Scroll to the bottom of the page for the Math Geek Adventures blog archive.

Wednesday, August 28, 2013

Outlaws of Dirt, Frisco, Colorado

Dirtjump culture. Alex is in the red pants.
During the last week of summer, we took Alex down to Frisco, Colorado for stop two of the Outlaws of Dirt, dirt jump contest. He also attended stop one, back in May, where he got second place.
Big air, Frisco bike park.

Alex's main passion in life is dirt jumping. It's cool to see him navigate the challenges that he faces, the primary one being that there are almost no dirt jumps, and very few dirt jumpers, in Montana. And all of the competitions are far away, in Canada and Colorado. Nonetheless, he soldiers on and rides hard every day. I tell him, "Everyone faces challenges, and if it isn't early on, it's later; adversity is universal. So if you can't figure out how to get past this obstacle, it wasn't meant to be." He's determined to figure it out.
Three sixty
So we drive 1000 miles and he competes against kids his age that have access to some of the best dirt jump parks in the U.S. and he does well. This contest, he stepped up and performed better than he ever has, and in the atmosphere of a competition no less. He's a competitor, and it's cool to watch. Nonetheless, there were other tough competitors there as well and he got fourth, but was in the mix for the top three.
This trip came at the end of a summer of a lot of travel for me. So it turned out not to be a vacation. In fact, it was more stressful than staying home and working would have been. I also ruined my cross bike driving it into a garage with it on the roof (ahhh!). And the drives were brutal: going both down and back, we banged out the 1000 mile trip in one shot. But it was worth the sacrifice to watch Alex compete and also to realize how much I love Montana. Colorado is staggeringly beautiful, but it isn't for me.

Thursday, August 22, 2013

Math Blog #4: Regularization from the spectral point of view

In Math blog #2, I presented regularization as the method for overcoming the difficulties with solving ill-posed problems of the form z=Au. We mentioned that the problem could be made well-posed by solving instead

(1) u_alpha = argmin_u {||Au-z||^2 + alpha*||u||^2},

which is also the solution of the 'normal equations' for (1); that is,

(A'*A+alpha*I)u_alpha = A'*z,

where I is the identity operator (or matrix in the discrete case) and alpha>0. This is known as Tikhonov regularization.

Tikhonov regularization can also be viewed spectrally. Recall from Math blog #3 that the inverse of A has a spectral representation via the singular value expansion (SVE) of A: 

(2) Ainv*z = \sum_i {V_i*<U_i,z>/s_i}.

The problem with (2) for ill-posed problems, as we've discussed previously, is that {1/s_i} is unbounded (or has extremely large maximum in the discrete case). 

Tikhonov regularization (1) can be equivalently expressed in terms of the SVE of A as follows:

(3) u_alpha = \sum_i {V_i*<U_i,z>/s_i^alpha}, 

where s_i^\alpha=(s_i^2+alpha)/s_i. Note that if alpha=0, (3) and (2) are the same, but for alpha>0, {1/s_i^alpha} is now bounded (or has a much smaller maximum than {1/s_i} in the discrete case), making z-->u_alpha a stable (or more stable in the discrete case) mapping.

Tuesday, August 13, 2013

Math Blog #3: A Spectral Characterization of Ill-Posed Inverse Problems

In Hadamard's definition of well-posedness of Au=z, there are three requirements: (1) existences of a solution u; (2) uniqueness of that solution; and (3) continuous dependence of the solution u on z.

Let's have a look at this spectrally, by computing the singular value expansion of A:S-->T: for any u,

A*u=\sum_i {s_i*U_i*<V_i,u>},

where {s_i} are the (nonnegative) singular values of A; {V_i} are the right (orthonormal) singular functions (vectors in finite dimensions) that span the Hilbert space (vector space) S containing u; <.,.> denotes the inner-product on S (or T, where appropriate); and {U_i} are the left (orthonormal) singular functions (vectors) that span the function (vector) space T containing all possible Au and z.

