Speeding up the problem solving process usually happens by doing two things, increasing the time spent on said project and/or involving more people on the project. Increasing the time spent on the project may allow more ideas to flow rather than having a quick brainstorming session and then having to quickly move on to the next task. The stop and go of this path can result in a lack of progress being made.If the current team is at a roadblock, gathering the opinions of outside eyes can be beneficial to looking at the problem from a new perspective. 

I think I would ask my team where we think the roadblock is coming from and see what I can do to help the situation. Problem solving sometimes is best done as a group and I wouldn't want my team to feel like they have to tackle an issue on their own. Being a project manager/team manager, it would be your duty to ensure your team has what they need to succeed. @anthonynjit above had a great idea about a brainstorming session to see what we can come up with, and definitely having multiple minds in that session can help to speed up the process. Personally it helps me a lot when I can gather other people to help me come up with a solution to a problem because I do tend to struggle a bit when on my own.

Depending on the deadline for the project, weekly or even daily team meetings are crucial for problem identification and solutions. These meetings are vital for members to identify and tackle obstacles. After a roadblock is identified, members should allocated more effort into solving the issue. This could be done by involving more members to target the challenge, therefore yielding quicker and more reliable solutions.

I agree with the point made above about increasing the time spent on the project. While elongating the deadline for the project may not be desired, it might be best for the project as it's important not to rush solutions especially when they are lingering and can compromise the project. Not only will increasing the time spent on the project will give enough time to understand and come up with multiple solutions for the problem, but there could also be enough time to improve the project as well. Another way to speed up problem-solving in a project team is involving the entire time as well as people outside in upper management for a fresh perspective on the issue. Another way to speed up the problem solving process in a project team is by using a tool such as failure mode element analysis. 

Many time speeding up a process create more problems than they solve. They also can inherently increase the risk of the project. Speeding up things many time means cutting time from somewhere else. This usually ends up effecting the quality control and is overall not a great idea, unless it is planned ahead of time. One way I would advise speeding up problem solving process is to work on problems in a group, many times an outsiders input can be great as it allows one to reframe the question and look at it from a different point of view.

It is important to identify the problems first. There needs to be a clear definition of the problem to determine how it will be dealt with. Is it a design problem, process problem, or resource problem? Whatever the problem may be, it should immediately be communicated to management. The project team can continue to work on the problem, while the manager can prepare a strategy to deal with it. An entire team can sometimes be dedicated to solving a problem, and can sometimes be a project of its own.

Solving problems in an accelerated time manner is often needed in most industries, especially on client based project where deadlines are extremely crucial. I agree to above posts that in order to reach to a speedy solution, maximizing resources- money, people, time, are all important depending on the project and problem type. Most important in my opinion is the number of people needed to involved. More brains will equate to more ideas. But it's equally important to involve the right people to ensure no one is wasting time. More people will also allow having check points on taking any wrong decisions and avoid delays. Having some extra emergency budget can be allocated to expedite some processes may also help depending on seriousness of situation. Lastly, as someone mentioned above, more meeting, hence appropriate communication will circulate the ideas around better and will lead to a speedy solution. 

As @MahdiaSChy mentioned, communication between involved cross functional teams is the best approach to problem solving. When I had to work to secure network configurations for several ultrasounds units, just being able to work through the problem with IT and Security to recognize the mac address so that it was not blocked by the firewall and then again having the PACs team aware so that I could provide them with all the information they needed to recognize the unit so that it could send studies through to their DICOM storage and worklist servers made a task that would have been difficult alone, a lot easier to accomplish after discussing with those most immediately involved. Another way to speed up problem solving could be to ask for a fresh perspective from someone just as or even more knowledgeable about the task that needs to be completed.

As stated in the initial prompt, challenges can cause serious consequences if they aren’t identified quickly and handled carefully. This means that the problem-solving process can be broken up into two parts; problem identification and problem solution. In order to speed up this process as a whole, one or both parts needs to be improved upon. To improve the problem identification, it is important to have a good project supervisor and make sure people are keeping each other accountable for their own actions. The better that the team works together, the easier that they will be able to identify mistakes quickly and not let them affect other parts of the project. In order to improve the project solution step, effective planning and communication will be needed by all members of the team. Being able to identify and have a plan on how to solve the issues being presented will lead to a smoother transition between the phases of the project and can really help the team meet all their deadlines. Overall, speeding up the problem-solving process can really increase efficiency and be an overall benefit to the entire team.

To speed up problem-solving in project management you must first define the problem correctly. The to define the problem will determine how to attempt to solve it.

Once the problem is defined, you need to start to determine what is causing it.  You can use a fishbone diagram to help you perform a cause-and-effect analysis. Once you've determined which solution you will implement, it's time to take action. If the solution involves several actions or requires action from others, it is a good idea to create an action plan. Using this approach can increase the effectiveness of your problem-solving skills.

I think there  are some steps that can be taken to speed up the problem-solving process in a project team:

The first step can be encouraging open communication to make sure team members are comfortable sharing their concerns and opinions. Create a safe and respectful environment where people feel free to express themselves. Then you need to make sure everyone is clear on what the problem is and what it entails. This will help you to focus on the issue at hand and avoid wasting time and effort on peripheral concerns.

Once the problem is defined, encourage the team to come up with as many potential solutions as possible. This will help you to explore a wide range of options and identify the most effective one. Evaluating and prioritize solutions is a very important step meaning that each solution needs to be Analyzed and weighed in terms of pros and cons. Prioritize the solutions based on their feasibility, impact, and cost or even time. Once a solution has been chosen, assign responsibilities to team members. Make sure everyone knows what is expected of them and what their role is in implementing the solution. In the end, progresses need to be monitored in order to make sure the solution is being implemented properly. If there are any issues, address them quickly to prevent the problem from recurring.