taylor expansion, well defined
can directly write ODE solution:
dx/dt = Ax
x = exp(tA)x_0
if A is diagonalizable, A = PvP^-1, v is diagonal matrix
exp(A) = P exp(v) P^-1
inverse is exp(-A)
A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0.
transformations don’t affect topology, are called homeomorphisms
tanh (and sigmoid and softplus but not ReLU) are continuous functions with continuous inverses
Qs:
how stiff
how many vars
how accrate
how sensitive(chaos)
how well-behaved is f(x,t)
how costly is f(x,t) and its jacobian
implicit,
more stable, solving stiff problems requires implicit methods
Euler’s method
trapezoidal
Newton-Raphson
stability criterion
x’(t) = a x(t)
forward euler is conditionally stable
backward euler is unconditionally stable
truncation error
Runge-Kutta methods
stage order
Multi-step methods
predictor-corrector method
richardson extrapolation
automatic selection
| [Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations | SIAM Journal on Scientific and Statistical Computing | Vol. 4, No. 1 | Society for Industrial and Applied Mathematics](https://epubs.siam.org/doi/pdf/10.1137/0904010?casa_token=sBjDZTSayFQAAAAA:XhlfyWkS4MRFNRnrZ6LmQff_UXAH7riLBkpcA58llDnYEJycmMMbMCli9cFkoYKRT7uNos94IpA) |
instability
zero stability
Re(lambda) < 0, x’(t) = lambda x(t) numerical solution decays to 0
region of absolute stability z = \lambda \delta t
all scaled eigenvalues of Jacobian should be in region, for ODE system
stability region contains the entire lefthalf plane
backward euler, implicit midpoint
no explicit one-step method can be A-stable
All explicit RK methods with r stages and of order r have the same stability region
if it is A-stable and it damps fast componentsof the solution
A stiff problem is one where ∆t has to be small even though the solution is smooth and a large ∆t is OK for accuracy
if the solution evolves on widely-separated timescales and the fast time scale decays (dies out) quickly
given linear ODE system, x’(t) = A x(t), decompose A to separate parts, x is formed by uncoupled n different y variables, each of n ODEs(y) is independent of the others
timestep of original system must be smaller than the smallest stability limits
the system is stiff if a strong separation of stability time scale (eigenvalue ratio)
Jacobian for non-linear system, complex eigenvalues
adjoint method
auto diff
半格是满足运算是幂等的和交换的半群。
半群是闭合于结合性二元运算之下的集合 S 构成的代数结构
集合S和其上的二元运算·:S×S→S。若·满足结合律，即：∀x,y,z∈S，有(x·y)·z=x·(y·z)，则称有序对(S,·)为半群
history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process
考虑任意一个偏序集合（L,≤），如果对集合L中的任意元素a,b，使得a,b在L中存在一个最大下界，和最小上界，则(L,≤)是一个格
一个格是完全的，如果它的所有子集都有一个交和一个并
a collection of random variables is heteroscedastic (or heteroskedastic;[a] from Ancient Greek hetero “different” and skedasis “dispersion”) if there are sub-populations that have different variabilities from others