Now, note that if any s_i=0, Au=U_i has no solution, so condition (1) fails. If s_i>0 and s_j=0, then Au=U_i has a solution, u=V_i/s_i, but it isn't unique since u=V_i/s_i+c*V_j is a solution for all scalars c. Hence in this case (2) fails.

Finally, if (1) and (2) hold (i.e. s_i>0 for all i), A is invertible with singular value expansion Ainv:T-->S: for any z,

Ainv*z = \sum_i {V_i*<U_i,z>/s_i}.

This mapping is discontinuous if the set of singular values {s_i} has a cluster point at 0. To see this, note that for any M>0, there exist s_i<1/M and Ainv*U_i=V_i/s_i, which has norm 1/s_i>M. Thus Ainv is discontinuous. In the finite dimensional case, Ainv is not discontinuous (since min{1/s_i}>0), but its singular values 1/s_i become extremely large, making the inverse mapping highly unstable.

Sunday, August 11, 2013

Bechler River, Yellowstone, August 2013

Add Lonestar Trailhead near Old Faithful: Dave, Penn, me, Alex.
 Dave Sumner and I have been meeting for an annual summer backpack for years now. Our first trip was back in 2000 and we've only missed a couple of years since. 
Night 1 camp along the upper Firehole River.
This year was the first in which we brought along any of our kids: our two oldest, Alex (16 years) and Penn (19 years). I chose the Bechler River in Yellowstone because it is lauded as a classic backpack, and at 30 miles over 5 days, with a net elevation loss and short walks on days 1 and 5, it was accessible for our group. 
Alex night 1. 
The way to do the trip is from Old Faithful, heading south toward and beyond Shoshone Lake, along the Bechler River to the Bechler Ranger Station in the southwest corner of the park. The shuttle is a bit brutal at 2.5 hours one way -- we dropped a car at Bechler. 
Upper Firehole with barely visible Sandhill Cranes flying away in the middle of the picture.
We got a late start from the trailhead at Lonestar. Our site for the first night was on the upper Firehole River, where it's only a small stream. Alex fly fished for the first time, catching a couple of small brookies.
Shoshone Lake
The next day we hiked up and over the Continental Divide, but took a side trip to Shoshone Lake and the Shoshone Geyser Basin. From there it was a few miles to our second night's camp at 8500 feet, in a sparse meadow.

Shoshone Geyser Basin: the fellas swimming with a boiling pot in the foreground.
Dave and Penn, night 2.
At our second night's camp, we met a group of 19 year old fellas, most of whom were pretty ill-prepared: two had forgotten sleeping pads and one of those also forgot a sleeping bag. And they were laden down with military MREs one of their dads had stored in the garage and an emergency surgery kit. We had some good laughs about their experience around their campfire.
Three fellas having an adventure.
The next day we joined the Bechler and lost steady elevation. At the halfway point, we side tripped up to Mr. Bubbles, a large and awesome back country hotspring.
Alex and Penn soaking in Mr. Bubbles.
On our way to our night 3 camp, there was a good swim spot along the Bechler.  
That's me in the Bechler just below Mr. Bubbles.
That night, near camp, we all fished and did well catching a bunch of cutthroat trout. Alex's fly fishing technique improved quickly.
Alex fishing at the night 3 sight, Bechler River.
On the fourth day, the highlights were: the waterfalls,
Colonade Falls
 the berries (huckleberries, wild raspberries, and thimble berries),
Alex picking thimble berries.
several river fords, this one the most beautiful with a great swimming hole, heading into Bechler Meadows, 
Ford heading into Bechler Meadows.
Bechler Meadows itself, with views of the Tetons,
Bechler Meadows with the Tetons in the background (the Grand just visible).
and the last night's awesome camp, with good fishing.
Last nights camp spot, lower Bechler. There was good fishing for rainbows here. 
 The last day was an easy walk (3.5 level miles) to Bechler Ranger Station.
The crew at the end of the trip.
The trip was one of our best, for sure. The country was awesome and it was great to have our boys (the next generation) along.