hybrid analog-digital solution of nonlinear partial...

Solve nonlinear ODE using analog accelerator

Operate on numbers encoded as voltage / current

Program connections & parameters for ODE

Configure circuit inside analog accelerator

Monitor / obtain analog waveform solving ODE

Extend ODE solving to help linear algebra & PDEs

Time independent PDE

Time dependent PDE

Partial differential equation (PDE)

Parabolic PDE (e.g., heat equation)

Hyperbolic PDE (e.g., wave equation)

Elliptic PDE (e.g., Poisson eq.)

Spatial discretization Spatial discretization

System of ordinary differential equations (ODE)

Explicit time stepping (e.g., RK4, analog)

Implicit time stepping (e.g., backward Euler)

Sparse system of linear equations (SLE)

Direct solvers (e.g., Cholesky, QR, SVD)

Iterative solvers (e.g., CG, analog)

Dense linear eqs. (e.g., optimization)

Nonlinear system of equations

Nonlinear solvers (e.g., Newton’s)

∫

∫

DAC ADC

AnalogInput

AnalogOutput

LUTsin(x)

-.5x

-5x

−𝑎𝑎10

−𝑎𝑎00

∫d𝑥𝑥1d𝑡𝑡 𝑥𝑥1(𝑡𝑡)

⍦

ADCDAC

∫d𝑥𝑥0d𝑡𝑡 𝑥𝑥0(𝑡𝑡)

⍦

ADCDAC

−𝑎𝑎11

−𝑎𝑎01

𝑏𝑏1

𝑏𝑏0

∫∫+F(u)

-JF(u)

∫∫u

δ≈JF-1(u)F(u)

-

0

50

100

150

200

0 200 400 600

conv

erge

nce

time

(s)

total grid points

digital CG

analog 20KHz

Linear (analog 20KHz)

Linear (analog 80KHz projection)

Linear (analog 320KHz projection)

Linear (analog 1.3MHz projection)

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

0.001 0.01 0.1 1 10

Tim

e to

con

verg

ence

(sec

onds

)

Reynolds number

16-by-16 Grid

Digital Time Analog Time

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

0.001 0.01 0.1 1 10

Tim

e to

con

verg

ence

(sec

onds

)

Reynolds number

8-by-8 Grid


1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

0.001 0.01 0.1 1 10

Tim

e to

con

verg

ence

(sec

onds

)

Reynolds number

4-by-4 Grid


• [email protected]• [email protected]

• Ning Guo, Yipeng Huang, Tao Mai, Sharvil Patil, Chi Cao, Mingoo Seok, Simha Sethumadhavan, Yannis Tsividis, “Low-Energy Hybrid Analog/Digital Approximate Computation in Continuous Time,” IEEE Journal of Solid-State Circuits (JSSC), 2016.

• Yipeng Huang, Ning Guo, Mingoo Seok, Yannis Tsividis, Simha Sethumadhavan, “Evaluation of an Analog Accelerator,” ACM/IEEE International Symposium on Computer Architecture (ISCA), 2016.

• Yipeng Huang, Ning Guo, Mingoo Seok, Yannis Tsividis, Kyle Mandli, Simha Sethumadhavan, “Hybrid Analog-Digital Solution of Nonlinear Partial Differential Equations,” IEEE/ACM International Symposium on Microarchitecture (MICRO), 2017.

• Awarded Defense Advanced Research Projects Agency (DARPA) Small Business Technology Transfer grant to investigate modern analog computing applications.

• Anticipated slowdown of speed and efficiency improvements in digital integrated circuits due to material and economic constraints

• Analog circuits deliver fast and efficient computation with existing silicon technology; analog computing history offers new directions

• Continuous-time, continuous-value hardware matches well with physical problems, which are often stochastic and nonlinear

• Downsides such as low precision and accuracy, limited scalability, and difficulty in programming can be mitigated

• Approximate solutions are useful in physical simulations and machine learning tasks for humanity’s grand challenge problems

0.08 0.07 0.08 0.07 0.080.15

0.09 0.10

0.81

0.06 0.06 0.06 0.06 0.06 0.08 0.08 0.08 0.05

0

0.5

1

1.5

2

0.01 0.02 0.03 0.06 0.13 0.25 0.50 1.00 2.00

Tim

e to

con

verg

ence

(s)

Reynolds number

Solution time vs. Reynolds numberfor digital and seeded digital solver

baseline digital solver analog seeded digital solver

Using negative feedback analog solves linear equations; just provide constants, coefficients, and initial guesses

For smaller problem sizes & lower precision solutions, analog compares favorably against digital iterative numerical methods

Analog linear algebra subroutine accelerates Newton’s method, useful for finite difference solution of nonlinear PDEs

Analog continuous Newton’s method takes infinitesimal steps, avoiding challenges in selecting correct initial guesses

When solving nonlinear equations resulting from Burgers’ PDE, analog solves faster if problems are large and more nonlinear

Analog approximate solutions for nonlinear equations are initial guesses for digital, which proceeds to high precision solution

As digital hardware improvements slow, analog holds promise in problems that are continuous, nonlinear, stochastic, approximate

Our group’s multiple generations of analog accelerator chips validate full-custom analog circuits & aid application discovery

Contact information and selected publications

Physically fabricated analog accelerator chips provide ground-truth timing, area, and power consumption measurements

20 KHz analog design bandwidth16 integrators & 32 multipliersConfigurable analog crossbar

8-bit analog-digital-analog conversionNonlinear function lookup tables

Analog inputs & outputs for multi-chip cooperation

1.2V 65nm TSMC manufacturing process3.7mm × 39mm dimensions

4.8 mW at full power0.06 W/cm2 very low power density

Hybrid Analog-Digital Solution of Nonlinear Partial Differential EquationsYipeng Huang, Ning Guo, Mingoo Seok, Yannis Tsividis, Kyle Mandli, Simha Sethumadhavan

Columbia University

hybrid analog-digital solution of nonlinear partial...

Documents