Note

Click here to download the full example code

# 01. The LinearOpeator¶

This first tutorial is aimed at easing the use of the PyLops library for both new users and developers.

Since PyLops heavily relies on the use of the
`scipy.sparse.linalg.LinearOperator`

class of SciPy, we will start
by looking at how to initialize a linear operator as well as
different ways to apply the forward and adjoint operations. Finally we will
investigate various *special methods*, also called *magic methods*
(i.e., methods with the double underscores at the beginning and the end) that
have been implemented for such a class and will allow summing, subtractring,
chaining, etc. multiple operators in very easy and expressive way.

Let’s start by defining a simple operator that applies element-wise
multiplication of the model with a vector `d`

in forward mode and
element-wise multiplication of the data with the same vector `d`

in
adjoint mode. This operator is present in PyLops under the
name of `pylops.Diagonal`

and
its implementation is discussed in more details in the Implementing new operators
page.

First of all we apply the operator in the forward mode. This can be done in four different ways:

`_matvec`

: directly applies the method implemented for forward mode`matvec`

: performs some checks before and after applying`_matvec`

`*`

: operator used to map the special method`__matmul__`

which checks whether the input`x`

is a vector or matrix and applies`_matvec`

or`_matmul`

accordingly.`@`

: operator used to map the special method`__mul__`

which performs like the`*`

opetator

We will time these 4 different executions and see how using `_matvec`

(or `matvec`

) will result in the faster computation. It is thus advised to
use `*`

(or `@`

) in examples when expressivity has priority but prefer
`_matvec`

(or `matvec`

) for efficient implementations.

```
# setup command
cmd_setup = """\
import numpy as np
import pylops
n = 10
d = np.arange(n) + 1.
x = np.ones(n)
Dop = pylops.Diagonal(d)
DopH = Dop.H
"""
# _matvec
cmd1 = "Dop._matvec(x)"
# matvec
cmd2 = "Dop.matvec(x)"
# @
cmd3 = "Dop@x"
# *
cmd4 = "Dop*x"
# timing
t1 = 1.0e3 * np.array(timeit.repeat(cmd1, setup=cmd_setup, number=500, repeat=5))
t2 = 1.0e3 * np.array(timeit.repeat(cmd2, setup=cmd_setup, number=500, repeat=5))
t3 = 1.0e3 * np.array(timeit.repeat(cmd3, setup=cmd_setup, number=500, repeat=5))
t4 = 1.0e3 * np.array(timeit.repeat(cmd4, setup=cmd_setup, number=500, repeat=5))
plt.figure(figsize=(7, 2))
plt.plot(t1, "k", label=" _matvec")
plt.plot(t2, "r", label="matvec")
plt.plot(t3, "g", label="@")
plt.plot(t4, "b", label="*")
plt.axis("tight")
plt.legend()
plt.tight_layout()
```

Similarly we now consider the adjoint mode. This can be done in three different ways:

`_rmatvec`

: directly applies the method implemented for adjoint mode`rmatvec`

: performs some checks before and after applying`_rmatvec`

`.H*`

: first applies the adjoint`.H`

which creates a new scipy.sparse.linalg._CustomLinearOperator` where`_matvec`

and`_rmatvec`

are swapped and then applies the new`_matvec`

.

Once again, after timing these 3 different executions we can see
see how using `_rmatvec`

(or `rmatvec`

) will result in the faster
computation while `.H*`

is very unefficient and slow. Note that if the
adjoint has to be applied multiple times it is at least advised to create
the adjoint operator by applying `.H`

only once upfront.
Not surprisingly, the linear solvers in scipy as well as in PyLops
actually use `matvec`

and `rmatvec`

when dealing with linear operators.

```
# _rmatvec
cmd1 = "Dop._rmatvec(x)"
# rmatvec
cmd2 = "Dop.rmatvec(x)"
# .H* (pre-computed H)
cmd3 = "DopH*x"
# .H*
cmd4 = "Dop.H*x"
# timing
t1 = 1.0e3 * np.array(timeit.repeat(cmd1, setup=cmd_setup, number=500, repeat=5))
t2 = 1.0e3 * np.array(timeit.repeat(cmd2, setup=cmd_setup, number=500, repeat=5))
t3 = 1.0e3 * np.array(timeit.repeat(cmd3, setup=cmd_setup, number=500, repeat=5))
t4 = 1.0e3 * np.array(timeit.repeat(cmd4, setup=cmd_setup, number=500, repeat=5))
plt.figure(figsize=(7, 2))
plt.plot(t1, "k", label=" _rmatvec")
plt.plot(t2, "r", label="rmatvec")
plt.plot(t3, "g", label=".H* (pre-computed H)")
plt.plot(t4, "b", label=".H*")
plt.axis("tight")
plt.legend()
plt.tight_layout()
```

Just to reiterate once again, it is advised to call `matvec`

and `rmatvec`

unless PyLops linear operators are used for
teaching purposes.

