive to hardware noises. Hence, we recommend using preconditioned techniques, such as the incomplete LU factorization, to reduce the matrix condition number 38 . The preconditioned techniques can also reduce the matrix bandwidth and will be bene cial for the in-memory SpMV.…”

Sparse matrix multiplication in a record-low power self-rectifying memristor array for scientific computing

“…These examples are obtained from the SuiteSparse matrix collection, a widely used dataset of SpMV benchmarks collected from a wide range of applications (table S3) (46). Noticeably, a preconditioning algorithm, i.e., the incomplete lower-upper factorization (ILU), is introduced to deal with the ill matrix condition (47). The preconditioning algorithms also approximate the arbitrary sparse matrix to the identity matrix, whereas the problem can be efficiently addressed using our approach after quantization (text S10).…”

Toward memristive in-memory computing: principles and applications

“…Le Gallo et al proposed a mixed-precision IMC architecture combing the low-precision in-memory processing unit and the high-precision digital unit to solve the matrix equation [99], namely where A is the known non-singular matrix, b is the column vector, and x is the n-dimensional unknown vector to be solved. The mixed-precision solver utilized the Richardson refinement method [103] to solve the matrix equation with high-precision. In the hybrid system as demonstrated in Fig.…”