We now go through some other *methods* and *special methods* that
are implemented in `scipy.sparse.linalg.LinearOperator`

(and
`pylops.LinearOperator`

):

`Op1+Op2`

: maps the special method`__add__`

and performs summation between two operators and returns a`pylops.LinearOperator`

`-Op`

: maps the special method`__neg__`

and performs negation of an operators and returns a`pylops.LinearOperator`

`Op1-Op2`

: maps the special method`__sub__`

and performs summation between two operators and returns a`pylops.LinearOperator`

`Op1**N`

: maps the special method`__pow__`

and performs exponentiation of an operator and returns a`pylops.LinearOperator`

`Op/y`

(and`Op.div(y)`

): maps the special method`__truediv__`

and performs inversion of an operator`Op.eigs()`

: estimates the eigenvalues of the operator`Op.cond()`

: estimates the condition number of the operator`Op.conj()`

: create complex conjugate operator

```
<10x10 LinearOperator with dtype=float64>
<10x10 LinearOperator with dtype=float64>
<10x10 LinearOperator with dtype=float64>
<10x10 LinearOperator with dtype=float64>
[1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
[10.+0.j 9.+0.j 8.+0.j]
(10.00000000000003+0j)
<10x10 _ConjLinearOperator with dtype=float64>
```

To understand the effect of `conj`

we need to look into a problem with an
operator in the complex domain. Let’s create again our
`pylops.Diagonal`

operator but this time we populate it with
complex numbers. We will see that the action of the operator and its complex
conjugate is different even if the model is real.

```
y = Dx = [0.+1.j 0.+2.j 0.+3.j 0.+4.j 0.+5.j]
y = conj(D)x = [0.-1.j 0.-2.j 0.-3.j 0.-4.j 0.-5.j]
```

At this point, the concept of linear operator may sound abstract.
The convinience method `pylops.LinearOperator.todense`

can be used to
create the equivalent dense matrix of any operator. In this case for example
we expect to see a diagonal matrix with `d`

values along the main diagonal

```
D = Dop.todense()
plt.figure(figsize=(5, 5))
plt.imshow(np.abs(D))
plt.title("Dense representation of Diagonal operator")
plt.axis("tight")
plt.colorbar()
plt.tight_layout()
```

At this point it is worth reiterating that if two linear operators are
combined by means of the algebraical operations shown above, the resulting
operator is still a `pylops.LinearOperator`

operator. This means
that we can still apply any of the methods implemented in the original
scipy class definition like `*`

, as well as those in our class
definition like `/`

```
Dop1 = Dop - Dop.conj()
y = Dop1 * x
print(f"x = (Dop - conj(Dop))/y = {Dop1 / y}")
D1 = Dop1.todense()
plt.figure(figsize=(5, 5))
plt.imshow(np.abs(D1))
plt.title(r"Dense representation of $|D - D^*|$")
plt.axis("tight")
plt.colorbar()
plt.tight_layout()
```

```
x = (Dop - conj(Dop))/y = [1.+0.j 1.+0.j 1.+0.j 1.+0.j 1.+0.j]
```

Finally, another important feature of PyLops linear operators is that we can always keep track of how many times the forward and adjoint passes have been applied (and reset when needed). This is particularly useful when running a third party solver to see how many evaluations of our operator are performed inside the solver.

```
Dop = pylops.Diagonal(d)
y = Dop.matvec(x)
y = Dop.matvec(x)
y = Dop.rmatvec(y)
print(f"Forward evaluations: {Dop.matvec_count}")
print(f"Adjoint evaluations: {Dop.rmatvec_count}")
# Reset
Dop.reset_count()
print(f"Forward evaluations: {Dop.matvec_count}")
print(f"Adjoint evaluations: {Dop.rmatvec_count}")
```

```
Forward evaluations: 2
Adjoint evaluations: 1
Forward evaluations: 0
Adjoint evaluations: 0
```

This first tutorial is completed. You have seen the basic operations that
can be performed using `scipy.sparse.linalg.LinearOperator`

and
our overload of such a class `pylops.LinearOperator`

and you
should be able to get started combining various PyLops operators and
solving your own inverse problems.

**Total running time of the script:** ( 0 minutes 1.025 seconds)